text:
\begin{document}

\title[Quantitative weak mixing for random substitution tilings]
      {Quantitative weak mixing for random substitution tilings}
      \author{Rodrigo Trevi\~no}
      \address{University of Maryland, College Park}
      \email{rodrigo@umd.edu}
      \date{\today}
      \begin{abstract}
        For $N$ compatible substitution rules on $M$ prototiles $t_1,\dots, t_M$, consider tilings and tiling spaces constructed by applying the different substitution rules at random. These give (globally) random substitution tilings. In this paper I obtain bounds for the growth on twisted ergodic integrals for the $\mathbb{R}^d$ action on the tiling space which give lower bounds on the lower local dimension of spectral measures. For functions with enough regularity, uniform bounds are obtained. The results here extend results of Bufetov-Solomyak \cite{BS:translation} to tilings of higher dimensions.
      \end{abstract}
      \maketitle

      \section{Introduction and statement of results}
      Let me start describing what \textbf{quantitative} weak mixing is for a minimal and uniquely ergodic flow $\varphi_t:\Omega\rightarrow \Omega$ on a compact metric space $\Omega$. Weak mixing is equivalent to the non-existence of $L^2$-eigenfunctions for the flow, that is, non-existence of functions $f$ that satisfy $f\circ \varphi_t = e^{2\pi i t\lambda}f$ for all $t\in\mathbb{R}$ and some $\lambda\in\mathbb{R}$ (the eigenvalue). One can quantify how far away a system is from having an eigenfunction: given a function $f\in L^2$, its \textbf{spectral measure} $\mu_f$ is obtained, by Bochner's theorem, by taking the Fourier transform of the correlation function $\langle f\circ \varphi_t, f\rangle$. In the case of $f$ being an eigenfunction with eigenvalue $\lambda$, the spectral measure $\mu_f$ is a Dirac measure at $\lambda$. Recall that the \textbf{lower and upper local dimensions} of a locally finite Borel measure $\mu$ are the quantities
      $$d^-_\mu(x) := \liminf_{r\rightarrow 0^+}\frac{\log \mu (B_r(x))}{\log r}\hspace{.4in}\mbox{ and }\hspace{.4in}d^+_\mu(x) := \limsup_{r\rightarrow 0^+}\frac{\log \mu (B_r(x))}{\log r}.$$
      In the case of a spectral measure $\mu = \mu_f$, the local dimensions will be denote by $d_f^\pm(x): = d_{\mu_f}^\pm(x)$.

      The upper and lower dimensions of the spectral measure of an eigenfunction are zero. Quantitative weak mixing therefore involves bounding the local dimension of spectral measures from below, that is, away from the case of Dirac measures, which appear for eigenfunctions.

      Over the past decade, a series of works by Bufetov and Solomyak \cite{BS:substitution, BS:genus2, BS:spectralCocycle, BS:translation} have studied the issue of quantitative weak mixing for an increasingly large class of minimal and uniquely ergodic flows, culminating with a very large class of non-stationary, one-dimensional tiling spaces, a subset of which models a typical flow on flat surfaces of genus greater than or equal to two. Recently \cite{forni:twist} Forni has developed an extension from his theory of flat surfaces in \cite{forni:deviation} through the use of twisted cohomology in order to obtain results on quantitative weak mixing for translation flows on typical higher-genus surfaces. Forni's contribution \cite{forni:twist} was done after the results of Bufetov-Solomyak \cite{BS:genus2} for translation flows on genus two, but before their results for flows on surfaces of arbitrary genus. In fact, the results  of \cite{BS:translation} cover a very general class of one-dimensional tiling spaces as well as translation flows on flat surfaces of infinite genus.

      Much less has been done in this direction for higher rank actions, in particular for actions of $\mathbb{R}^d$. Spaces coming from tilings are natural candidates for these questions. Given that tilings have some sort of physical significance as models of quasicrystals, many spectral results for tilings are known. For example, Solomyak \cite{SolomyakEigenfunctions} has characterized weak mixing for tiling spaces which are constructed from a single substitution rule. Emme \cite{Emme:spectral} has studied the behavior of spectral measures at zero for self-similar tilings of $\mathbb{R}^d$. These results are extended  in the (unpublished) Master's thesis of Marshall \cite{Marshall:master}, with results on quantitative weak mixing for a class of self-similar tilings of $\mathbb{R}^d$. Although it does not directly addresses questions related to eigenfunctions, the work \cite{BGM:correlation} brings forth methods to determine the absence of absolute continuity of spectral measures in a large class of self-similar tilings of any dimension.

      This paper is a contribution to the expanding list of systems for which there are results on quantitative weak mixing. In \cite{ST:random, T:TTT}, a method of studying (globally) random substitution tilings was introduced and developed. The idea is as follows: suppose there are $N$ different compatible substitution rules $S_1,\dots, S_N$ on $M$ prototiles $t_1,\dots, t_M$, and that a tiling was created from these by applying the different substitution rules at random over all the tiles. This procedure would create a tiling whose structure would not be self-similar, but the structure would be represented by the order in which the substitutions were applied to construct the tiling. In other words, some of the structure would be recorded by a point $x\in \Sigma_N$ in the $N$-shift. This is very different from the (locally) random constructions where not only are the substitutions chosen at random, but so are the tiles on which they are applied. Systems of globally random substitution similar to the one here sometimes go by the name S-adic systems or mixed substitution systems \cite{GM:mixed}.

      The study of (globally) random substitution tilings is made possible through the use of Bratteli diagrams (see \S \ref{subsec:brat} for definitions): they hold the information of both how substitutions are applied and in which order. More specifically, given $x = (x_1,x_2,\dots)\in\Sigma_N^+$, the level $n$ of the Bratteli diagram $B_x$ encodes the substitution $S_{x_n}$. There is a continuous map $\Delta_x:X_{B_x}\rightarrow \Omega_x$ from the path space of $B_x$ to a tiling space $\Omega_x$. Very importantly, with this construction come homeomorphisms of tiling spaces $\Phi_x:\Omega_x\rightarrow \Omega_{\sigma(x)}$. These maps drive the renormalization dynamics on which every result relies.

      As in the works of Bufetov-Solomyak and Forni, the route to proving estimates of local dimension for spectral measures are via the study of \textbf{twisted ergodic integrals}. That is, suppose $\mathbb{R}^d$ acts on $\Omega$ minimally and in an uniquely ergodic way. For some $f\in L^2$ and $\lambda\in\mathbb{R}^d$, the twisted integral of $f$ by $\lambda$ is
      $$\mathcal{S}_R^{x}(f,\lambda):=\int_{C_R(0)}e^{-2\pi i \langle \lambda, t\rangle }f\circ \varphi_t(x)\, dt$$
      where $C_R(0)$ is the $d$-cube of sidelength $2R$ centered around the origin. The relation between twisted integrals and quantitative weak mixing is given by the following fact, which goes back to Hof \cite{Hof:scaling} (see \S \ref{sec:proofs} for the proof).
      \begin{lemma}
        \label{lem:dimension}
        Suppose $\mathbb{R}^d$ acts on $\Omega$ preserving a Borel probability measure $\mu$. Suppose that for some $\lambda\in\mathbb{R}^d$, $R_0>0$, $f\in L^2(\Omega,\mu)$ and $\alpha\in(0,d)$
  $$\left\|\mathcal{S}_R(f,\lambda)\right\|_{L^2}\leq C_1R^{d-\alpha}$$
        for all $R>R_0$. Then
        $$\mu_f(B_r(\lambda))\leq Cr^{2 \alpha}$$
        for all $r<1/2R_0$ and some $C>0$. In particular, the lower local dimension of the spectral measure satisfies
        $$d^-_f(\lambda)\geq 2 \alpha.$$
      \end{lemma}

      Given the topological nature of tiling spaces, the methods used here are more in line with those of Bufetov-Solomyak's than those of Forni. However, there is a crucial difference between the work here and both of those works: both \cite{BS:translation} and \cite{forni:twist} make use of a renormalization cocycle which describes \emph{both} the hierarchical structure of the foliated space as well as the structure of return vectors to a special transversal because, for $\mathbb{R}$ actions, both of these objects coincide, since all return vectors are colinear. This cocycle has been used to study the rates of deviation of ergodic integrals for functions with some regularity.

      This no longer holds for $\mathbb{R}^d$ actions when $d>1$, the higher rank case. More precisely, there is a cocycle which relates the hierarchical structures of tiling spaces $\{\Omega_{\sigma^k(x)}\}$ along the orbit of $x$. This is called the \textbf{trace cocycle} in \cite{T:TTT}, where it was shown that its behavior describes the behavior of ergodic integrals of the $\mathbb{R}^d$ action on tiling spaces.

      But there is another cocycle at play. Given a transversal $\mho_x\subset \Omega_x$ in a tiling space, there is a set of vectors $\Lambda_x\subset\mathbb{R}^d$ consisting of return vectors to the transversal. The group $\Gamma_x:=\mathbb{Z}[\Lambda]$ generated by integer combinations of these vectors is a finitely generated Abelian group whenever the tilings in $\Omega_x$ have finite local complexity. The renormalization maps $\Phi_{\sigma(x)}:\Omega_{x}\rightarrow \Omega_{\sigma(x)}$ induce a map from $\Gamma_x$ to $\Gamma_{\sigma(x)}$ which, under a careful choice of generators for $\Gamma_x$ and $\Gamma_{\sigma(x)}$, is represented by an integer matrix. This map is called the \textbf{return vector cocycle}, and its behavior gives the estimates necessary to bound spectral measures from below. Thus, in order to obtain quantitative weak mixing results for higher rank actions, the simultaneous understanding of two renormalization cocycles is needed.

      How can these two different renormalization cocycles be understood? A quick answer is cohomologically. It can be shown (see \S \ref{subsec:RetVecCoh}) that every return vector $v\in\mathbb{R}^d$ gives a cohomology class $[v]\in \check H^1(\Omega;\mathbb{Z})$ (more generally, in $H^1$ of the Lie-algebra cohomology of the $\mathbb{R}^d$ action). Whereas the usual renormalization cocycle describes the action of the renormalization maps on the top level cohomology of the tiling space $H^d(\Omega;\mathbb{R})$ (more generally, $H^d$ of the Lie-algebra cohomology of the $\mathbb{R}^d$ action), the return vector cocycle describes the action of the renormalization maps on $H^1(\Omega;\mathbb{R})$. Thus, when $d>1$, these are different cocycles and both of these are needed for quantitative weak mixing results. The return vector cocycle in fact projects of the cocycle $\Phi^*_x$ induced by the renormalization maps on the first cohomology $H^1(\Omega;\mathbb{R})$ of the tiling space (see \S\ref{subsec:RetVecCoh} for details).

      One way in which one can pick tilings at random is to choose the sequence of substitutions at random, parametrized by a point $x\in\Sigma_N$ according to some shift-invariant probability measure on $\Sigma_N$. But there is another way of randomizing the construction of a tiling, and that is through deformations (see \S\ref{subsec:deform} for definitions related to deformations). For any tiling $\mathcal{T}$ of finite local complexity the set $H^1(\Omega_\mathcal{T};\mathbb{R}^d)$ is a natural space of deformations of the tiling, a subset of which can be thought of as a moduli space for the tiling space $\Omega_\mathcal{T}$. Therefore, in addition to picking the sequences of substitutions at random, one can pick the geometry of the tilings at random from $H^1(\Omega;\mathbb{R}^d)$. In dimension 1 this amounts to picking the length of the tiles in the tiling. However, for dimension greater than 1 the geometry of the tiles can be varied in much more interesting ways. The standard language to use in the deformation of orbits is that of \textbf{time changes}. For higher rank actions, and especially in tiling spaces, the language of \textbf{shape changes} is also used.    The goal of this paper is to give sufficient conditions for when random choices of hierarchical structure (i.e. order of substitutions) and geometry (i.e. deformation parameter or time change) give tilings and tiling spaces which are quantitatively weakly mixing. The results here can be seen as a quantitative, non-stationary and measure-theoretic generalization of the topological results for self-similar tilings in \cite{ClarkSadun:shape} (see (vi) below in Remark \ref{rem:1}).
      \subsection{Statement of results}
      Let me describe everything needed to understand the main result. Let $S_1,\dots, S_N$ be a collection of substitution rules on $M$ prototiles $t_1,\dots, t_M$. A substitution rule is called \textbf{uniformly expanding} if there is a $\theta\in(0,1)$ such that every prototile is expanded by $\theta_i^{-1}$ under the substitution rule $S$. See \S \ref{subsec:tilings} for the necessary background on tilings and tiling spaces. Here it will be assumed that the substitution rules are \textbf{compatible} in the sense of \cite{GM:mixed} (see \S \ref{sec:RandSub}): any arbitrary combination of them produces tilings in which tiles meet edge to edge and have finite local complexity. That this is a large and interesting class of tilings at least in dimension two is evident from the experimental results in \cite{GKM:computer}.

      As mentioned in the previous section, by picking a element of $x = (x_1,x_2,\dots)\in\Sigma_N^+$ one specifies a sequence of substitutions $S_{x_1},S_{x_2},\dots$ from which one can try to construct a tiling space $\Omega_x$. Naturally, some conditions need to be imposed so that, given a set of substitution rules, a choice of $x$ will in fact produce a tiling space $\Omega_x$. The construction involves first creating a Bratteli diagram $B_x$, an infinite directed graph, which records not only the order of the substitutions given by $x$, but also some accompanying combinatorial information about the substitution. This combinatorial information can be expressed in a sequence of integer valued, non-negative matrices $\{A_k\}_k$, where $A_i$ encodes some information from the substitution $S_{x_i}$ (in fact $A_i$ is the substitution matrix $F_{x_i}$ for $S_{x_i}$). The basic necessary condition in order to construct an unambiguous tiling space $\Omega_x$ from $x$ is requiring that the sequence $\{A_k\}$ has the property that for every $m\in\mathbb{N}$ there is an $n>m$ such that the matrix $A_n\cdots A_m$ has all positive entries. Note that this property is $\sigma$-invariant: if the matrices for $B_x$ have this property then the matrices for $B_{\sigma(x)}$ have this property. A $\sigma$-invariant measure is called \textbf{minimal} if for $\mu$-almost every $x$, the matrices for $B_x$ have this property. See \S\ref{sec:RandSub} for more details.

      A further condition is needed to obtain quantitative weak mixing results. A word $w  = w_1w_2\dots w_n$ is \textbf{simple} if $w_i\cdots w_n\neq w_1\cdots w_{n-i+1}$ for all $1<i\leq n$. A word $w$ is \textbf{positively simple} if it is a simple word and in addition:
      \begin{enumerate}
      \item it breaks up into two subwords $w= w^-w^+ = w_1^-\cdots w_{n^-}^-w_1^+\cdots w_{n^+}^+$, and
      \item each entry in both matrices $Q^\pm := F_{w_{n^\pm}^\pm}\cdots F_{w_1^\pm}$ is strictly greater than 1, where $F_{w_j}$ is the substitution matrix for $S_{w_j}$.
      \end{enumerate}
      Let $C([w_1.w_2])\subset \Sigma_N$ be the cylinder set with the word $w_1w_2$ around the origin. A $\sigma$-invariant probability measure $\mu$ on $\Sigma^+_N$ for which $\mu(C([w^-.w^+]))>0$ for a positively simple word $w$ is called a \textbf{positively simple measure}. Any such measure is a minimal measure. The existence of a positively simple word which occurs infinitely often in $x$ guarantees the existence of a finite set $\Lambda_w\subset \mathbb{R}^d$ of vectors which model good return vectors of arbitrarily large scale for the $\mathbb{R}^d$ action on $\Omega_x$. This is covered in \S\ref{sec:LFreturn}.

      Let $r_x$ be the rank of the group $\Gamma_x=\mathbb{Z}[\Lambda_x]$ generated by integer combinations of return vectors. Chosing a set of generators $\{v_i\}$ of $\Gamma_x$ one can define the \textbf{address map} $\alpha:\Gamma_x\rightarrow \mathbb{Z}^{r_x}$ by sending the generators $\{v_i\}$ to the canonical generators. As such, every element $\tau\in\Lambda_x$ has an address $\alpha(\tau)$. A positively simple measure $\mu$ is called \textbf{postal} if $\alpha(\Lambda_w)$ generates $\mathbb{Z}^{r_x}$, that is, if there are enough special return vectors in $\Lambda_w$ so that any address can be found with these vectors through the address map.

      Let $\mu$ be a positively simple, postal, $\sigma$-invariant, ergodic probability measure on $\Sigma_N$ such that the return vector cocycle is integrable (the return vector cocycle is defined in \S\ref{subsec:RV} ). For any point $x\in\Sigma_N$ consider the corresponding tiling space $\Omega_x$. As mentioned above, the geometry of the tilings in $\Omega_x$ are given by an element $\Rd_x\in H^1(\Omega_x;\mathbb{R}^d)$, which is a finite dimensional vector space carrying a natural Lebesgue measure. There is an open subset $\mathcal{M}_x\subset H^1(\Omega_x;\mathbb{R}^d)$ which parametrizes all deformations (equivalently, time changes) of tilings in $\Omega_x$ (see \S\ref{subsec:deform}) which inherits a natural absolutely continuous measure from $H^1(\Omega_x;\mathbb{R}^d)$. Any tiling space obtained by deforming the tiling space $\Omega_x$ by deforming through $\Rd\in \mathcal{M}_x$ will be denoted by $\Omega_x^{\Rd}$. The set $\mathcal{M}_x$ can be thought of a moduli space of $\Omega_x$.

      The shift $\sigma:\Sigma_N\rightarrow \Sigma_N$ induces a homeomorphism of tiling spaces $\Phi_x:\Omega_x\rightarrow \Omega_{\sigma(x)}$ which generalizes the hyperbolic self-homeomorphisms which appear in self-similar tiling spaces. Such maps induce an action $\Phi_x^*:H^*(\Omega_{\sigma(x)};\mathbb{Z})\rightarrow H^*(\Omega_{x};\mathbb{Z})$ on cohomology. Under the assumption that $\log \|\Phi_x^*\|$ is $\mu$-integrable for some $\sigma$-invariant and ergodic $\mu$, the Oseledets theorem gives a decomposition of the first cohomology space $H^1(\Omega_x;\mathbb{R}) = E^+_x\oplus E^-_x$, where $E^+_x$ is the Oseledets space of strictly positive Lyapunov exponents. Throughout the paper, $C_R(0) := [-R,R]^d\subset \mathbb{R}^d$.
      \begin{theorem}
        \label{thm:main}
        Let $S_1,\dots, S_N$ be a collection of $N$ uniformly expanding compatible substitution rules on the $M$ prototiles $t_1,\dots, t_M$ of dimension $d$ and $\mu$ a $\sigma$-invariant positively simple, postal ergodic probability measure on $\Sigma_N$ such that $\log \|\Phi_x^*\|\in L^1_\mu$, where $\Phi^*_x$ is the map on the first cohomology of $\Omega_x$ induced by the shift. If $\mathrm{dim}(E^+_x)>d$ then there exists a constant $\alpha_\mu\in(0,1]$ such that for $\mu$-almost every $x\in\Sigma_N$, for almost every $\Rd\in\mathcal{M}_x$, any $B>1$, any Lipschitz function $f:\Omega_x^{\Rd}\rightarrow \mathbb{R}$, $\mathcal{T}\in\Omega_x^{\Rd}$, $\varepsilon>0$ and $\lambda\in\mathbb{R}^d$ with $0< \|\lambda\|\leq B$,
        $$\left|\int_{C_R(0)}e^{-2\pi i \langle \lambda, t\rangle }f\circ \varphi_t(\mathcal{T})\, dt\right|\leq C_{f,B} R^{d-\alpha_\mu+\varepsilon},$$
        for all $R$ large enough. In particular, the lower local dimension of the spectral measures is bounded from below:
        $$2\alpha_\mu\leq d_f^-(\lambda).$$
      \end{theorem}
      I should point out that in this setting it is always true that $\mathrm{dim}\, E^+_x \geq d$ for a tiling of dimension $d$. Thus the theorem requires one extra expanding Oseledets subspace. The consequence is that a generic time change of a self-similar, aperiodic tiling with large-enough unstable space does not have any eigenfunctions (see Remark (vi) below).
      \begin{corollary}
        \label{cor:main}
        Let $S$ be a primitive substitution rule on the $M>1$ prototiles $t_1,\dots, t_M$ of dimension $d$. If $\mathrm{dim}(E^+_x)>d$ then there exists a constant $\alpha_\mu\in(0,1]$ such that for almost every $\Rd\in\mathcal{M}_x$, for any $B>1$, any Lipschitz function $f:\Omega_x\rightarrow \mathbb{R}$, $\mathcal{T}\in\Omega_x$, $\varepsilon>0$ and $\lambda\in\mathbb{R}^d$ with $0< \|\lambda\|\leq B$,
          $$\left|\int_{C_R(0)}e^{-2\pi i \langle \lambda, t\rangle }f\circ \varphi_t(\mathcal{T})\, dt\right|\leq C_{f,B} R^{d-\alpha_\mu+\varepsilon},$$
          for all $R$ large enough. In particular, the lower local dimension of the spectral measures is bounded from below:
        $$2\alpha_\mu\leq d_f^-(\lambda).$$
      \end{corollary}
      \begin{remark}
        \label{rem:1}
        Some remarks:
        \begin{enumerate}
        \item The main theorem does not imply all of the results of \cite{BS:translation}, which cover the case $d=1$. The reason is that the hypotheses here assume that the substitutions are uniformly expanding, which is not an assumption in \cite{BS:translation}. Removing this assumption for $d>1$ may give tilings of arbitrarily many scales, which significantly complicates the analysis of the group of return vectors and cohomology spaces. This is not an issue for $d=1$ because all boundaries of tiles are the same. On the other hand, \cite{BS:translation} assumes the invertibility of the substitution matrices, which is not an assumption made here, although it seems to me like the results in \cite{BS:translation} can do without such assumption. It seems like first removing the compatibility assumption adopted here may be a more approachable extension of the results here.
        \item The value of $\alpha_\mu$ can be explicitly given in terms of the dynamical behavior of the return vector cocycle; see Proposition \ref{prop:Veech} in \S\ref{sec:criterion}.
        \item That $\alpha_\mu$ is bounded above by $1$ is a consequence that boundary effects appear in ergodic integrals for higher rank actions; see \S \ref{sec:twisted}.
        \item The ``simple'' part of a positively simple word implies that this word has no overlaps. These types of words always exist in a subshift as long as there are periodic orbits of arbitrary high period (see e.g. \cite[Lemma 2.5]{Yang:normal}). Therefore, measures of positive entropy on $\Sigma_N$ are positively simple measures. This was what was used in \cite{BS:translation} to apply their quantitative weak mixing results to flat surfaces: Rauzy-Veech induction has as base dynamics a mixing subshift of finite type, and so the Masur-Veech measure has the property that makes it a positively simple measure.
        \item It is unclear whether, in general, there is any relationship between the spectrum of the trace cocycle and the spectrum of that of the return vector cocycle. In the case of single substitutions there seems to be some; see Lemma \ref{lemma:OFK} and Appendix \ref{app:1}.
        \item In \cite[Theorem 1.7]{ClarkSadun:shape}, Clark and Sadun give a cohomological condition so that for almost any choice of deformation parameter $\Rd$ the system is topologically weak mixing when the starting system is self-similar. By a result of Solomyak \cite[Theorem 3.3]{SolomyakEigenfunctions}, this implies measure theoretic weak mixing for typical deformation parameters in the self-similar case. Theorem \ref{thm:main} above can be seen as a non-stationary, quantitative, and measure-theoretic generalization of their result. Corollary \ref{cor:main} here requires more conditions than \cite[Theorem 1.7]{ClarkSadun:shape} but has stronger conclusion.
        \end{enumerate}
      \end{remark}
      Weak mixing is defined as the convergence in the average of correlations of functions of zero mean. The quantitative weak mixing results yield uniform rates of weak mixing for functions with sufficiently regularity in the leaf direction. More precisely, let $\Omega$ be a tiling space whose elements are repetitive, aperiodic tilings of dimension $d$ of finite local complexity. Assume that the $\mathbb{R}^d$ action on $\Omega$ is uniquely ergodic. Fixing an orthonormal frame $X_1,\dots, X_d$ for the Lie algebra $\mathbb{R}^d$, denote by $X_if$ the leafwise derivative of $f$ by $X_i$ and by $C^{\infty}(\Omega)$ the set of Lipschitz functions which are $C^\infty$ in the leaf direction, that is, Lipschitz functions such that $X^\alpha f \in C^{\infty}(\Omega)$ for any multi-index $\alpha = (\alpha_1,\dots, \alpha_d)\in\mathbb{N}^d$ with $X^\alpha = X_1^{\alpha_1}\cdots X_d^{\alpha_d}$. For $k\in\mathbb{N}$ let
$$\|f\|_{(k)} := \sum_{\alpha\in \mathbb{N}^d:|\alpha|= s}\|X^\alpha f\|_{L^2},$$
where $|\alpha| = \alpha_1+\cdots+\alpha_d$. For any $r>1$ define
$$\RI_r(\Omega) = \left\{ f\in C^\infty(\Omega):\hspace{.3in} \substack{ \displaystyle \sum_{k=1}^\infty \frac{\| f\|_{(k)}}{r^{k}}<\infty  \;\hspace{.4in}\mbox{ and }\hspace{.4in}\; \sup_k \frac{\|X^{\alpha_k} f\|_{L^\infty}}{r^{k}}<\infty  \hspace{.2in} \\ \mbox{ for any sequence $\alpha_k\in \mathbb{N}^d$ with $|\alpha_k|=k$ for all $k$}.}\right\}$$
and note that $\RI_r(\Omega)\subset \RI_s(\Omega)$ whenever $r>s>1$.
\begin{theorem}
  \label{thm:main2}
  Let $S_1,\dots, S_N$ be a collection of $N$ uniformly expanding compatible substitution rules on the $M$ prototiles $t_1,\dots, t_M$ of dimension $d$ and $\mu$ a $\sigma$-invariant positively simple, postal ergodic probability measure on $\Sigma_N$ such that $\log \|\Phi_x^*\|\in L^1_\mu$, where $\Phi^*_x$ is the map on the first cohomology of $\Omega_x$ induced by the shift. If $\mathrm{dim}(E^+_x)>d$ then there exists a constant $\alpha'_\mu\in(0,1]$ such that for $\mu$-almost every $x\in\Sigma_N$, for almost every $\Rd\in\mathcal{M}_x$, $\varepsilon>0$, and any functions $f,g\in \RI_r(\Omega_x^{\Rd})$ of zero average,
    \begin{equation}
      \label{eqn:avgCorr}
  \int_{C_R(0)}|\langle f\circ \varphi_t,g\rangle|\, dt \leq C_{f,g,r,\varepsilon} R^{d-\frac{\alpha'_\mu}{2}+\varepsilon}
    \end{equation}
    for all $R>1$, where
    $$\alpha'_\mu = \min\left\{\alpha_\mu,d\left(1-\frac{\lambda_2}{\lambda_1}\right),1\right\},$$
    $\alpha_\mu$ is the constant given in Theorem \ref{thm:main}, and $\lambda_1>\lambda_2$ are the top two Lyapunov exponents for the trace cocycle. Moreover, for all $\lambda\in \mathbb{R}^d$:
    \begin{equation}
      \label{eqn:uniform}
      \left|\int_{C_R(0)}e^{-2\pi i \langle \lambda, t\rangle }f\circ \varphi_t(\mathcal{T})\, dt\right|\leq C_{f,r,\varepsilon}' R^{\frac{d+\lambda^*}{2}+\varepsilon},
    \end{equation}
        for all $R$ large enough, where $\lambda^* = \max\{d-1,d\frac{\lambda_2}{\lambda_1}\}$. In particular, the lower local dimension of the spectral measures is bounded from below:
        $$\min\left\{1,d\left(1-\frac{\lambda_2}{\lambda_1}\right)\right\}\leq d_f^-(\lambda).$$
\end{theorem}
\begin{remark}
Some remarks:
  \begin{enumerate}
  \item In general one expects $\alpha_\mu$ to be much smaller than $\min\{ 1, d(1-\lambda_2/\lambda_1)\}$, so the uniform bounds in Theorem \ref{thm:main2} for functions with enough regularity are better than the bounds in Theorem \ref{thm:main}.
  \item The bound depends on both the Lyapunov spectrum of the trace cocycle and the Lyapunov spectrum of the return vector cocycle. In cohomology terms, the bound in Theorem \ref{thm:main2} depends both on the spectral gap of the Lyapunov spectrum on the cocycle on $H^d(\Omega_x;\mathbb{R})$ as well as the spectrum on the cocycle on $H^1(\Omega_x;\mathbb{R})$.
  \item The proof for the uniform bounds in Theorem \ref{thm:main2} obtained from non-uniform bounds in Theorem \ref{thm:main} for smooth functions apply to the non-uniform bounds in \cite{BS:translation}, thus giving uniform estimates for their results in dimension 1, bringing them closer to the uniform bounds of Forni in \cite{forni:twist}. In fact, the proof of (\ref{eqn:avgCorr}) follows Forni's argument in \cite[Lemma 9.3]{forni:twist}. The uniform bounds in (\ref{eqn:uniform}) follow from (\ref{eqn:avgCorr}) using an argument of Venkatesh \cite[Lemma 3.1]{venkatesh:sparse}.
  \item Theorem \ref{thm:main2} is of more relevance to the mathematical physics literature than Theorem \ref{thm:main} given the importance of correlation functions in the theory of diffraction.
  \end{enumerate}
\end{remark}
Given the uniform bounds in Theorem \ref{thm:main2}, an argument of \cite[\S 2]{BS:products} caries over \emph{verbatim} after replacing $\mathbb{R}$ with $\mathbb{R}^d$ for product systems. Besides applying to the uniform results here, it also applies to the uniform results of Forni \cite{forni:twist}.
\begin{corollary}
  Let $S_1,\dots, S_N$ be a collection of $N$ uniformly expanding compatible substitution rules on the $M$ prototiles $t_1,\dots, t_M$ of dimension $d$ and $\mu$ a $\sigma$-invariant positively simple, postal ergodic probability measure on $\Sigma_N$ such that $\log \|\Phi_x^*\|\in L^1_\mu$, where $\Phi^*_x$ is the map on the first cohomology of $\Omega_x$ induced by the shift. If $\mathrm{dim}(E^+_x)>d$ then for $\mu$-almost every $x\in\Sigma_N$, for almost every $\Rd\in\mathcal{M}_x$, $\varepsilon>0$, for any ergodic action $\phi_t:X\rightarrow X$ of $\mathbb{R}^d$ on a standard probability space $(X,\nu)$, for any functions $f\in \RI_r(\Omega_x^{\Rd})$ and $g\in L^2(X,\nu)$ of zero average,

  $$     \left|\int_{C_R(0)}  \left\langle \left(f\otimes g\right)\circ \left(\varphi_t\times \phi_t\right),\left(f\otimes g \right)\right\rangle \, dt\right|\leq C_{f,r,\varepsilon,g} R^{d-\lambda^*+\varepsilon},$$
  where $\lambda^* = \max\{d-1,d\frac{\lambda_2}{\lambda_1}\}$ as in Theorem \ref{thm:main2}.

\end{corollary}
The bounds for a twisted integral $\mathcal{S}_R(f,\lambda)$ with a spectral parameter $\lambda$, and thus the bounds for the lower local dimension of spectral measures at $\lambda$, is obtained by bounding traces applied to specially-constructed elements of locally finite algebras (see \S \ref{sec:LFreturn}). As detailed in Remark \ref{rem:spectralCocycle}, this can be rewritten in terms of growth of a family of cocycles $\mathbb{M}_x^{(k)}(\lambda)$, parametrized by spectral parameters $\lambda\in\mathbb{R}^d$, over the shift $\sigma:\Sigma_N\rightarrow \Sigma_N$, and which coincides with the trace cocycle (also, the substitution matrix cocycle) when $\lambda=0$. This cocycle goes by several names in the literature: it is the spectral cocycle of Bufetov-Solomyak \cite{BS:spectralCocycle}, the Fourier matrix cocycle of Baake-G\"ahler-Ma\~nibo \cite{BGM:correlation}, or the twisted cocycle of Forni \cite{forni:twist}. In this paper, the growth properties are bounded from above in order to obtain lower bounds for the local dimension of spectral measures; upper bounds for the local dimension of spectral measures will be established in a future paper.

For reasons that I do not fully understand, there has been some contention as to what the definition of a crystal should be \cite{lifshitz:what}. A prevailing view is that a crystal should have long-range positional order which can be infered from the existence of Braggs peaks in the Fourier spectrum of the solid. If tilings serve as mathematical models of crystals, then this means that tilings with some discrete spectrum are candidates for being models for crystals. The results in this paper show that, left to our own devices and constructing tilings at random, we are almost surely not constructing models for any type of crystal. I think this makes both crystals and tilings constructed at random special.

      This paper is organized as follows. In \S\ref{sec:background} the necessary background for tilings and Bratteli diagrams is covered. This is followed in \S\ref{sec:RandSub} by a review of the construction of random substitution tilings using graph iterated function systems, which was a construction introduced in \cite{ST:random}. The basic theory of locally finite dimensional algebras and their traces is covered in \S \ref{sec:algebras}. The ideas of renormalization are introduced in \S\ref{sec:renorm}, where the behavior of the groups of return vectors under the substitution maps (the return vector cocycle) is analyzed, as well as the induced action on traces. In \S\ref{sec:LFreturn} elements of the LF algebras instroduced in \S\ref{sec:algebras} are constructed to organize the set of inner products $\langle \lambda,\tau\rangle$ of spectral parameters $\lambda\in\mathbb{R}^d$ and return vectors $\tau$. This allows to give bounds of the norms of traces applied to these elements, which are crucial to estimating the growth of twisted integrals, which is done in \S\ref{sec:twisted}. A criterion is proved in \S\ref{sec:criterion} giving bounds of the spectral measures of functions in terms of the quantitative behavior of the return vector cocycle. This type of result goes by the name of a \textbf{quantitative Veech criterion}, refering to the criterion by Veech for weak mixing in \cite{Veech:metric1}, and the criterion here is a generalization to higher rank of the criterion in \cite{BS:translation}. Cohomology is introduced in \S\ref{sec:cohomology} and the basic theory of deformations is introduced. A link between the return vector cocycle and the induced action on cohomology is established from which a cohomological quantitative Veech criterion is derived in \S \ref{sec:CQVC}. In \S\ref{sec:dimension}, Bufetov and Solomyak's version of the Erd\H{o}s-Kahane argument is used to show that the set of deformation parameters which are bad is very small, allowing the proof of Theorem \ref{thm:main} to be finished, which is done in \S\ref{sec:proofs}. The uniform bounds are derived from Theorem \ref{thm:main} in \S \ref{sec:uniform}. There are two appendices which include results which are not directly related to the proof of quantitative weak mixing, but they are \emph{not} unrelated to the ideas floating around. Appendix \ref{sec:plotTwist} introduces twisted cohomology for tiling spaces basic properties are derived. Appendix \ref{app:1} presents a generalization of Kenyon's theorem on the structure of return vectors in the non-stationary setting here.
      \begin{ack}
        This work was supported by the NSF grant DMS-1665100. I am also very grateful to Alejandro Maass and the Centro de Modelamiento Matem\'atico in Santiago, where some of this work was carried out, for outstanding hospitality during my visit. I also want to thank Giovanni Forni for fruitful discussions on twisted integrals, especially suggesting how Theorem \ref{thm:main2} may follow from the (non-uniform) results on quantitative weak mixing, as well as bringing \cite{BS:products} to my attention. The suggestion to use Venkatesh's argument to obtain uniform bounds came from Osama Khalil, for which I am grateful. Finally, I am grateful to Boris Solomyak for giving valuable feedback which led to improving the exposition of the paper.
      \end{ack}
      \section{Background: Tilings and Bratteli diagrams}
      \label{sec:background}
      \subsection{Tilings and tiling spaces}
      \label{subsec:tilings}
      A \textbf{tiling} $\mathcal{T}$ of $\mathbb{R}^d$ is a covering of $\mathbb{R}^d$ by compact sets $t$, called \textbf{tiles}, such that any two tiles of $\mathcal{T}$ may only intersect along their boundaries. The translation $\varphi_\tau(\mathcal{T})$ of a tiling $\mathcal{T}$ is the tiling obtained by translating each of the tiles $t$ of $\mathcal{T}$ by the vector $\tau\in\mathbb{R}^d$. If there exist compact sets $t_1,\dots, t_M$ such that every tile $t\in\mathcal{T}$ is translation-equivalent to some $t_i$, then we call the collection $\{t_1,\dots, t_M\}$ the \textbf{prototiles} of $\mathcal{T}$. A union $\mathcal{P}\subset \mathcal{T}$ of a finite number of tiles of $\mathcal{T}$ is called a \textbf{patch} of $\mathcal{T}$. For any bounded set $A\subset\mathbb{R}^d$ define the patch
$$\mathcal{O}^-_\mathcal{T}(A) = \mbox{ largest patch of $\mathcal{T}$ completely contained in $A$}.$$

For two tiles $t,t'\in\mathcal{T}$, let $t\sim t'$ denote translation-equivalence, that is, if there exists a $\tau\in\mathbb{R}^d$ such that $\varphi_\tau(t) = t'$, then one writes $t\sim t'$.

A tiling is \textbf{repetitive} if for any patch $\mathcal{P}$ in $\mathcal{T}$ there is an $R_\mathcal{P}>0$ such that the ball $B_{R_\mathcal{P}}(x)$ contains a patch $\mathcal{P}'$ which is translation-equivalent to $\mathcal{P}$ for any $x\in\mathbb{R}^d$. A tiling has \textbf{finite local complexity} if for any $R>0$ there is a finite list of patches $\mathcal{P}_1^R,\mathcal{P}_2^R,\dots, \mathcal{P}_{N(R)}^R$ such that for any $x\in\mathbb{R}^d$, $\mathcal{O}^-_\mathcal{T}(B_r(x))\sim \mathcal{P}^R_j$ for some $j$.

Given a finite collection of tiles $t_1,\dots, t_M$, a \textbf{substitution rule} is a collection of affine maps $\{f_{ijk}\}$ such that
$$t_i = \bigcup_{j=1}^M\bigcup_{k=1}^{\kappa(i,j)} f_{ijk}(t_j).$$
The maps $f_{ijk}$ are assumed to be dilation and translation, so in fact they are of the form $f_{ijk}(x) = \theta_{ijk}x + \tau_{ijk}$ for some $\theta_{ijk}\in (0,1]$ and $\tau_{ijk}\in \mathbb{R}^d$. The substitution rule is \textbf{uniformly expanding} if there is a $\theta\in(0,1)$ such that $\theta_{i,j,k}=\theta$ for all $i,j,k$.

  There is a metric on the set $\{\varphi_t(\mathcal{T})\}_{t\in\mathbb{R}^d}$ of translates of $\mathcal{T}$. First, define
  $$\bar{d}(\mathcal{T},\varphi_t(\mathcal{T})) = \inf\left\{ \varepsilon>0: \mathcal{O}^-_\mathcal{T}(B_{1/\varepsilon}(0)) = \mathcal{O}^-_{\varphi_{t+s}\mathcal{T}}(B_{1/\varepsilon}(0))\mbox{ for some }\|s\|\leq \varepsilon\right\},$$
  and from this define
  \begin{equation}
    \label{eqn:metric}
    d(\mathcal{T},\varphi_t(\mathcal{T})) = \min\left\{1,\bar{d}(\mathcal{T},\varphi_t(\mathcal{T}))\right\}.
  \end{equation}
  That (\ref{eqn:metric}) is metric is a standard fact. Denote by
  $$\Omega_\mathcal{T} := \overline{\{\varphi_t(\mathcal{T}):t\in\mathbb{R}^d\}}$$
  the metric completion of the set of all translates of $\mathcal{T}$ with respect to the metric (\ref{eqn:metric}). This set is called the \textbf{tiling space} of $\mathcal{T}$. Note that if $\mathcal{T}$ is repetitive then we have that $\Omega_{\mathcal{T}'} = \Omega_{\mathcal{T}}$ for any other $\mathcal{T}'\in\Omega_\mathcal{T}$. By chosing a marked point in the interior of each prototile $t_i$ it can be assumed that every tile in $\mathcal{T}$ has a marking which is translation equivalent to the marking on prototiles by the corresponding translation equivalence of tiles with prototiles. The \textbf{canonical transversal} of $\Omega_\mathcal{T}$ is the set
  \begin{equation}
    \label{eqn:transversal}
    \mho_\mathcal{T} := \{\mathcal{T}\in\Omega_\mathcal{T}:\mbox{ the marked point in the tile containing the origin is the origin}\}.
  \end{equation}
  Let $\mathcal{P}\subset \mathcal{T}$ be a patch and $t\in\mathcal{P}$ a tile in the patch. Then
  $$\mathcal{C}_{\mathcal{P},t} = \{\mathcal{T}'\in\Omega_\mathcal{T}: \mbox{ $\mathcal{P}$ is a patch in $\mathcal{T}'$ and the distinguished point in the tile $t$ is the origin} \}.$$
  is the $(\mathcal{P},t)$-cylinder set, and for $\varepsilon>0$ the $(\mathcal{P},t,\varepsilon)$-cylinder set is
  $$\mathcal{C}_{\mathcal{P},t,\varepsilon} := \bigcup_{\|\tau\|<\varepsilon}\left\{ \varphi_\tau(\mathcal{T'}):\mathcal{T}'\in  \mathcal{C}_{\mathcal{P},t}\right\}\subset \Omega_\mathcal{T}.$$
  The tiling space is compact if $\mathcal{T}$ has finite local complexity. Moreover, if $\mathcal{T}$ is a repetitive tiling of finite local complexity then the topology of $\Omega_\mathcal{T}$ is generated by cylinder sets of the form $\mathcal{C}_{\mathcal{P},t,\varepsilon}$ for arbitrarily small $\varepsilon>0$. This gives $\Omega_\mathcal{T}$ a the local product structure of $\mathcal{C}\times B_\varepsilon$, where $\mathcal{C}$ is a Cantor set and $B_\varepsilon\subset \mathbb{R}^d$ is an open ball.

  There is a natural action of $\mathbb{R}^d$ on $\Omega_\mathcal{T}$ by translation and this action is minimal if $\mathcal{T}$ is repetitive. The only case considered here is the case when this action is uniquely ergodic. That is, there exists a unique Borel probability measure $\mu$ which is invariant under the action of $\mathbb{R}^d$. By the local product structure $\mathcal{C}\times B_\varepsilon$ of $\Omega_\mathcal{T}$, the  measure $\mu$ has a natural product structure $\nu\times \mbox{Leb}$, where $\nu$ is a transverse measure to the action of $\mathbb{R}^d$ and invariant under holonomy. Since open sets along the totally disconnected transversal $\mho_\mathcal{T}$ are given by cylinder sets of patches, the measure $\nu$ satisfies $\nu(\mathcal{C}_{\mathcal{P},t}) = \mbox{freq}(\mathcal{P})$, where $\mbox{freq}(\mathcal{P})$ is the asymptotic frequency of translation copies of the patch $\mathcal{P}$ in the tiling $\mathcal{T}$.
  \subsubsection{Functions}
  \label{subsubsec:functions}
  \begin{definition}
    Let $\mathcal{T}$ be a tiling of $\mathbb{R}^d$. A function $h:\Omega_\mathcal{T} \rightarrow \mathbb{R}$ is \textbf{transversally locally constant} (TLC) if for any $\mathcal{T}$ if there is an $R>0$ such that if $\mathcal{O}^-_{\mathcal{T}'}(B_R(0)) = \mathcal{O}^-_{\mathcal{T}}(B_R(0))$ for any $\mathcal{T}'\in\Omega_\mathcal{T}$ then $h(\mathcal{T}) = h(\mathcal{T}')$.
  \end{definition}
  \begin{definition}
    Let $\mathcal{T}$ be a tiling of $\mathbb{R}^d$. A function $f:\mathbb{R}^d \rightarrow \mathbb{R}$ is \textbf{$\mathcal{T}$-equivariant} if there is an $R>0$ such that if $\mathcal{O}^-_\mathcal{T}(B_R(x)) = \mathcal{O}^-_\mathcal{T}(B_R(y))$ then it follows that $f(x) = f(y)$.
  \end{definition}
  There is a correspondence between TLC functions on $\Omega_\mathcal{T}$ and $\mathcal{T}'$-equivariant functions on $\mathbb{R}^d$. Given a TLC function on $\Omega_\mathcal{T}$ and $\mathcal{T}'\in\Omega_\mathcal{T}$, let $i_{\mathcal{T}'}(h):\mathbb{R}^d\rightarrow \mathbb{R}$ be the function
  \begin{equation}
    \label{eqn:imap}
    i_{\mathcal{T}'}(h)(t) = h\circ \varphi_t(\mathcal{T}')
  \end{equation}
  for all $t\in\mathbb{R}^d$. This map in fact gives an algebra isomorphism between TLC functions and $\mathcal{T}$-equivariant functions \cite[Theorem 20]{KP:RS}.
  \subsubsection{Return vectors}
  \label{subsec:retVectors}
  Let $\mathcal{T}$ be a repetitive tiling of finite local complexity. Define
  $$\Lambda_i^\mathcal{T} = \left\{\tau\in \mathbb{R}^d: \varphi_\tau(t_1) = t_2,\;\;\;\; \mbox{ for some }\;\;\;\; t_1\sim t_2\sim t_i \right\}$$
  for $i = 1,\dots, M$, and
  $$\Lambda_\mathcal{T} = \bigcup_i \Lambda_i^\mathcal{T}.$$
  The set $\Lambda_i^\mathcal{T}$ is the set of return vectors to a transversal between two tiles which is of the same translation equivalence type. Note that by repetitivity this set is independent of the tiling in the tiling space and, as such, it is defined by the tiling space.

    Let $\Gamma_\mathcal{T}$ be the group generated by the Delone set $\Lambda_\mathcal{T}$ (sometimes called the \textbf{Lagarias group}). This is defined as the group generated by linear combination of integer multiples of elements of $\Lambda_\mathcal{T}$:
    $$\Gamma_\mathcal{T} := \mathbb{Z}[\Lambda_\mathcal{T}].$$
    By finite local complexity, this is a finitely generated free Abelian group. Let $r$ be the rank of this group and choose a basis $\{v_i,\dots, v_r\}$ for $\Gamma_\mathcal{T}$. This choice defines the \textbf{address map} $\alpha:\Gamma_\mathcal{T}\rightarrow \mathbb{Z}^r$ by $\alpha(v_i)=e_i$ for all $i= 1,\dots, r$ and extending in a way to yield an isomorphism. By a result of Lagarias \cite{Lagarias:GeomMod1} this map is Lipschitz when restricted to $\Lambda_\mathcal{T}$.

      \subsection{Bratteli Diagrams}
      \label{subsec:brat}
A \textbf{Bratteli diagram} is an infinite directed graph $B= (V,E)$ with
$$V = \bigsqcup_{k\geq 0} V_k\,\hspace{1in}\mbox{ and }\hspace{1in}\, E = \bigsqcup_{k>0}E_k$$
and range and source maps $r,s:E\rightarrow V$ with $r(E_k) = V_k$ and $s(E_{k}) = V_{k-1}$ for $k\in\mathbb{N}$. It is assumed that $|E_k|$ and $|V_k|$ are finite for every $k$.  A Bratteli diagram $B$ can also be defined by a sequence of matrices $\{A_k\}$, where $A_k$ is a $|V_{k}|\times |V_{k-1}|$ matrix and $(A_k)_{ij}$ is the number of edges $e$ with $s(e) = v_j\in V_{k-1}$ and $r(e) = v_i\in V_{k}$.

      A finite path $\bar{e}$ is a finite collection of edges $(e_m,\dots, e_{m+n})$ with the property that with $e_i\in E_i$ and $r(e_i) = s(e_{i+1})$  for $i=m, \dots, m+n-1$. The range and source maps $r,s$ can be extended to the set of finite paths by setting $s((e_m,\dots, e_{m+n}) ) = s(e_m)$ and $r((e_m,\dots, e_{m+n}) ) = r(e_{m+n})$. The set $E_{m,n}$ is then the set of finite paths $\bar{e}$ such that $s(\bar{e})\in V_m$ and $r(\bar{e})\in V_n$. For two vertices $v\in V_m$ and $w\in V_n$ with $m<n$, $E_{vw}$ is the set of finite paths $\bar{e}$ such that $s(\bar{e})=v$ and $r(\bar{e})=w$. A Bratteli diagram is \textbf{minimal} is for every $k\geq 0$ there is a $\ell\in\mathbb{N}$ such that $E_{vw}\neq\varnothing$ for every $v\in V_k$ and $w\in V_{k+\ell}$.

      An infinite path $\bar{e}$ is a path of infinite length with $s(\bar{e})\in V_0$. The set $X_B$ is a set of all infinite paths, and it is topologized by a basis of cylinder sets of the form
      $$C_p = \{\bar{e}' = (e_1',e_2',e_3', \dots)\in X_B: e_i' = e_i \mbox{ for all }i\leq k\}$$
      for a finite path $p = (e_1,\dots, e_k)\in E_{0,k}$.

      Two paths $e,e'\in X_B$ are \textbf{tail-equivalent} (denoted by $e\sim e'$) if there exists a $k\geq 0$ such that $e_i = e_i'$ for all $i\geq k$. This relation is an equivalence relation and the tail-equivalence class of a path $\bar{e}$ is denoted by $[\bar{e}]$. Note that if a Bratteli diagram $B$ is minimal then $\overline{[\bar{e}]} = X_B$ for any $\bar{e}\in X_B $. A Borel measure $\mu$ on $X_B$ is \textbf{invariant} (or invariant for the tail-equivalence relation) if for any two finite paths $p,q\in E_{0,k}$ with $r(p) = r(q)$ it holds that $\mu(C_{p})=\mu(C_q)$. For such measures, for $v\in V_k$, denote $\mu(v) :=  \mu(C_p)$ for any $p\in E_{0,k}$ with $r(p)=v$.
      \subsubsection{Bi-infinite diagrams}
      The notion of a Bratteli diagram is easily generalized to a bi-infinite graph.
      \begin{definition}
A \textbf{bi-infinite Bratteli diagram} is an infinite directed graph $\mathcal{B}= (\mathcal{V},\mathcal{E})$ with
$$\mathcal{V} = \bigsqcup_{k\in\mathbb{Z}} \mathcal{V}_k\,\hspace{1in}\mbox{ and }\hspace{1in}\, \mathcal{E} = \bigsqcup_{k\in\bar{\mathbb{Z}}}\mathcal{E}_k,$$
where $\bar{\mathbb{Z}} = \mathbb{Z}-\{0\}$, along with range and source maps $r,s:\mathcal{E}\rightarrow \mathcal{V}$ with $r(\mathcal{E}_k) = \mathcal{V}_k$ and $s(\mathcal{E}_{k}) = \mathcal{V}_{k-1}$ for $k\in\mathbb{N}$, and $r(\mathcal{E}_k) = \mathcal{V}_{k-1}$ and $s(\mathcal{E}_{k}) = \mathcal{V}_{k}$ for $k<0$.  It is assumed that $|\mathcal{E}_k|$ and $|\mathcal{V}_k|$ are finite for every $k$.
      \end{definition}
      The notational convention adopted here is that $B,V,E, A$ will denote objects corresponding to Bratteli diagrams as in \S \ref{subsec:brat}, whereas $\mathcal{B},\mathcal{V},\mathcal{E},\mathcal{A}$ are the objects corresponding to bi-infinite diagrams as defined above. The sets of paths $\mathcal{E}_{m,n}$ are analogously defined as in $E_{m,n}$ in the obvious way, and so is the concept of minimality for a bi-infinite diagram $\mathcal{B}$.

      A bi-infinite diagram $\mathcal{B}$ can be thought of a two one-sided diagrams $B^+$ and $B^-$ as follows. The \textbf{positive part} $\mathcal{B}^+$ of the Bratteli diagram $\mathcal{B}$ is the restriction of $\mathcal{B}$ to its vertices and edges with non-negative indices. The \textbf{negative part} $\mathcal{B}^-$ of $\mathcal{B}$ is the one-sided Bratteli diagram obtained by considering the vertices and edges of $\mathcal{B}$ with non-positive indices and multiplying the indices by $-1$ to obtain a one-sided diagram.  Equivalently, if $\{\mathcal{A}_k\}_{k\in\bar{\mathbb{Z}}}$ is the sequence of matrices defining $\mathcal{B}$, then $\mathcal{B}^+$ is defined by the matrices $A^+_k = \mathcal{A}_k$ for $k>0$, and $\mathcal{B}^-$ is defined by the matrices $A^-_k = \mathcal{A}^T_{-k}$ for $k>0$. Note that $\mathcal{B}^+$ and $\mathcal{B}^-$ share the same level of vertices at level 0. As such, the space of all infinite paths $X_{\mathcal{B}}$ on $\mathcal{B}$ has a local product structure given by the identification
      $$X_{\mathcal{B}} = \left\{(\bar{e}^+,\bar{e}^-)\in X_{\mathcal{B}^+}\times X_{\mathcal{B}^-}: s(\bar{e}^-) = s(\bar{e}^+)\right\}. $$
      \section{Random substitution tilings}
      \label{sec:RandSub}
      Let $S_1,\dots, S_N$ be $N$ substitution rules on the prototiles $t_1,\dots, t_M$, where it is assumed without loss of generality that each prototile contains the origin in the interior. Recall that each substitution rule $S_\ell$ gives a collection of affine maps $\{f^\ell_{ijk}\}$ with the property that
      \begin{equation}
        \label{eqn:subsRule}
        t_i = \bigcup_{j=1}^M\bigcup_{k=1}^{\kappa_\ell(i,j)} f^\ell_{ijk}(t_j).
      \end{equation}
      Let $\Sigma_N = \{1,\dots, N\}^{\bar{\mathbb{Z}}}$ be the full $N$-shift, where $\bar{\mathbb{Z}} = \mathbb{Z}\backslash \{0\}$ and it inherits the natural order from $\mathbb{Z}$. For $x\in\Sigma_N$, define the Bratteli diagram $\mathcal{B}_x$ to be the diagram with $M$ vertices at every level, and, for $n>0$, having $\kappa_{x_n}(i,j)$ edges going from $v_j\in \mathcal{V}_{n-1}$ to $v_i\in\mathcal{V}_n$ for all $i,j = 1,\dots, M$. For $n<0$, having $\kappa_{x_n}(i,j)$ edges going from $v_j\in \mathcal{V}_{n}$ to $v_i\in\mathcal{V}_{n+1}$ for all $i,j = 1,\dots, M$. As such, since there is a natural bijection at level $n$ between edges and maps $\{f^{x_n}_{ijk}\}$, we denote by $f_e$ the corresponding affine linear map corresponding to the edge $e\in \mathcal{E}_n$. For a finite path $\bar{e} = (e_m,\dots, e_n)$ define $f_{\bar{e}} = f_{e_n}\circ \cdots\circ f_{e_m}$ and $f_{\bar{e}}^{-1} = f_{e_m}^{-1}\circ \cdots \circ f_{e_n}^{-1}$. Finally, for an infinite path $(\dots, e_{-1},e_1,e_2,\dots)=(\bar{e}^-,\bar{e}^+)=\bar{e} \in X_{\mathcal{B}_x}$ denote by $ \bar{e}|_k= \bar{e}^+|_k$ the truncation of $\bar{e}^+$ after level $k$. Thus $f_{\bar{e}|_k} = f_{\bar{e}^+|_k} = f_{e_k}\circ \cdots \circ f_{e_1}$.

      Following the constructions in \cite{ST:random, T:TTT}, one can build a tiling $\mathcal{T}_{\bar{e}}$ from a path $\bar{e} = (\bar{e}^-,\bar{e}^+)\in X_{\mathcal{B}_x}$ as follows. The $k^{th}$-approximant of $\mathcal{T}_{\bar{e}^+}$ is the set
      \begin{equation}
        \label{eqn:approximant}
        \mathcal{P}_k(\bar{e}^+) = \bigcup_{\substack{\bar{e}'\in \mathcal{E}_{0,k}\\ r(\bar{e}') = r(\bar{e}|_k)}}f_{\bar{e}|_k}^{-1}\circ f_{\bar{e}'}(t_{s(\bar{e}')}).
      \end{equation}
      It can be quickly verified that $\mathcal{P}_k(\bar{e}^+)\subset \mathcal{P}_{k+1}(\bar{e}^+)$ as a subpatch, and so by considering larger and larger indices one obtains
      $$\mathcal{T}_{\bar{e}}^+ = \bigcup_{k>0}\mathcal{P}_k(\bar{e}^+).$$
      This tiling is tiled by the tiles $f_{\bar{e}|_k}^{-1}\circ f_{\bar{e}'}(t_{s(\bar{e}')})$ for $\bar{e}'\in \mathcal{E}_{0,k}$ with $r(\bar{e}') = r(\bar{e}|_k)$ and arbitrary $k$. This tiling may or may not cover all of $\mathbb{R}^d$, but this potential problem will addressed below in Proposition \ref{prop:sum}. Let
      $$\tau_{\bar{e}}^- = \bigcap_{k<0} f_{(\bar{e}_k,\dots, \bar{e}_{-1})}(t_{s(e_{k})})$$
      and assume for the moment that this intersection is a single point. Then the tiling
      $$\mathcal{T}_{\bar{e}} = \varphi_{\tau_{\bar{e}}^-}(\mathcal{T}^+_{\bar{e}})$$
      is the tiling constructed from $\bar{e}$ as long as 1) $\mathcal{T}^+_{\bar{e}}$ covers all of $\mathbb{R}^d$ and 2) $\tau_{\bar{e}}$ is a single point. As will be discussed below in Proposition \ref{prop:sum}, both of these conditions will generically hold true. Note that if one defines
      $$\mathcal{P}_k(\bar{e}) = \varphi_{\tau^-_{\bar{e}}}(\mathcal{P}_k(\bar{e}^+))$$
      then $\mathcal{P}_k(\bar{e})\subset \mathcal{P}_{k+1}(\bar{e})$ for all $k>0$ and
      $$\mathcal{T}_{\bar{e}} = \bigcup_k \mathcal{P}_k(\bar{e}).$$

      Suppose $\mathcal{B}_x$ is minimal and let $\varnothing\neq \mathring{X}_{\mathcal{B}_x}\subset X_{\mathcal{B}_x}$ be the set of paths $\bar{e}$ such that $\mathcal{T}_{\bar{e}}$ covers all of $\mathbb{R}^d$. Then the construction $\bar{e}\mapsto \mathcal{T}_{\bar{e}}$ defines a surjective map $\Delta_x:\mathring{X}_{\mathcal{B}_x}\rightarrow \Omega_{\mathcal{T}_{\bar{e}}} = \Omega_x$ by \cite[Lemma 5]{ST:random}, called the \textbf{Robinson map}. That is, the tiling space $\Omega_{\mathcal{T}_{\bar{e}}}$ obtained is independent of the path $\bar{e}$ used. A \textbf{level-$k$ supertile} of a tiling $\mathcal{T}\in\Omega_x$ is a patch $\mathcal{P}$ of $\mathcal{T}$ which is translation equivalent to a level-$k$ approximant $\mathcal{P}_k(\bar{e}^+)$ in (\ref{eqn:approximant}) for some $\bar{e}^+\in X_{\mathcal{B}_x^+}$ not only as a set, but as a tiled patch.

      Since each of the substitution maps is of the form, $f_{ijk}^\ell(x) = \theta_{ijk\ell}x+\tau_{ijkl}$ for $\theta_{ijk\ell}\in(0,1]$, we can write $f_e = \theta_e x+\tau_e$. Define
        $$S^\pm_x(\bar{e}) = \lim_{n\rightarrow \pm \infty} \frac{1}{|n|}\sum_{i=1}^n\log \theta_{e_n}$$
        and, if the collection of substitutions are uniformly expanding,
        $$S^\pm_x(\bar{e}) = \lim_{n\rightarrow \pm \infty} \frac{\log \theta_{(n)_x}}{|n|},$$
        where $\theta_{(\pm n)_x}:= (\theta_{x_{\pm1}}\cdots \theta_{x_{\pm n}})$. If $X_{\mathcal{B}_x}^\pm$ admits a single tail-invariant probability measure, then $S^\pm_x:X^\pm_{\mathcal{B}_x}\rightarrow \mathbb{R}$ is constant. Moreover, we have that
        $$S^\pm_x(\bar{e}) = S^\pm_{\sigma(x)}(\sigma(\bar{e})),$$
        i.e., the functions $S^\pm_x$ are invariant under a shift of indices of the Bratteli diagram $\mathcal{B}_x$. As such, if $\mu$ is an $\sigma$-invariant ergodic probability measure on $\Sigma_N$ then $S^\pm_{x}$ are constant along the orbits.
        \begin{definition}
          An invariant probability measure $\mu$ is \textbf{contracting} if the $\mu$-almost every $x$ we have that $S^\pm_x(\bar{e})<0$ for all $\bar{e}\in X^\pm_{\mathcal{B}_x}$.
        \end{definition}
        Thus if the collection of substitution rules are uniformly contracting any invariant measure is contracting. The following comes from \cite{GM:mixed}.
        \begin{definition}
          A set of subtitution rules $S_1,\dots, S_N$ on the prototiles $t_1,\dots, t_M$ is \textbf{compatible} if each prototile $t_i$ has a CW structure such that for any $x\in\Sigma_N$, $(\bar{e}^-,\bar{e}^+)\in  X_{\mathcal{B}_X}$ and any $k\in\mathbb{N}$, the $k^{th}$ approximant $\mathcal{P}_k(\bar{e}^+)$ in (\ref{eqn:approximant}) is tiled by copies of prototiles such any intersections of two tiles is a union of subcells.
        \end{definition}
        Compatibility in dimension 1 is automatic. In higher dimension is harder to enforce but, as shown in \cite{GM:mixed}, there are still plenty of interesting examples in higher dimensions. The following is a collection of results from \cite{ST:random}.
        \begin{proposition}
          \label{prop:sum}
          Let $S_1,\dots, S_N$ be a collection of $N$ compatible substitution rules on the prototiles $t_1,\dots, t_M$ and let $\mu$ be a minimal, contracting, $\sigma$-invariant ergodic probability measure. Then for $\mu$-almost every $x$ there are unique invariant mesures  $\mu^\pm_x$ on $X^\pm_{\mathcal{B}_x}$ with $\mu_x = \mu^+_x\times \mu^-_x$ a measure on $X_{\mathcal{B}_x}$ with $\mu_x(\mathring{X}_{\mathcal{B}_x}) = 1$. The map $\Delta_x$ sends $\mu_x$ to the unique invariant measure on $\Omega_x$ which is invariant under the action of $\mathbb{R}^d$, locally as $(\Delta_x)_*\mu_x = (\Delta_x)_*(\mu_x^+\times \mu_x^-) = \nu_x\times\mbox{Leb}_x$, where $\nu_x$ is the unique $\mathbb{R}^d$-invariant probability measure $\mho_x\subset \Omega_x$ which is invariant under the holonomy of $\mathbb{R}^d$.
        \end{proposition}
        \subsection{Functions}
        \begin{definition}
          Let $S_1,\dots, S_N$ be a family of compatible substitution rules on the prototiles $t_1,\dots, t_M$, $x\in \Sigma_N$ and $\mathcal{B}_x$ the Brattelli diagram constructed from $x$ and the family of substitutions. Suppose that $\mathcal{B}_x$ is minimal and $S^\pm_x(\bar{e})<0$ for all $\bar{e}\in X_{\mathcal{B}_x}$. If $\Delta_{\bar{e}}:\mathring{X}_{\mathcal{B}_x}\rightarrow\Omega_x $ is the surjective Robinson map, then $h:\Omega_x\rightarrow \mathbb{R}$ is a \textbf{level-}$k$ transversally locally constant (TLC) function if there exist compactly supported functions $\psi_i:t_i\rightarrow \mathbb{R}$, $i=1,\dots, M$, such that
          $$h(\Delta_x(\bar{e})) =   \psi_i\left( \bigcap_{i<k} f_{(\bar{e}_i,\dots, \bar{e}_{k-1})}(t_{s(e_{i})})\right)$$
        if $r(\bar{e}_k) = v_i\in\mathcal{V}_k$.
        \end{definition}
        In words: if $\mathcal{T}'\in\Omega_x$ then there is a path $\bar{e}' = (\dots, e_{-1},e_1,e_2,\dots)$ with $\Delta_x(\bar{e}') = \mathcal{T}'$. Now, the level-$k$ supertile associated to $\mathcal{T}'$ is the patch associated to the level-$k$ approximant $\mathcal{P}_k(\bar{e}')$ which contains the origin, and this approximant, as a set, is a comformally-equivalent to some prototile $t_i$. Then $h(\mathcal{T}')$ is the value of $\psi_i$ at the point in $t_i$ corresponding to where the origin sits in $\mathcal{P}_k(\bar{e}')$. That the intersection is a single point is a consequence of the assumption $S^\pm_x(\bar{e})<0$.

        Having the parameter $k$ to describe TLC functions allows one to organize the set of all TLC functions as introduced in \S\ref{subsubsec:functions}. The same idea motivates the following definition, which is a modification to the current setting of the definition in \S\ref{subsubsec:functions}.
        \begin{definition}
          Let $\mathcal{T}\in\Omega_x$ be an aperiodic, repetitive tiling of finite local complexity. A function $g:\mathbb{R}^d\rightarrow \mathbb{R}$ is a \textbf{level-$k$ $\mathcal{T}$-equivariant function} if the value of $g$ at $x$ only depends on the level-$k$ supertile of $\mathcal{T}$ which contains $x$. More specifically, there is a level-$k$ TLC function $h:\Omega_\mathcal{T}\rightarrow \mathbb{R}$ such that $g(x) = h\circ \varphi_x(\mathcal{T})$.
        \end{definition}
        \begin{lemma}
          \label{lem:LipApprox}
          Let $S_1,\dots, S_N$ be a family of uniformly expanding compatible substitution rules on the prototiles $t_1,\dots, t_M$ and let $x\in\Sigma$ so that $\mathcal{B}_x$ is minimal and $\Omega_x$ has a uniquely ergodic $\mathbb{R}^d$ action. Let $f\in\Omega_x$ be Lipschitz function with Lipschitz constant $L_f$. Then for every $k\in\mathbb{N}$ there exists a level-$k$ TLC function $f_k$ such that
        $$\|f-f_k\|_\infty\leq L_f \max_{v\in\mathcal{V}_k}\mu_x^+(v)$$
        on a full measure set.
      \end{lemma}
      \begin{proof}
        The tiling space $\Omega_x$ is partitioned (up to a set of measure zero) into sets $\Omega_x^{k,1},\dots, \Omega_x^{k,M}$ with the property that any $\mathcal{T}\in\Omega_x^{k,i}$ has as its level-$k$ supertile containing the origin a supertile of type $i$. Pick any  $\mathcal{T}_i\in \Omega_{x}^{k,i}$ for $i=1,\dots, M$. Define $f_k$ as follows. If $\mathcal{T}\in\Omega_x^{k,i}$, then it is at a distance less than or equal to $\mu_x^+(v_i)$ from a translate $\mathcal{T}'\in \Omega_{x}^{k,i}$ of $\mathcal{T}_i$. Define in this case $f_k(\mathcal{T}) = f(\mathcal{T}')$. The approximation estimate follows.
      \end{proof}

      \subsection{Choices, canonical supertiles and control points}
      \begin{definition}
        \label{def:choice}
        Let $S_1,\dots, S_N$ be a collection of compatible substitution rules on $M$ prototiles, and $x\in \Sigma_N$. By the construction in this section we obtain a Bratteli diagram $\mathcal{B}_x=(\mathcal{V},\mathcal{E})$. A \textbf{choice function} is any map $c:\mathcal{V}\rightarrow \mathcal{E}$ that satifies
        \begin{enumerate}
        \item $c(v)\in r^{-1}(v)$ for all $v\in\mathcal{V}$,
        \item if $x_k = x_{k'}$ then $c(v) = c(v')$ for $v\in \mathcal{V}_{k}$ and $v'\in \mathcal{V}_{k'}$ under the obvious bijection between the sets of edges $\mathcal{E}_k$ and $\mathcal{E}_{k'}$.
        \end{enumerate}
        Given a choice function $c$, $k>0$ and $v\in\mathcal{V}_k$, the \textbf{choice path} $\bar{e}_{c,v}\in\mathcal{E}_{0,k}$ is the unique path $(e_1,\dots, e_k)\in\mathcal{E}_{0,k}$ satisfying $c(v) = e_k$, and $e_{i} = c(s(e_{i+1}))$ for all $i = 1,\dots, k-1$.
      \end{definition}
      What choice functions give is a canonical choice of tile in any supertile. More specifically, by (\ref{eqn:approximant}), any level-$k$ supertile is associated to a vertex $v\in \mathcal{V}_k$, and the super tile is obtained through the union of a collection of tiles in bijection with the set of paths $\bar{e}\in\mathcal{E}_{0,k}$ with $r(\bar{e}) = v$. Thus, since a choice function $c$ picks an element $\bar{e}_{c,v}\in\mathcal{E}_{0,k}$, then this corresponds to a choice of tile in the level-$k$ supertile.

Let $X_{c}^x\subset X_{\mathcal{B}_x}$ be the set of \textbf{maximal choice paths}. This is any path $\bar{e} = (\dots, e_{-1},e_1,e_2,\dots)\in X_{\mathcal{B}_x}$ such that $e_i = c(r(e_i))$ for all $i$. By compactness of $\mathcal{B}_x^+$, $X_{c}^x\neq \varnothing$. Note that by construction one has that $X_{c}^x\subset\mho_x$. For $v_i\in\mathcal{V}_k$, $k>0$, the choice function $c$ gives a unique choice path $\bar{e}_{c,v_i}\in\mathcal{E}_{0,k}$ with $r(\bar{e}_{c,v_i}) = v_i$ and thus a unique cylinder set $C_{c,v_i} := C_{\bar{e}_{c,v_i} }$.
\begin{definition}

  For $k>0$, $i\in \{1,\dots, M\}$ and $v_i\in\mathcal{V}_k$, the $M$ \textbf{canonical level-$k$ supertiles}

$$\mathcal{P}_k(v_i) := \mathcal{P}_k(\bar{e}^+),$$

  each of which is independent of the choice $\bar{e}\in C_{c,v_i}$
\end{definition}
\subsubsection{Control points}
\label{subsubsec:controlPts}
Let $S_1,\dots, S_N$ be $N$ compatible substitution rules on the prototiles $t_1,\dots, t_M$. As discussed in \S \ref{sec:RandSub}, a choice of point in the interior of each of the prototiles defines a canonical transversal (see (\ref{eqn:transversal})). This choice is arbitrary, but if $x\in\Sigma_N$ with $\mathcal{B}_x$ minimal, we can make a choice which will prove to be advantageous. According to \cite{SolomyakEigenfunctions}, this idea goes back to Thurston.

      Given a choice function $c$ and $v\in\mathcal{V}_0$, let $\bar{e}_{c,v}^{-\infty} = (\dots, e_{-2}^{c,-\infty},e_{-1}^{c,-\infty}) = \in\mathcal{E}_{-\infty,0}$ be the unique path $(\dots, e_{-k},\dots, e_{-1})\in\mathcal{E}_{-\infty,0}$ satisfying $e_{i-1} = c(s(e_{i}))$ for all $i<0$ and $r(e_{-1}) = v$. In other words, this path starts at $v$ and traces back a path by going along the edges given by the choice function $c$.
      \begin{definition}
        Let $S_1,\dots, S_N$ be $N$ compatible substitution rules on the prototiles $t_1,\dots, t_M$ and $x\in\Sigma_N$ such that $\mathcal{B}_x$ is minimal. The control points on the prototile $t_i$ corresponding to the vertex $v_i\in\mathcal{V}_0$ is given by
\begin{equation}
\label{eqn:controlPt}
\bigcap_{k<0} f_{(\bar{e}_k^{c,-\infty},\dots,e_{-2}^{c,-\infty},e_{-1}^{c,-\infty})}(t_{s(e_{k})})\in t_i
\end{equation}
The collection of the $M$ control points gives a marking in teach tile $t\in\mathcal{T}\in\Omega_x$ for any $\mathcal{T}\in\Omega_x$.
\end{definition}
      Let $\mathcal{C}_\mathcal{T}\subset\mathbb{R}^d$ be the set of points given by the markings on every tile of $\mathcal{T}$. Then it is immediate to see that
      \begin{equation}
      \label{eqn:CtrlPt1}
      \Lambda_\mathcal{T}\subset \mathcal{C}_\mathcal{T}-\mathcal{C}_\mathcal{T}.
      \end{equation}
      \section{Algebras and traces}
      \label{sec:algebras}
      \subsection{Algebras}
      \label{subsec:algebras}
      A \textbf{multi-matrix algebra} $\mathcal{A}$ is a $*$-algebra of the form
      $$\mathcal{A} = M_{n_1}\oplus \cdots \oplus M_{n_k},$$
      where $M_\ell$ is the algebra of $\ell\times \ell$ matrices over $\mathbb{C}$. Any finite-dimensional C$^*$-algebra is isomorphic to a multimatrix algebra \cite{davidson:book}. Let $\mathcal{M}_1 = M_{\ell_{1,1}}\oplus \cdots \oplus M_{\ell_{n,1}}$ and $\mathcal{M}_2 = M_{\ell_{1,2}}\oplus \cdots \oplus M_{\ell_{n,2}}$ be multi-matrix algebras and suppose $\phi:\mathcal{M}_1\rightarrow \mathcal{M}_2$ is a unital homomorphism of $\mathcal{M}_1$ into $\mathcal{M}_2$. Then $\phi$ is determined up to unitary equivalence in $\mathcal{M}_2$ by a $\ell_{n,2}\times \ell_{n,1}$ non-negative integer matrix $A_\phi$ \cite[\S III.2]{davidson:book}. It follows that the inclusion of a multi-matrix algebra $\mathcal{M}_0$ into a larger multimatrix algebra $\mathcal{M}_1$ is determined up to unitary equivalence by a matrix $A_0$ which roughly states how many copies of a particular subalgebra of $\mathcal{M}_0$ goes into a particular subalgebra of $\mathcal{M}_1$.

      Let $\mathcal{B}$ be a Bratteli diagram and let $A_k^+$, $k\in\mathbb{N}$, be the connectivity matrix at level $k$. In other words, $A_k(i,j)^+$ is the number of edges going from $v_j\in \mathcal{V}_{k-1}$ to $v_i\in\mathcal{V}_k$. An analogous matrix $A^-_k$ can be defined for $k<0$. Starting with $\mathcal{M}_0 = \mathbb{C}^{|\mathcal{V}_0|}$ the matrices $A_k^\pm$ define two families of inclusions $i_{|k|}^\pm:\mathcal{M}_{|k|-1}^\pm\rightarrow \mathcal{M}_{|k|}^\pm$ (up to unitary equivalence), one for $+$ and one for $-$, where each $\mathcal{M}_k^\pm$ is a multimatrix algebra. More explicitly, if
$$\mathcal{M}_k^\pm = M_{n_1^\pm}\oplus \cdots \oplus M_{n_k^\pm}$$
then starting with the vector $h^0=(1,\dots, 1)^T\in \mathbb{C}^{|\mathcal{V}_0|}$ and defining $h^{k,+} = A_k^+h^{k-1,+} = A_k^+\cdots A_{1}^+ h^0$ for $k\geq 0$ and $h^{k,-} =  h^{k-1,-}A_k^- =  h^0 A_1^-\cdots A_{k}^- $ for $k\leq 0$, we have that
$$\mathcal{M}_k^+ = M_{h_1^{k,+}}\oplus \cdots \oplus M_{h_{n_k}^{k,+}}\hspace{.5in}\mbox{ and }\hspace{.5in}\mathcal{M}_k^- = M_{h_1^{k,-}}\oplus \cdots \oplus M_{h_{n_k}^{k,-}}$$
and the inclusions $i_{|k|}^\pm:\mathcal{M}_{|k|-1}^\pm\rightarrow \mathcal{M}_{|k|}^\pm$ are defined up to unitary equivalence by the matrices $A_k^\pm$. The vectors $h^{k,\pm}$ are called the \textbf{height vectors} of the positive and negative parts of $\mathcal{B}$, respectively.

With these systems of inclusions one can define the inductive limits
\begin{equation}
 LF(\mathcal{B}^+) := \bigcup_k\mathcal{M}_k^+ = \lim_{\rightarrow}(\mathcal{M}_k^+,i_k^+) \hspace{.7in}  LF(\mathcal{B}^-) := \bigcup_k\mathcal{M}_k^- = \lim_{\rightarrow}(\mathcal{M}_k^-,i_k^-)
\end{equation}
which are $*$-algebras called the \textbf{locally finite (LF) algebras} defined by $\mathcal{B}$. Their $C^*$-completion
$$AF(\mathcal{B}^+) := \overline{LF(\mathcal{B}^+)},\hspace{.8in AF(\mathcal{B}^-) := \overline{LF(\mathcal{B}^-)}}$$
are the \textbf{approximately finite-dimensional (AF) algebras} defined by $\mathcal{B}$.
\subsection{Traces}
A \textbf{trace} on a $*$-algebra $\mathcal{A}$ is a linear functional $\tau:\mathcal{A}\rightarrow \mathbb{C}$ which satisfies $\tau(ab) = \tau(ba)$ for all $a,b\in\mathcal{A}$. The set of all traces of $\mathcal{A}$ forms a vector space over $\mathbb{C}$ and it is denoted by $\tr(\mathcal{A})$. There is no assumption that traces are positive (that is, $\tau(aa^*)>0$). A \textbf{cotrace} $\tau'$ is an element of the dual vector space $\tr^*(\mathcal{A}):=\tr(\mathcal{A})^*$.

For $M_\ell$, the algebra of $\ell\times\ell$ matrices, $\tr(M_\ell)$ is one-dimensional and generated by the trace $ a\mapsto \sum_{i=1}^\ell a_{ii}$. For a multimatrix algebra $\mathcal{M} = M_{\ell_1}\oplus\cdots\oplus M_{\ell_n}$, the dimension of $\tr(\mathcal{M})$ is $n$ and is generated by the canonical traces in each summand, so $\mathrm{Tr}(\mathcal{M}) =  \tr(M_{\ell_1})\oplus\cdots\oplus \tr(M_{\ell_n})$.

Let $i_k^+:\mathcal{M}_{k-1}^+\rightarrow \mathcal{M}_k^+$ be the family of inclusions defined by the positive part of a Bratteli diagram $\mathcal{B}^+$. Then there is a dual family of inclusions $i_k^*:\tr(\mathcal{M}_k^+)\rightarrow \tr(\mathcal{M}_{k-1}^+)$ (and an analogous family $i_k^*:\tr(\mathcal{M}_k^-)\rightarrow \tr(\mathcal{M}_{k-1}^-)$). The trace spaces of the LF algebras defined by a Bratteli diagram $\mathcal{B}$ are then the inverse limits
\begin{equation}
  \begin{split}
    \tr(\mathcal{B}^+) &:= \tr(LF(\mathcal{B}^+)) = \lim_{\leftarrow} (i^*_k,\tr(\mathcal{M}^+_k)) \\
    \tr(\mathcal{B}^-) &:= \tr(LF(\mathcal{B}^-)) = \lim_{\leftarrow} (i^*_k,\tr(\mathcal{M}^-_k))
    \end{split}
  \end{equation}
which are vector spaces. The respective spaces of cotraces are then
$$\tr^*(\mathcal{B}^+)  = \lim_{\rightarrow} ((i^*_k)^*,\tr^*(\mathcal{M}^+_k))\hspace{.4in}\mbox{ and } \hspace{.4in} \tr^*(\mathcal{B}^-)  = \lim_{\rightarrow} ((i^*_k)^*,\tr^*(\mathcal{M}^-_k)).$$

For a Bratteli diagram $\mathcal{B}^+$ the canonical generating traces for $\mathrm{Tr}(\mathcal{M}_k^+)$ will be denoted by $\mathfrak{t}_{k,1},\dots, \mathfrak{t}_{k,|\mathcal{V}_k|}$ so that
\begin{equation}
    \label{eqn:canTraces}
  \left\langle \mathfrak{t}_{k,1},\dots,  \mathfrak{t}_{k,|\mathcal{V}_k|}\right\rangle = \mathrm{Tr}(\mathcal{M}_k^+)\hspace{.5in}\mbox{ and }\hspace{.5in} \mathfrak{t}_{k,j}(\mathrm{Id}) = h^{k}_{j}
\end{equation}
for $j\in\{1,\dots, |\mathcal{V}_k|\}$.
\section{Renormalization}
\label{sec:renorm}
Let $\sigma:\Sigma_N\rightarrow \Sigma_N$ be the shift map and $V\in\Sigma_N$ an open set.
\begin{definition}
A \textbf{maximal sequence of return times} to $V$ for $x\in\Sigma_N$ under $\sigma$ is a sequence $k_n\rightarrow \infty$ such that $\sigma^{k_n}(x)\in V$ for all $n$ and such that if $\{k'_m\}$ satisfies $\sigma^{k'_m}(x)\in V$ for all $m$ then $\{k'_m\}\subset \{k_n\}$.
\end{definition}

Let $S_1,\dots, S_N$ be a collection of uniformly expanding compatible substitution rules on the prototiles $t_1,\dots, t_M$. For $x\in\Sigma_N$, supposed that the resulting Bratteli diagram $\mathcal{B}_x$ is minimal and denote by $\Omega_x$ the resulting tiling space. The (right) shift map $\sigma:x\mapsto \sigma(x)$ induces a homeomorphism $\Phi_x:\Omega_x\rightarrow \Omega_{\sigma(x)}$ giving the time-change equation
      \begin{equation}
        \label{eqn:conj}
\Phi_x\circ \varphi_{\theta_{x_{-1}}t}(\mathcal{T}) = \varphi_{t}\circ \Phi_x(\mathcal{T})
        \end{equation}
      for any $\mathcal{T}\in\Omega_x$. Recall the notion of control points from \S \ref{subsubsec:controlPts}. From (\ref{eqn:controlPt}) it immediately follows that for any $\mathcal{T} = \Delta(\bar{e})\in\Omega_x$,
      \begin{equation}
        \label{eqn:CtrlPt2}
\theta_{x_{-1}}^{-1} \mathcal{C}_{\mathcal{T}}\subset \mathcal{C}_{\Phi_x(\mathcal{T})}.
        \end{equation}
      Therefore, since the choice function determines the canonical transversal via the control points, it follows that
      \begin{equation}
        \label{eqn:CtrlPt3}
        \Phi_x(\mho_x)\subset \mho_{\sigma(x)}\hspace{.3in}\mbox{ and if }\hspace{.3in} \bar{e}\in X_c^x,\hspace{.3in}\mbox{ then }\hspace{.3in}\Phi^{\pm (\pm k)}_x(\mathcal{T}_{\bar{e}})\in\mho_{\sigma^k(x)}
      \end{equation}
      for any $k\in\mathbb{Z}$.

      More generally, denoting the homeomorphism $\Phi_x^{(k)} := \Phi_{\sigma^{k-1}(x)}\circ \cdots \circ \Phi_x:\Omega_x\rightarrow \Omega_{\sigma^k(x)}$, it follows that
      \begin{equation}
        \label{eqn:conj2}
\Phi_x^{(k)}\circ \varphi_{\theta_{(-k)_x}t}(\mathcal{T}) = \varphi_{t}\circ \Phi_x^{(k)}(\mathcal{T}),
      \end{equation}
      where $\theta_{(-k)_x}:= \theta_{x_{-k}}\cdots\theta_{x_{-1}}$. The following fact will be used later; the proof is left to reader.
      \begin{lemma}
        \label{lem:TLCfunctions}
        Let $h:\Omega_x\rightarrow \mathbb{R}$ be a level-$k$ TLC function. Then $\Phi_{\sigma^{-y}(x)}^{(y)*}h:\Omega_{\sigma^{-y}(x)}\rightarrow \mathbb{R}$ is a level-$k'$ TLC function, where $k' = \max\{0,k-y\}$.
      \end{lemma}
      \subsection{Return vectors}
      \label{subsec:RV}
Throughout this section it will be assumed that $\mathcal{B}_x$ is minimal. Recalling the definition of return vectors in \S\ref{subsec:retVectors}, note that since the sets $\Lambda_i^\mathcal{T}$ defined in \S\ref{subsec:retVectors} do not depend on $\mathcal{T}$ by repetitivity, so denote by $\Lambda_{x,i}^{(0)}$ the set of return vectors to the subset $\mho_x^i\subset \mho_x$ of the canonical transversal made up of tilings whose tile containing the origin is translation equivalent to $t_i$:
$$\Lambda_{x,i}^{(0)} := \{ \tau\in\mathbb{R}^d: \varphi_\tau(\mathcal{T})\in \mho_x^i\ \mbox{ for some }\mathcal{T}\in\mho^i_x \} \hspace{.5in}\mbox{ and }\hspace{.5in}  \Lambda_x^{(0)} := \bigcup_i \Lambda_{x,i}^{(0)}.$$
Now, for every $k\in\mathbb{N}$, let $\Lambda_{x,i}^{(k)}$ be the set of vectors taking level-$k$ supertiles of type $i$ to level-$k$ supertiles of type $i$:
$$\Lambda_{x,i}^{(k)} := \{ \tau\in\mathbb{R}^d: \mbox{ there are patches }\mathcal{P},\mathcal{P}'\subset \mathcal{T}\in\Omega_x \mbox{of the type $i$ such that } \varphi_\tau(\mathcal{P})= \mathcal{P}' \} $$
and
$$\Lambda_x^{(k)} := \bigcup_i \Lambda_{x,i}^{(k)}.$$
All of these sets are well-defined for any $x\in\Sigma_N$ as long as $\mathcal{B}_x$ is minimal. In particular, it is well-defined along the orbit $\{\Omega_{\sigma(x)}\}$ of $x$. Note now that for every $i\in\{1,\dots, M\}$ and $k\in\mathbb{N}$,
\begin{equation}
  \label{eqn:vecScale}
  \Lambda_{x,i}^{(k)} = \theta^{-1}_{(k)_x} \Lambda_{\sigma^{-k}(x),i}^{(0)}\hspace{.4in} \mbox{ and so }\hspace{.4in}\Lambda_{x}^{(k)} = \theta^{-1}_{(k)_x} \Lambda_{\sigma^{-k}(x)}^{(0)}.
  \end{equation}
Because any $\tau\in\Lambda_{x,i}^{(k)}$ is a translation vector between level-$k$ supertiles, it is also a translation between the tiles which tile the supertile. As such, by repetitivity/minimality, for all large enough $k$ one has that $\Lambda_{x,i}^{(k)}\subset \Lambda_x^{(0)}$ for any $i$ and so
\begin{equation}
  \label{eqn:inclusionRet}
\Lambda_{x}^{(k)} \subset \Lambda_x^{(0)}.
\end{equation}
Denote by $\Gamma_x^{(k)} := \mathbb{Z}(\Lambda_{x}^{(k)})$ the subgroups of $\mathbb{R}^d$ that these point sets generate, with $\Gamma_x := \Gamma_x^{(0)}$. By final local complexity, these are finitely generated subgroups of $\mathbb{R}^d$. Let $r_x^{(k)}$ be their respective ranks, with $r_x := r_x^{(0)}$. Thus, by (\ref{eqn:vecScale}) and (\ref{eqn:inclusionRet}) we have that
\begin{equation}
  \label{eqn:inclusionRet2}
  \Gamma_x^{(k)}= \theta_{(k)_x}^{-1} \Gamma_{\sigma^{-k}(x)} \subset   \Gamma_x.
  \end{equation}

For any $k\in\mathbb{N}$, consider $v_i\in\mathcal{V}_k$ and paths $p,q\in\mathcal{E}_{0,k}$ with $r(p) = r(q)$. Since $\mathcal{P}_k(v_i)$ is tiled by $|\mathcal{E}_{0,{v_i}}|$ tiles, there is a vector $\tau(p,q)$ which describes the translation between the marked point of the tile corresponding to $p$ in $\mathcal{P}_k(v_i)$ to the marked point on the tile corresponding to the path $q$. As such, $\tau(q,p) = -\tau(p,q)$.
\begin{lemma}
\label{lem:generators}
  Let $\mathcal{B}_x$ be minimal. For any $R>0$ there exists a $k\in\mathbb{N}$ such that
  $$\Lambda_x\cap B_R(0) \subset \left\{\tau(p,q)\in\mathbb{R}^d: p,q\in\mathcal{E}_{0,k} \mbox{ with }r(p) = r(q)\right\}.$$
\end{lemma}
\begin{proof}Each $\tau\in \Lambda_x\cap B_R(0)$ comes from the translation within a patch of a diameter bounded by some constant which depends on $R$ and $x$. By finite local complexity, there are finitely many such patches. By repetitivity there exists a $R'>0$ such that a ball of radius $R'$ contains a copy of each of these patches. Let $r_{min}$ be the minimum of the injectivity radii of the prototiles $t_1,\dots, t_M$. If $k$ is such that $\theta_x^{-(k)}r_{min}>R'$ then each level-$k$ supertile contains a ball of radius greater than $R'$ and thus all of the patches which give the vectors in $\Lambda_x\cap B_R(0)$.
\end{proof}
\begin{lemma}
  \label{lem:recurrence}
Let $x\in\Sigma_N$ be recurrent and $\mathcal{B}_x$ minimal. Then $r_x = r_{\sigma^k(x)}$ for all $k\in\mathbb{N}$.
\end{lemma}
\begin{proof}
Let $k_n\rightarrow\infty$ be such that $\sigma^{k_n}(x)\rightarrow x$ as $n\rightarrow \infty$. By (\ref{eqn:inclusionRet2}) it follows that
  $$r_x\geq r_{\sigma(x)} \geq \cdots \geq r_{\sigma^{k_n}(x)}\geq\cdots\geq  r_{\sigma^{k_{n+1}}(x)}\geq\cdots.$$
Choosing generators $v_1,\dots , v_{r_x}\in\Lambda_x$, by Lemma \ref{lem:generators} and (\ref{eqn:inclusionRet2}), for all large enough $n$, we have that $r_{\sigma^{k_n}(x)} = r_x$, so $r_x = r_{\sigma^{k}(x)}$ for all $k\in\mathbb{N}$.
  \end{proof}
The following is a consequence of the Poincar\'e recurrence theorem.
\begin{corollary}
Let $\mu$ be a minimal, $\sigma$-invariant ergodic probability measure. Then $r_x$ is constant $\mu$-almost everywhere.
\end{corollary}
\begin{definition}
The \textbf{rank} of a minimal, $\sigma$-invariant ergodic probability measure $\mu$ is the $\mu$-almost everywhere constant value $r_x$, and it is denoted by $r_\mu$.
\end{definition}
\begin{lemma}
  There exists a collection of open subsets $\{U_i\}$ of $\Sigma_N$ such that for any two $x,x'\in U_i$ with $\mathcal{B}_x$ and $\mathcal{B}_{x'}$ minimal and $r_x  =  r_{x'}$ then $\Gamma_x  = \Gamma_{x'}$.
\end{lemma}
\begin{proof}
Let $\mathcal{B}_x$ be minimal and consider a generating set $v_1,\dots, v_{r_x}\in\Lambda_x$ of $\Gamma_x$. Let $R$ be the maximum of the norm of the vectors in this generating set. By Lemma \ref{lem:generators} there is a $k$ such that the vectors in the generating set are all of the form $\tau(p,q)$ with $p,q\in\mathcal{E}_{0,k}$. Let $x'\in\Sigma$ be such that $x_i=x_i'$ for all $0<i\leq k$ and such that $\mathcal{B}_{x'}$ is minimal. Then the vectors $v_1,\dots, v_{r_x}$ are also elements of $\Lambda_{x'}$ and they generate a subgroup $\Gamma_{x'}\subset \Gamma_{x}$. Since $r_x = r_{x'}$ it follows that $\Gamma_x = \Gamma_{x'}$.
  \end{proof}
\begin{corollary}
  \label{cor:2}
Let $\mathcal{R}_k\subset \Sigma_N$ be the set of recurrent $x$ such that $r_x = k$ and $\mathcal{B}_x$ is minimal. For each $k\in\mathbb{N}$ such that $\mathcal{R}_k\neq \varnothing$ there exists a collection of disjoint subsets $\{U_i^k\}_i$ of $\mathcal{R}_k$ such that if $x\in U_i^k$ and $v_1,\dots, v_k \in \Lambda_x$ is a generator for $\Gamma_x$, then $v_1,\dots, v_k$ generates $\Gamma_{x'}$ for any $x'\in U_i^k$. In particular, $\Gamma_x = \Gamma_{x'}$ for any two $x,x'\in U_i^k$.
\end{corollary}
Let $x\in\Sigma_N$ be recurrent and $\mathcal{B}_x$ minimal and choose a set of generators $v_1^i,\dots, v_{r_x}^i$ in each $U_i^{r_x}$. For $x\in U_i^{r_x}\subset  \Sigma_N$, let $V_x$ be the $d\times r_x$ matrix with column vectors $v_1^i,\dots, v_{r_x}^i$ so that for each $\tau\in\Lambda_x$ there is a unique $\alpha_x(\tau)\in\mathbb{Z}^{r_x}$ such that $\tau = V_x\alpha_x(\tau)$. Here $\alpha_x:\Gamma_x\rightarrow \mathbb{Z}^{r_x}$ is the localized address map, which is well defined by the previous corollary. Note that both the matrix $V_x$ and address map $\alpha_x$ are locally constant in $\mathcal{R}_{r_x}$.

Given that $\theta_{x_{-1}}^{-1}\tau \in\Lambda_{\sigma(x)}$, there exists a $r_x\times r_x $ matrix $G_x$ such that $\theta_{x_{-1}}^{-1}V_{x} =  V_{\sigma(x)}G_x$ and, in general,
\begin{equation}
  \label{eqn:vectorCocycle}
  \begin{split}
  &\theta_{(-n)_x}^{-1}V_{x} =  V_{\sigma^n(x)}G_{\sigma^{(n-1)}(x)}\cdots G_{\sigma(x)} G_x, \hspace{.5in}\mbox{ and thus } \\
&\theta_{(-n)_x}^{-1}\tau = \theta_{(-n)_x}^{-1} V_{x}\alpha_x(\tau) =  V_{\sigma^n(x)} G_x^{(n)} \alpha_x(\tau) = V_{\sigma^n(x)} \alpha_{\sigma^n(x)}\left(\theta_{(-n)_x}^{-1}\tau\right)

  \end{split}
\end{equation}
for any $\tau\in\Gamma_x$, where $G^{(k)}_x :=  G_{\sigma^{k-1}(x)}\cdots G_{\sigma(x)} G_x$. Note that the map $x\mapsto G_x$ is locally constant on $\mathcal{R}_{r_x}$.

For $x\in U_i^k$, let $\tau_1,\dots, \tau_{r_x}$ denote the set of canonical generators for $\Gamma_{x'}$ for all $x'\in U_i^k$, and $v_j:= \alpha_x(\tau_j)\in \mathbb{Z}^{r_x}$ be their image under the address map, where it is assumed that $v_j$ is the $j^{th}$ canonical generator in $\mathbb{Z}^{r_x}$. Consider the real vector spaces generated by $v_1,\dots, v_{r_x}$
$$\mathfrak{R}_x:= \left\{\sum_{i=1}^{r_x} a_i v_i: a_i \in \mathbb{R}\right\}\hspace{.7in}\mbox{ and } \hspace{.7in}\mathfrak{R}_x^*:= \left\{\sum_{i=1}^{r_x} a_i v_i^*: a_i \in \mathbb{R}\right\}$$
with $\langle v_j,v^*_\ell\rangle = \delta_{j\ell}$. Here $\mathfrak{R}_x$ (and its dual) is regarded as an abstract vector space and not as a subspace of $\mathbb{R}^d$. In other words, $\mathfrak{R}_x$ is a vector space of real dimension $r_x$ and so is its dual. By Corollary \ref{cor:2}, given $k$, since one can pick a canonical choice of generators in each $U_i^{k}$, one can identify all the vector spaces $\mathfrak{R}_{x'}$ for any $x'\in U_i^k$. In particular, one can chose norm $\|\cdot\|$ for all the spaces $\mathfrak{R}_{x'}$.

The locally constant maps $G_x$ define dual maps $\bar{G}_x$ in a natural way: since $\theta_{x_1}^{-1}:\Gamma_{\sigma^{-1}(x)} \rightarrow \Gamma_x$ and, by (\ref{eqn:vectorCocycle}), $G_{\sigma^{-1}(x)}:\mathfrak{R}_{\sigma^{-1}(x)}\rightarrow \mathfrak{R}_x$, the dual maps satisfy $\bar{G}_x:= G^*_{\sigma^{-1}(x)}:\mathfrak{R}^*_x\rightarrow \mathfrak{R}^*_{\sigma^{-1}(x)} $.

\begin{definition}
  Let $S_1,\dots, S_N$ be a collection of $N$ compatible substitution rules on prototiles $t_1,\dots, t_M$. The \textbf{return vector bundle} is the map $p_R:\mathfrak{R}_N\rightarrow \Sigma_N$ where the fibers are the vector spaces\footnote{This is only defined for $x\in\Sigma$ such that $\Omega_x$ is well defined. If the substitution rules $S_i$ are all primitive, then this is defined for all $x\in\Sigma_N$.} $p^{-1}_R(x) =  \mathfrak{R}_x^*$. The \textbf{return vector cocycle} is the map $\mathfrak{G}:(x,v)\mapsto (\sigma^{-1}(x), \bar{G}_x v)$. The cocycle will be denoted $\bar{G}^{(n)}_x:= \bar{G}_{\sigma^{n-1}(x)}\circ\cdots\circ \bar{G}_x$, and set $\bar{G}^{(m,n)}_x:= \bar{G}_{\sigma^{m-1}(x)}\cdots\bar{G}_{\sigma^{n}(x)} $.
\end{definition}

It is not clear whether the maps $G_x$ are invertible. In any case, one can invoke the semi-invertible Oseledets theorem \cite{FLQ:coherent}.
\begin{theorem}[Semi-invertible Oseledets Theorem]
  \label{thm:oseledetsReturn}
  Let $\mu$ be a minimal $\sigma$-invariant ergodic probability measure. Suppose that $\log^+\|G_x\|\in L^1(\Sigma_N,\mu)$. Then there exists numbers $\omega_1^+ > \omega_2^+ > \cdots >\omega^+_{\sh^+_\mu} > 0 >\omega_1^- > \omega_2^- > \cdots > \omega^-_{\sh^-_\mu}$  and a measurable, $\mathfrak{G}$-invariant family of subspaces $Y_j^\pm(x), Y^\infty(x)\subset \mathfrak{R}_x$ such that for $\mu$-a.e. $x$:
  \begin{enumerate}

  \item The decomposition $\mathfrak{R}_x =Y^\infty_x \oplus Y^+_x\oplus Y^-_x$ holds with
    $$ Y^+_x = \bigoplus_{i=1}^{\sh_\mu^+} Y^+_i(x)\hspace{.8in}\mbox{ and }\hspace{.8in}Y^-_x=\bigoplus_{i=1}^{\sh_\mu^-} Y_i^-(x),$$
  \item \hspace{.3in} $G_x Y_i^\pm(x) = Y_i^\pm(\sigma^{-1}(x))$ \hspace{.3in} and \hspace{.3in} $G_x Y^\infty(x)\subset Y^\infty(\sigma^{-1}(x))$,
  \item For all $\xi\in Y_i^\pm(x)\backslash \{0\}$ and $\xi_\infty\in Y^\infty(x)\backslash \{0\}$, one has
    $$\lim_{n\rightarrow \infty} \frac{1}{n}\log\|G^{(n)}_x\xi\| = \omega_i^\pm\hspace{.7in}\mbox{ and } \hspace{.7in} \lim_{n\rightarrow \infty} \frac{1}{n}\log\|G_x^{(n)}\xi\| = -\infty.$$
\end{enumerate}
\end{theorem}
The numbers $\omega_i^\pm$ are the \textbf{Lyapunov exponents} of the return vector cocycle.
\subsection{The trace cocycle}
This section recalls the construction from \cite{T:TTT}. Let $\mathcal{B}$ be a bi-infinite Bratteli diagram. By the definition of $LF(\mathcal{B}^+_x)$, there is a countable family of multimatrix algebras $\mathcal{M}_k^x := \mathcal{M}_k^{\mathcal{B}^+_x}$, and thus a countable family of vector spaces of traces $\tr(\mathcal{M}_k^x) = \left\langle \mathfrak{t}^k_{x,i}\right\rangle_{i=1}^{|\mathcal{V}_k|}$, where $\mathfrak{t}^k_{x,j}$ is the canonical trace in the $j^{th}$ matrix algebra of $\mathcal{M}_k^x$ as defined in (\ref{eqn:canTraces}).

The map $i^*_1:\tr(\mathcal{M}_1^x)\rightarrow \tr(\mathcal{M}_0^x)$ can be explicitly computed as follows. Denote the standard basis of $\mathbb{C}^{M} = \mathcal{M}_0^x$ by $\{\delta_1,\dots, \delta_{M}\}$ and by $\{\mathfrak{t}^1_{x,1},\dots, \mathfrak{t}^1_{x,M}\}$ the canonical basis for $\tr(\mathcal{M}_1^x)$. Then every element in $a\in \mathcal{M}_0^x$ and any element $\mathfrak{t}\in \tr(\mathcal{M}_1^x)$ is of the form
$$a = \sum_{j=1}^{M}\alpha_j \delta_j,\hspace{1in}\mbox{ and } \hspace{1in} \mathfrak{t}  = \sum_{i = 1}^{M} \beta_i \mathfrak{t}^1_{x,i},$$
respectively, where $\alpha_j,\beta_i\in\mathbb{C}$. Then
\begin{equation}
  \label{eqn:tracemap}
  \begin{split}
    \mathfrak{t}(i_1(a)) &= \mathfrak{t}\left(i_1\left(\sum_{i=1}^{M}\alpha_j\delta_j\right)\right) = \sum_{j=1}^{M}\beta_i \mathfrak{t}^x_{1,i}\circ i_1 \left(\sum_{i=1}^{M}\alpha_j\delta_j\right) \\
    &= \sum_{j=1}^{M}\beta_i  \sum_{i=1}^{M}\alpha_j\left(\mathfrak{t}^x_{1,i}\circ i_1\left(\delta_j\right)\right)  = \sum_{j=1}^{M}\beta_i  \sum_{i=1}^{M}\alpha_j (\mathcal{A}^x_1)_{i,j} = \langle (\mathcal{A}^x_1)^* \mathfrak{t},a\rangle = (i_1^*\mathfrak{t})(a).
  \end{split}
\end{equation}
So the induced map $i^*_1$ is given by the matrix $(\mathcal{A}^x_1)^*$ and the dual map $(i_1^*)^*:\tr^*(\mathcal{M}_0^x)\rightarrow \tr^*(\mathcal{M}_1^x)$ is given by the matrix $\mathcal{A}^x_1$.
\begin{lemma}
  \label{lem:induced}
  The left shift $\sigma^{-1}:\Sigma_N\rightarrow \Sigma_N$ induces a linear map $\sigma_*^{-1}:\tr^*(\mathcal{M}_0^x)\rightarrow \tr^*(\mathcal{M}^{\sigma^{-1}(x)}_0)$ given by the matrix $\mathcal{A}^x_1$ after the canonical identifications $\tr^*(\mathcal{M}_0^x) = \mathbb{C}^{M}=\tr^*(\mathcal{M}_0^{\sigma^{-1}(x)})$.
\end{lemma}
\begin{proof}
  Define $\mathfrak{t}^x_1:\mathcal{M}_1^x\rightarrow \mathbb{C}^{M}= \mathcal{M}_0^{\sigma^{-1}(x)}$ to be the map defined by
  $$a\mapsto \left(\mathfrak{t}^x_{1,1}(a), \mathfrak{t}^x_{1,2}(a), \dots,\mathfrak{t}^x_{1,M}(a)\right)$$
  for all $a\in\mathcal{M}_1^x$. Now, for $a\in\mathcal{M}_0^x$, consider the composition of $\mathfrak{t}^x_1$ with the inclusion map $i_1$, $\sigma_*^{-1}:=\mathfrak{t}^x_1\circ i_1: \mathcal{M}_0^x\rightarrow \mathcal{M}^{\sigma^{-1}(x)}_0$. Note that the dual map $\mathfrak{t}^{x*}_1:\tr(\mathcal{M}_0^{\sigma^{-1}(x)})\rightarrow \tr( \mathcal{M}_1^x )$ is represented by the identity matrix when using the canonical basis in both $\tr(\mathcal{M}_0^{\sigma^{-1}(x)})$ and $\tr( \mathcal{M}_1^x )$. By this fact and (\ref{eqn:tracemap}) we have that the dual map $\sigma^{-1*}: \tr(\mathcal{M}^{\sigma^{-1}(x)}_0) \rightarrow\tr( \mathcal{M}_0^x )$ is represented by the matrix $(\mathcal{A}^x_1)^*$ in canonical coordinates, which defines a map $\sigma_*^{-1}:\tr^*(\mathcal{M}_0^x)\rightarrow \tr^*(\mathcal{M}^{\sigma^{-1}(x)}_0)$ through the matrix $\mathcal{A}^x_1$.
\end{proof}
\begin{definition}
  Let $S_1,\dots, S_N$ be $N$ compatible substitution rules on the prototiles $t_1,\dots, t_M$. The \textbf{trace bundle} $\tr^+(\Sigma_N)$ is the set of pairs $(x,s)$ where $x\in\Sigma_N$ and $s\in \tr^*(\mathcal{M}_0^{\mathcal{B}^+_x})$. The \textbf{trace cocycle} is fiberwise linear map $G_+:\tr^+(\Sigma_N)\rightarrow \tr^+(\Sigma_N)$ over the left shift $\sigma^{-1}:\Sigma_N\rightarrow \Sigma$ with fiber map $\sigma_*^{-1}$ as obtained from Lemma \ref{lem:induced}.  The composition $\sigma_*^{-1}\circ \cdots \circ \sigma_*^{-1}$ $k$ times will be denoted by $\sigma^{-(k)}_*:\tr^*(\mathcal{M}_0^\mathcal{B})\rightarrow \tr^*(\mathcal{M}_0^{\sigma^{-k}(\mathcal{B})})$ .
\end{definition}
Since the fiberwise linear maps $\sigma_*^{-1}$ agree, under a canonical choice of coordinates, with the linear maps given by the matrix $\mathcal{A}^x_1$, the cocycle can be written in these coordinates, for $m< n$, as
\begin{equation}
  \label{eqn:cocycleCoord}
  \Theta_x^{(m,n)} := \mathcal{A}_n^x \cdots \mathcal{A}_{m+1}^x\hspace{.5in} \mbox{ and }\hspace{.5in}\Theta_x^{(k)}:= \Theta_x^{(0,k)}.
\end{equation}
Oseledets theorem will not be necessary for the trace cocycle, only the existence of the leading Lyapunov exponent $\eta_1$. This is guaranteed by the theorem of Furstenberg and Kesten \cite{FK:matrices} as long as $\log^+\|\sigma_*^{-1}\|\in L^1(\Sigma_N,\mu)$. Any point for which the theorem of Furstenberg and Kesten result holds will be called \textbf{Furstenberg-Kesten-regular}.

\begin{comment}
Although the base dynamics $\sigma$ are invertible, the cocycle $G_+$ is only governed by a semi-group. There are some special instances where this can be extended to a group action, but in general it does not. Fortunately, in this case one can summon the semi-invertible Oseledets theorem \cite{FLQ:coherent}.
\begin{theorem}[Semi-invertible Oseledets Theorem]
  \label{thm:oseledets}
  Let $\mu$ be a minimal $\sigma$-invariant ergodic probability measure. Suppose that $\log^+\|\sigma_*^{-1}\|\in L^1(\Sigma_N,\mu)$. Then there exists numbers $\lambda_1^+>\dots>\lambda_{n_\mu^+}^+> 0\geq \lambda_1^->\cdots>\lambda_{n_\mu^-}^-$, multiplicities $m_1^\pm,\dots, m_{n_\mu}^\pm,m_\mu^\infty$ with $\mathrm{dim}\tr^*(\mathcal{M}_0^x) = m_1^++\cdots +m_{n_\mu}^++ m_1^-+\cdots +m_{n_\mu}^- + m_\mu^\infty$, and a measurable, $G_+$-invariant family of subspaces $V_j^\pm(x), V^\infty(x)\subset \tr^*(\mathcal{M}_0^x)$ such that for $\mu$-a.e. $x$:
  \begin{enumerate}
  \item \hspace{.3in} $\dim V_i^\pm(x) = m_i^\pm$ \hspace{.3in} and \hspace{.3in}  $\dim V^\infty(x) = m_\mu^\infty$
  \item $$\tr^*(\mathcal{M}_0^x) =V^\infty(x)\bigoplus_{i=1}^{n_\mu^+} V_i^+(x) \bigoplus_{i=1}^{n_\mu^-} V_i^-(x),$$
  \item \hspace{.3in} $\sigma_*V_i^\pm(x) = V_i^\pm(\sigma(x))$ \hspace{.3in} and \hspace{.3in} $\sigma_*V^\infty(x)\subset V^\infty(\sigma^{-1}(x))$,
  \item For all $\xi\in V_i^\pm(x)\backslash \{0\}$ and $\xi_\infty\in V^\infty(x)\backslash \{0\}$, one has
    $$\lim_{n\rightarrow \infty} \frac{1}{n}\log\|\sigma_*^{-(n)}\xi\| = \lambda_i^\pm\hspace{.7in}\mbox{ and } \hspace{.7in} \lim_{n\rightarrow \infty} \frac{1}{n}\log\|\sigma_*^{-(n)}\xi\| = -\infty.$$
\end{enumerate}
\end{theorem}
The numbers $\lambda_i$ are called the \textbf{Lyapunov exponents} for the measure $\mu$. Since $\tr^*(\mathcal{B}^+_x)$ is identified with the subspace of $\tr^*(\mathcal{M}_0^{x})$ composed of $v\in \tr^*(\mathcal{M}_0^{x})$ such that $F_k^x\cdots F_1^xv\neq 0$ for all $k$, part (ii) of the semi-invertible Oseledets theorem gives a $G_+$-invariant fiberwise decomposition
\begin{equation}
  \label{eqn:fibre-decomp}
  \tr^*(\mathcal{B}^+_x) = E^+_x\oplus E^-_x\hspace{.7in}\mbox{ and }\hspace{.7in} \tr^*(\mathcal{M}_0^x) = V^\infty(x)\oplus\tr^*(\mathcal{B}^+_x)
  \end{equation}
into unstable and stable bundles, where the unstable bundle corresponds to Oseledets subspaces in (ii) corresponding to positive Lyapunov exponents, while the stable to the non-positive Lyapunov exponents. Denote by $\lambda_1\geq \lambda_2\geq \cdots \geq \lambda_{d^+_\mu}>0$ all the positive Lyapunov exponents of $\mu$ for the trace cocycle and decompose $E^+_x$ further into one-dimensional, $G_+$-invariant subspaces
\begin{equation}
  \label{eqn:unstableDecomp}
  E^+_x = \bigoplus_{i=1}^{d^+_\mu}E^{+}_{i}(x).
\end{equation}
Note that the Oseledets decomposition of the cotrace bundle gives an Oseledets decomposition of $\tr^*(\mathcal{M}_k^x)$ for any $k\geq 0$ by the inclusion properties of the direct limit $\tr^*(\mathcal{B}^+_x)$.

Likewise, there is a dual $\sigma^{-1*}$-invariant decomposition
$$\tr(\mathcal{B}^+_x) = T^+_x\oplus T^-_x\hspace{.7in}\mbox{ and }\hspace{.7in} \tr(\mathcal{M}_0^\mathcal{B}) = V^\infty(\mathcal{B})^*\oplus\tr(\mathcal{B}^+_x),$$
where $T^\pm_x = (E^\pm_x)^* $ are the dual spaces to the stable and unstable bundles. The dual unstable bundle further decomposes into one-dimensional subspaces
\begin{equation}
  \label{eqn:dualDecomp}
T^+_x = \bigoplus_{i=1}^{d^+_\mu}T_{i}^+(x),
  \end{equation}
where $T_i^+(x) = (E_i^+(x))^*$. Again, by the projection properties of the inverse limit $\tr(\mathcal{B}^+_x)$ the dual decomposition given by Oseledets theorem gives an Oseledets decomposition of $\tr(\mathcal{M}_k^x)$ for all $k\geq 0$.

Denote by $\{\mathfrak{t}^x_{0,1},\dots, \mathfrak{t}^x_{0,M}\}$ the canonical basis of $\tr(\mathcal{M}_0^x)$ and by $\{\delta_1,\dots, \delta_{M}\}$ a dual basis for $\tr^*(\mathcal{M}_0^x)$. For any Oseledets-regular $x$, by (\ref{eqn:fibre-decomp}), we can identify $\tr^*(\mathcal{B}^+_x)$ with a subspace of $\tr^*(\mathcal{M}_0^x)$, and so any cotrace in $\tr^*(\mathcal{B}^+_x)$ can be written as
\begin{equation}
\label{eqn:cotraceExp}
\delta = \sum_{i=1}^{M} \beta_{i}(\delta)\delta_{i}.
\end{equation}
Define the map
\begin{equation}
  \label{eqn:jMap}
  j_x^+:\tr^*(\mathcal{B}^+_x)\longrightarrow LF(\mathcal{B}^+_x)\subset AF(\mathcal{B}^+_x)
\end{equation}
as follows. For $\delta\in \tr^*(\mathcal{B}^+)$, the image $a_{\delta} = j^+_{x}(x)$ is defined through
\begin{equation}
  \label{eqn:jMap2}
  j^+_{x}(\delta) = (\beta_1(\delta),\dots,\beta_{M}(\tau^*))\in \bigoplus_{i=1}^{M}\mathbb{C}  = \mathcal{M}_0^x,
\end{equation}
which is well-defined by the expression (\ref{eqn:cotraceExp}). We denote by $[a_{\delta}] = [j^+_{x}(\delta)]$ its class in $LF(\mathcal{B}^+_x)$. Note that by $G_+$-invariance, the map $j^+_x$ is equivariant with the trace cocycle. That is, for any $v\in \tr^*(\mathcal{M}^x_0)$ we have by (\ref{eqn:tracemap})
\begin{equation*}
  \begin{split}
    j^+_{\sigma^{-1}(x)}\left(\sigma_*^{-1}v\right) &= \left(\mathfrak{t}^{\sigma^{-1}(x)}_{1,1}( i_1[j^+_{x}(v)]),\dots,\mathfrak{t}^{\sigma^{-1}(x)}1_{1,M}( i_1[j^+_{x}(v)])\right) \\
    &= \left(i^*_1\mathfrak{t}^{\sigma^{-1}(x)}_{1,1}( [j^+_{x}(v)]),\dots, i^*_1\mathfrak{t}^{\sigma^{-1}(x)}_{1,M}( [j^+_{x}(v)])\right)

  \end{split}
  \end{equation*}
which generalizes to
\begin{equation}
  \label{eqn:equivariance}
  \begin{split}
    j^+_{\sigma^{-k}(x)}\left(\sigma_*^{-(k)}v\right)&=\left(\mathfrak{t}^{\sigma^{-k}(x)}_{1,1}(i_k\cdots i_1[j^+_{x}(v)]),\dots,\mathfrak{t}^{\sigma^{-k}(x)}_{1,M}(i_k\cdots i_1[j^+_{\sigma^{-k}}(v)]) \right) \\
    &=\left(i_1^*\cdots i_k^*\mathfrak{t}^{\sigma^{-k}(x)}_{1,1}([j^+_{x}(v)]),\dots,i_1^*\cdots i_k^*\mathfrak{t}^{\sigma^{-k}(x)}_{1,M}([j^+_{\sigma^{-k}}(v)]) \right) \\
    &=\left((\mathcal{A}^x_k\cdots \mathcal{A}^x_1)^*\mathfrak{t}^{\sigma^{-k}(x)}_{1,1}([j^+_{x}(v)]),\cdots,(\mathcal{A}^x_k\cdots \mathcal{A}^x_1)^*\mathfrak{t}^{\sigma^{-k}(x)}_{1,M}([j^+_{x}(v)]) \right)\in\mathcal{M}_0^{\sigma^{-k}(x)} .
    \end{split}
\end{equation}
\end{comment}
\section{LF algebras and return vectors}
\label{sec:LFreturn}
Let $S_1,\dots, S_N$ be $N$ uniformly expanding substitutions on the prototiles $t_1,\dots, t_M\subset \mathbb{R}^d$, $x\in\Sigma_N$ and suppose that $\mathcal{B}_x$ is minimal. Recall the sequence of height vectors $h^k_x$ for the positive part $\mathcal{B}_x^+$ from \S \ref{subsec:algebras}. Let $\{\mathcal{A}^x_k\}$ be the sequence of matrices defined by $\mathcal{B}_x^+$. Starting with $h^0_x=(1,\dots, 1)^T$, the height vectors are defined by $h^k_x = \mathcal{A}^x_k\cdots \mathcal{A}^x_1 h^0_x$. Note that $h_{x,i}^k$ also is the number of paths from $\mathcal{V}_0$ to $v_i\in\mathcal{V}_k$.

Recall also that $\mathcal{B}^+_x$ defines a sequence of inclusions (up to unitary equivalence), which yields the LF-algebra $LF(\mathcal{B}^+_x)$ via the direct limit of the inclusions $i_k : \mathcal{M}^x_{k-1}\hookrightarrow \mathcal{M}^x_k$, where
$$\mathcal{M}_k^x = M_{h_{x,1}^k}\oplus\cdots\oplus M_{h_{x,M}^k}.$$
From \S \ref{sec:RandSub}, the number $h_{x,i}^k$ also denotes the number of tiles inside the patch defined by the canonical level-$k$ supertile $\mathcal{P}_k(v_i)$. Thus, chosing a sequence $\{i_k\}$ of inclusions gives a bijection between the set of paths from $\mathcal{V}_0$ to $v_i\in\mathcal{V}_k$ and the set $\{1,\dots, h^k_{x,i}\}$. That is, for any $\ell\in\{1,\dots, h_{x,i}^k\}$ there is a unique path $p_\ell\in \mathcal{E}_{v_i}$ Under this correspondence, define the sets of indices
$$I^k_{x,i,j} = \left\{\ell\in \{1,\dots, h^k_{x,j}\}:\substack{\mbox{ the path $p_\ell\in\mathcal{E}_{0,v_j}$ corresponding to this index satisfies} \\\mbox{ $s(p) = v_i\in\mathcal{V}_0,\; r(p) = v_j\in\mathcal{V}_k$}} \right\},$$
and note that $\sum_i|I^k_{x,i,j}| = h^k_{x,j}$. Define for each $i\in\{1,\dots, M\},k\in\mathbb{N}$ and $\lambda\in\mathbb{R}^d$ the element
$$a^k_{x,i,\lambda} = \left(a^k_{x,i,1,\lambda},\dots,a^k_{x,i,M,\lambda}\right) \in \mathcal{M}^x_k$$
by defining $a^k_{x,i,j,\lambda}\in M_{h^k_{x,j}}$ to be the diagonal matrix which is only non-zero for the diagonal elements with index in $I^k_{x,i,j}$ and, for an indicex $\ell\in I^k_{x,i,j}$, the entry is
\begin{equation}
  \label{eqn:diagonal}
  \left(a^{k}_{x,i,j,\lambda}\right)_{\ell\ell} = e^{-2\pi i \langle\lambda, \tau_\ell\rangle},
\end{equation}
where $\tau_\ell\in\mathbb{R}^d$ is the vector corresponding to the control point inside the tile of type $t_i$ associated to the index $\ell\in I^k_{x,i,j}$ in the canonical supertile $\mathcal{P}_k(v_j)$. Thus the element $a^k_{x,i,\lambda}\in LF(\mathcal{B}_x^+)$ tracks the return vectors to the subtransveral $\mho_x^i$ inside of level-$k$ supertile $\mathcal{P}_k(v_j)$. That is, it records how tiles of type $i$ sit inside canonical level-$k$ $\mathcal{P}_k(v_j)$ supertiles.

There is a relationship between canonical level-$k$ super tiles and the level-$(k-1)$ supertiles that tile it. But first, consider the relationship between level-$1$ supertiles with the tiles that tile it. The first set of edges $\mathcal{E}_1$ of the diagram $\mathcal{B}_x$ is determined by the element $x_1$. Taking any vertex $v\in\mathcal{V}_1$, its associated level-1 supertile $\mathcal{P}_1(v)$ is tiled by $|r^{-1}(v)|$ tiles, each of which is marked by its own control point and one of which contains the control point of the level-$1$ supertile $\mathcal{P}_1(v)$. As such, there are vectors $\bar{0} = \tau_1,\dots, \tau_{|r^{-1}(v)|}\in\mathbb{R}^d$, one for each edge in $r^{-1}(v)$, which describe the translations between the control points on these $|r^{-1}(v)|$ tiles. Since there are finitely many choices for $x_1$, there is a set of $N$ sets of vectors $\{\mathcal{W}^\ell\}_{\ell=1}^N$, one for each substitution rule $S_\ell$, with
\begin{equation}
  \label{eqn:canVectors}
  \mathcal{W}^\ell = \bigcup_{j=1}^M\mathcal{W}^\ell_{j} = \bigcup_{j=1}^M\bigcup_{i=1}^M \mathcal{W}^\ell_{i,j} \hspace{.6in}\mbox{ and }\hspace{.6in} |\mathcal{W}^\ell_{i,j}| = |r^{-1}(v_i)\cap s^{-1}(v_j)|.
\end{equation}
It should be emphasized again that the zero vector is an element of $\bigcup_i\mathcal{W}^\ell_{i,j}$ for any $j,\ell$. This corresponds to the trivial translation from the tile which contains the control point in $\mathcal{P}_1(v_j)$ to itself.
\begin{figure}[t]
  \centering
  \includegraphics[width = 5.5in]{canonical}
  \caption{\small{This is a canonical level-$k$ supertile tiled by level-$(k-1)$ supertiles. The solid dot is the control point of the supertiles that contain it. The other circles represent the control points of some level-$(k-1)$ supertiles of the same type. The vectors $u_e, u_{e'}, u_{e''}$ represent the translations for the level-$(k-1)$ supertiles which are not in canonical position to be in canonical position. This system of references allows to express translations between tiles of the same type, such as $u_e-u_{e'}$ as depicted here.}}
  \label{fig:canonical}
\end{figure}
More generally, for $k>0$ and $i,j\in \{1,\dots, m\}$ there is a collection of vectors $\mathcal{U}^x_{i,j,k}$ with $|\mathcal{U}^x_{i,j,k}| = |r^{-1}(v_i)\cap s^{-1}(v_j)| = (\mathcal{A}_k^x)_{i,j}$ whose elements are described as follows. Consider the canonical level-$k$ supertile $\mathcal{P}_k(v_i)$. It is tiled by $|r^{-1}(v_i)|$ level-$(k-1)$ supertiles $\mathcal{P}_{k-1}(\bar{p}_e)$, $e\in r^{-1}(v_i)$, all but one of which are not in canonical position. Therefore for each edge $e\in r^{-1}(v_i)$ there is a vector $u_e\in\mathbb{R}^d$ such that $\varphi_{u_e}(\mathcal{P}_{k-1}(\bar{p}_e)) = \mathcal{P}_{k-1}(v_j)$ for $e\in r^{-1}(v_i)\cap s^{-1}(v_j)$, that is, so that translating the level-$(k-1)$ supertile $\mathcal{P}_{k-1}(\bar{p}_e)$ is translated into canonical position. Thus $\mathcal{U}^x_{i,j,k}$ is the collection of all such vectors $u_e$ for $e\in r^{-1}(v_i)\cap s^{-1}(v_j)$, and difference vectors of the form $u_e-u_{e'}$ describe translations between level-$(k-1)$ supertiles inside level-$k$ supertiles (see Figure \ref{fig:canonical} for an auxiliary illustration). Thus, by (\ref{eqn:canVectors}), for $k>1$,
\begin{equation}
  \label{eqn:canVectors2}
  \mathcal{U}^x_{i,j,k} = \bar\theta_{x_{k-1}}\cdots\bar\theta_{x_1}\mathcal{W}^{x_k}_{i,j}.
  \end{equation}

 Now consider the element $(i_k(a^{k-1}_{x,i,\lambda}))_j\in M_{h^k_{x,j}}$. It is a diagonal matrix with entries expressing how tiles of type $i$ are located in all the level-$(k-1)$ supertiles contained inside $\mathcal{P}_k$, but it does not describe how the tiles of type $i$ are located inside the level-$k$ supertile $\mathcal{P}_k(v_j)$. Thus there is a diagonal matrix $M_{k,j,\lambda}\in M_{h^k_{x,j}}$ with entries of the form $e^{-2\pi i \langle\lambda , u_e\rangle}$, where $u_e\in \mathcal{U}^x_{i,j,k}$, such that
 $$a^k_{x,i,j,\lambda}= \left(i_k(a^{k-1}_{x,i,\lambda})\right)_j  M_{k,j,\lambda}$$
 and, more generally, there are diagonal elements $M_{k,\lambda} = (M_{k,1,\lambda},\dots, M_{k,M,\lambda})\in\mathcal{M}^x_k$ such that
 $$ a^k_{x,i,\lambda}= i_k\left(a^{k-1}_{x,i,\lambda}\right)  M_{k,\lambda},$$
 and so recursively applying the same argument there are $k$ diagonal matrices $M_{1,\lambda},\dots, M_{k,\lambda}$ with $M_{\ell,\lambda}$ having entries of the form $e^{2\pi i \langle\lambda , u_e\rangle}$, where $u_e\in \mathcal{U}^x_{i,j,\ell}$, such that
 \begin{equation}
   \label{eqn:aDecomp}
   a^k_{x,i,\lambda} = i_k\left(i_{k-1}\left(i_{k-2}\left( \cdots i_1(a^0_{x,i,\lambda})M_{1,\lambda}\cdots    \right) M_{k-2,\lambda}\right) M_{k-1,\lambda}\right) M_{k,\lambda}.
 \end{equation}

 Following (\ref{eqn:canTraces}), denote by $\mathfrak{t}_{k,1}^x,\dots, \mathfrak{t}_{k,M}^x$ the canonical traces for $\mathcal{M}^x_k$. The goal now is to find good bounds for $|\mathfrak{t}_{k,j}^x(a^k_{x,i,\lambda})|$ for any $i,j$. First, note that since all the entries of the diagonal matrices $M_{\ell,\lambda}$ have unit norm, by (\ref{eqn:canTraces}) and (\ref{eqn:aDecomp}),
 \begin{equation}
   \label{eqn:traceUpBnd}
\sum_{i=1}^M|\mathfrak{t}_{k,j}^x(a^k_{x,i,\lambda})| \leq h^k_j.
\end{equation}

 Now consider
 $$\sum_{i=1}^M a^1_{x,i,\lambda} = \sum_{i=1}^M i_1(a^0_{x,i,\lambda})M^{-1}_{1,\lambda} =  i_1(\mathrm{Id})M^{-1}_{1,\lambda}.$$
 Applying the $j^{th}$ canonical trace at level 1:
 $$\sum_{i=1}^M \mathfrak{t}_{1,j}^x(a^1_{x,i,\lambda}) =  \mathfrak{t}_{1,j}^x(i_1(\mathrm{Id})M^{-1}_{1,\lambda}) = \sum_{e\in r^{-1}(v_j)}e^{2\pi i \langle \lambda, u_e\rangle} \mathfrak{t}_{0,s(e)}^x(\mathrm{Id})$$
 is the sum of $h^1_j$ exponential terms including, if $|r^{-1}(v_j)|>1$, a term for the form $1+e^{2\pi i \langle \lambda, w_e\rangle}$ for some $w_e\in \mathcal{W}^{x_1}_{j}-\mathcal{W}^{x_1}_{j}$. It follows that
 $$|\mathfrak{t}_{1,j}^x(i_1(\mathrm{Id})M^{-1}_{1,\lambda})|\leq h^1_j - 2 + |1+e^{2\pi i \langle \lambda, w_e\rangle}|$$
and, from the inequality $|1+e^{2\pi i \omega}| \leq 2 - (1/2)\|\omega\|^2_{\mathbb{R}/\mathbb{Z}}$ for any $\omega\in\mathbb{R}$, that
\begin{equation}
  \label{eqn:traceBnd1}
 |\mathfrak{t}_{1,j}^x(i_1(\mathrm{Id})M^{-1}_{1,\lambda})|\leq h^1_j - \frac{1}{2} \| \langle \lambda, w_e\rangle\|_{\mathbb{R}/\mathbb{Z}}^2
\end{equation}
for any $w_e\in\mathcal{W}^{x_1}_j$. The goal now is to generalize this type of estimate to traces applied to any of the elements $a^k_{x,i,\lambda}$.

First, note that $\mathfrak{t}_{k,j}^x(a^k_{x,i,\lambda})$ can be written as
\begin{equation}
  \label{eqn:traceExp}
  \begin{split}
    \mathfrak{t}_{k,j}^x(a^k_{x,i,\lambda}) &=  \mathfrak{t}_{k,j}^x(i_k(a^{k-1}_{x,i,\lambda} )M_{k,\lambda})=\sum_{e\in r^{-1}(v_j)}e^{2\pi i \langle \lambda, u_e\rangle} \mathfrak{t}_{k-1,s(e)}^x(a^{k-1}_{x,i,\lambda})\\
    &= \sum_{e\in r^{-1}(v_j)}e^{2\pi i \langle \lambda, u_e\rangle} \sum_{\substack{e'\in\mathcal{E}_{k-1}:\\r(e') = s(e)}}e^{2\pi i \langle \lambda, u_{e'}\rangle} \mathfrak{t}_{k-2,s(e')}^x(a^{k-2}_{x,i,\lambda}) =\cdots \\
&=\sum_{e_k\in r^{-1}(v_j)} e^{2\pi i \langle \lambda, u_{e_k}\rangle}  \sum_{e_{k-1}\in r^{-1}(s(e_k))}e^{2\pi i \langle \lambda, u_{e_{k-1}}\rangle}\cdots  \sum_{e_{1}\in r^{-1}(s(e_2))}e^{2\pi i \langle \lambda, u_{e_1}\rangle} \mathfrak{t}_{0,s(e_1)}^x(a^{0}_{x,i,\lambda}),
\end{split}
\end{equation}
where $u_{e_\ell}\in \mathcal{U}^x_{i,j,\ell}$ for all $\ell\in\{1,\dots,k\}$ in the last expression.
\begin{remark}
  \label{rem:spectralCocycle}
The expression in (\ref{eqn:traceExp}) can be viewed from a different perspective. Let $\mathbb{M}_x^k$ be the matrix-valued function which assigns, for each $\lambda\in\mathbb{R}^d$, the matrix $\mathbb{M}_x^k(\lambda)$ with $i,j$ entry
$$\mathbb{M}_{x}^k(\lambda)_{i,j} := \sum_{e\in \mathcal{W}_{i,j}^{x_k}}e^{-2\pi i \langle \lambda, \tau_e\rangle}.$$
As such, define family of matrix products parametrized by $\lambda\in\mathbb{R}^d$
\begin{equation}
  \label{eqn:SpecCocycle0}
  \mathbb{M}_x^{(k)}(\lambda):=\mathbb{M}_{x}^{k}\left(\theta_{(k)_x}^{-1}\lambda\right) \cdot \mathbb{M}_{x}^{k-1}\left(\theta_{(k-1)_x}^{-1}\lambda\right) \cdots  \mathbb{M}_{x}^{1}\left(\theta_{(1)_x}^{-1}\lambda\right)
\end{equation}
for each $k\in\mathbb{N}$ and $\lambda\in\mathbb{R}^d$. As such, it follows that
\begin{equation}
  \label{eqn:SpecCocycle}
  \mathfrak{t}_{k,i}^x(a^k_{x,j,\lambda}) = \mathbb{M}_x^{(k)}(\lambda)_{i,j}
\end{equation}

and that $\mathbb{M}^{(k)}_x(0) = \Theta_x^{(k)}$ for any $k\in\mathbb{N}$.
The matrices $\mathbb{M}^k_{x}(\lambda)$ are called the \textbf{Fourier matrices} in \cite{BGM:correlation}, where the growth of the cocycle $\mathbb{M}_x^{(k)}$ in (\ref{eqn:SpecCocycle0}) was used to rule out existence of absolutely continuous parts of diffraction measures. In the work of Bufetov-Solomyak \cite{BS:spectralCocycle}, this is called the \textbf{spectral cocycle}, which has been used to extablish quantitative weak-mixing results. In my take here, (\ref{eqn:SpecCocycle}) aims to extend these types of results.
\end{remark}
\begin{definition}
  Let $S_1,\dots, S_N$ be a family of uniformly expanding compatible substitution rules on the prototiles $t_1,\dots, t_M$. A word $w  = w_1w_2\dots w_n$ is \textbf{simple} if $w_i\cdots w_n\neq w_1\cdots w_{n-i+1}$ for all $1<i\leq n$. A word $w$ is \textbf{positively simple} if it is a simple word and in addition:
  \begin{enumerate}
  \item it breaks up into two subwords $w= w^-w^+ = w_1^-\cdots w_{n^-}^-w_1^+\cdots w_{n^+}^+$, and
  \item each entry in both matrices $Q^\pm = A_{w_{n^\pm}^\pm}\cdots A_{w_1^\pm}$ is strictly greater than 1.
  \end{enumerate}
  A $\sigma$-invariant probability measure $\mu$ for which $\mu([w])>0$ for a positively simple word $w$ is called a \textbf{positively simple measure}.
\end{definition}
Note that a positively simple measure is a minimal measure.
\begin{proposition}
  \label{prop:TracesBound}
  Let $S_1,\dots, S_N$ be a family of uniformly expanding compatible substitution rules on the prototiles $t_1,\dots, t_M$ and a positively simple word $w = w^-w^+$. Suppose  $x\in\Sigma_N$ has infinitely many occurences of the word $w$, both in the future and the past. Denote by $k_n\rightarrow \infty$ the maximal sequence of return times of $x$ to $C([w^-.w^+])$ under $\sigma^{-1}$. Then there exists a $\ell\in\{1,\dots, M\}$, a finite set of vectors $\bar{\Lambda}_x\subset \Lambda_\ell^x$, $c>0$, and $k_x>0$ such that for any $m\in\mathbb{N}$
  \begin{equation}
    \label{eqn:twistBnd1}
    \begin{split}
      &\left|\mathfrak{t}^x_{m,j}(a^{m}_{x,i,\lambda})\right|  = |\mathbb{M}_x^{(m)}(\lambda)_{i,j}|\\
      &\hspace{.4in}\leq \left\{
      \begin{array}{ll}
        \displaystyle\left\| \Theta_x^{(m)}\right\|\prod_{n<N}\left( 1 - c\max_{\tau\in \bar\Lambda_x} \|\langle\lambda,\bar\theta_{(k_n+k_x)_x}\tau\rangle \|^2_{\mathbb{R}/\mathbb{Z}}\right) & \mbox{ if }\,k_{N}\leq m < k_N+k_x  \\
        \displaystyle\left\| \Theta_x^{(m)}\right\|\prod_{n \leq N}\left( 1 - c\max_{\tau\in \bar\Lambda_x} \|\langle\lambda,\bar\theta_{(k_n+k_x)_x}\tau\rangle \|^2_{\mathbb{R}/\mathbb{Z}}\right) & \mbox{ if }\, k_N+k_x\leq m \leq k_{N+1}
      \end{array} \right.
    \end{split}
  \end{equation}
  for any $i,j$.
\end{proposition}
\begin{proof}
  Since the positively simple word $w = w^-w^+$ occurs infinitely often, there exists (possibly empty) words $\{y_n\}_n$, none of which contains $w$ as a subword, such that
  $$x^+ = y_0w^-w^+y_1 w^-w^+y_2w^-w^+y_3w^-w^+y_4\cdots .$$
  Let $k_n\rightarrow \infty$ be defined as $k_1 = |y_0| + |w^-|$ and recursively $k_{n+1} = k_n+|w|+|y_n|$. This is a maximal sequence of return times to $C([w^-.w^+])$. Let $k_x = |w^+|$.

Consider the tiling space $\Omega_{\sigma^{-k_1}(x)}$ and $i\in \{1,\dots, M\}$. Consider the canonical level-$k_x$ supertile $\mathcal{P}_{k_x}(v_i)$ for tilings in this tiling space. Since $Q^+_{i,j}>1$, there are at least two tiles of type $t_{j}$ inside this supertile. As such, there is at least one vector $\tau\in\Lambda_{j}^{\sigma^{-k_1}(x)}$ which describes the translation between two tiles of type $t_j$. Let $\bar\Lambda_x$ be the union of all vectors $\Lambda_{j}^{\sigma^{-k_1}(x)}$ describing a translation equivalence between tiles of type $t_{j}$ in the canonical supertile $\mathcal{P}_{k_n}(v_j)$ for $\Omega_{\sigma^{-k_1}(x)}$, for all $j$; this is a finite set of vectors.

First suppose that $k_{n-1}+k_x\leq m\leq k_{n}$ for some $n>1$. Then as in (\ref{eqn:traceExp}),
\begin{equation}
  \label{eqn:traceExp2}
  \begin{split}
    \mathfrak{t}_{k_n,j}^x(a^{k_n}_{x,i,\lambda}) &= \sum_{\substack{p\in\mathcal{E}_{k_{n-1}+k_x,m}:  \\ r(p) = v_j}}e^{2\pi i \langle\lambda, u_p\rangle} \mathfrak{t}_{k_{n-1}+k_x,s(p)}^x(a^{k_{n-1}+k_x}_{x,i,\lambda}) \\
    &=\sum_{\substack{p\in\mathcal{E}_{k_{n-1}+k_x,m}: \\  r(p) = v_j}}e^{2\pi i \langle\lambda, u_p\rangle} \sum_{\substack{p'\in\mathcal{E}_{k_{n-1},k_{n-1}+k_x}: \\ r(p')= s(p)}} e^{2\pi i \langle\lambda, u_{p'}\rangle }    \mathfrak{t}_{k_{n-1},s(p')}^x(a^{k_{n-1}}_{x,i,\lambda}),
\end{split}
\end{equation}
where every vector $u_p$ above describes the position of a level-$k_{n-1}+k_x$ supertile inside a level-$k_n$ supertile relative to their control points, and the vectors $u_{p'}$ similarly describe the position of level $k_{n-1}$ supertiles inside level-$k_{n-1}+k_x$ supertiles. As such, in every term above of the form
$$\sum_{\substack{p'\in\mathcal{E}_{k_{n-1},k_{n-1}+k_x}: \\  r(p')= s(p)}} e^{2\pi i \langle\lambda, u_{p'}\rangle }\mathfrak{t}_{k_{n-1},s(p')}^x(a^{k_{n-1}}_{x,i,\lambda}),$$
which depends only on $Q^+$, the vector $u_{p'}$ is of the form $\bar\theta_{(k_{n-1}+k_x)_x}u$ for some $u\in \bar\Lambda_x$. Using the fact that $Q^+_{i,j}>1$ for all $i,j$, this term can be bounded. Fixing $v_\ell\in \mathcal{V}_{k_{n-1}+k_x}$:
\begin{equation}
  \label{eqn:traceExp3}
  \begin{split}
&\left| \sum_{\substack{p'\in\mathcal{E}_{k_{n-1},k_{n-1}+k_x} \\ r(p')=v_\ell}} e^{2\pi i \langle\lambda, u_{p'}\rangle }
\mathfrak{t}_{k_{n-1},s(p')}^x(a^{k_{n-1}}_{x,i,\lambda}) \right| = \left|\sum_{j=1}^M \sum_{\substack{p'\in\mathcal{E}_{k_{n-1},k_{n-1}+k_x}: \\ s(p')=v_j \in\mathcal{V}_{k_{n-1}}\\ r(p')=v_\ell}}  e^{2\pi i \langle\lambda, u_{p'}\rangle } \mathfrak{t}_{k_{n-1},j}^x(a^{k_{n-1}}_{x,i,\lambda})\right| \\
&\leq  \sum_{j=1}^M \left|\sum_{\substack{p'\in\mathcal{E}_{k_{n-1},k_{n-1}+k_x}: \\ s(p')=v_j \in\mathcal{V}_{k_{n-1}}\\ r(p')=v_\ell}}  e^{2\pi i \langle\lambda, u_{p'}\rangle } \mathfrak{t}_{k_{n-1},j}^x(a^{k_{n-1}}_{x,i,\lambda})\right|\leq  \sum_{j=1}^M \left( Q_{\ell,j}^+ -2 + |1+e^{2\pi i \langle\lambda,u_{p'}\rangle}| \right)  |\mathfrak{t}_{k_{n-1},j}^x(a^{k_{n-1}}_{x,i,\lambda})| \\
&\hspace{.25in}\leq  \sum_{j=1}^M \left(  Q^+_{\ell,j} - \frac{1}{2}\| \langle\lambda, u_{p'}\rangle\|^2_{\mathbb{R}/\mathbb{Z}}\right)  \left| \mathfrak{t}_{k_{n-1},j}^x(a^{k_{n-1}}_{x,i,\lambda})\right|\\
&\hspace{.5in}\leq  \sum_{j=1}^M \left( 1 - c_w\| \langle\lambda, u_{p'}\rangle\|^2_{\mathbb{R}/\mathbb{Z}}\right)     Q^+_{\ell,j}   \left| \mathfrak{t}_{k_{n-1},j}^x(a^{k_{n-1}}_{x,i,\lambda})\right| \\
&\hspace{.75in}\leq  \left( 1 - c_w\max_{\tau\in \bar\Lambda_x}\| \langle\lambda, \theta_{(k_{n-1}+k_x)_x}^{-1}\tau\rangle\|^2_{\mathbb{R}/\mathbb{Z}}\right) \sum_{j=1}^M \left| \mathfrak{t}_{k_{n-1},j}^x(a^{k_{n-1}}_{x,i,\lambda})\right|   Q^+_{\ell,j} ,
  \end{split}
\end{equation}
where $c_w = (2M\max_{i,j}Q^+_{i,j})^{-1}$, where the third inequality follows from a bound of the type done in (\ref{eqn:traceBnd1}), and where the last one follows from the fact that this estimate works for any good return vector of the form $\bar\theta_{(k_{n-1}+k_x)_x}u$ for some $u\in \bar\Lambda_x$. The bound (\ref{eqn:traceExp3}) can now be used in bounding (\ref{eqn:traceExp2}):
\begin{equation*}
  \label{eqn:traceExp4}
  \begin{split}
   & \left| \mathfrak{t}_{m,j}^x(a^{m}_{x,i,\lambda})\right| \\
   & \hspace{.25in}\leq   \sum_{\substack{p\in\mathcal{E}_{k_{n-1}+k_x,m}: \\  r(p) = v_j}}e^{2\pi i \langle\lambda, u_p\rangle}   \left( 1 - c_w\max_{\tau\in \bar\Lambda_x}\| \langle\lambda, \bar\theta_{(k_{n-1}+k_x)}\tau\rangle\|^2_{\mathbb{R}/\mathbb{Z}}\right)\sum_{\ell=1}^M Q^+_{s(p),\ell}  \left| \mathfrak{t}_{k_{n-1},m}^x(a^{k_{n-1}}_{x,i,\lambda})\right| \\
    &\hspace{.25in}\leq \left( 1 - c_w\max_{\tau\in \bar\Lambda_x}\| \langle\lambda, \bar\theta_{(k_{n-1}+k_x)}\tau\rangle\|^2_{\mathbb{R}/\mathbb{Z}}\right)\sum_{\ell=1}^M \left|\left\{\substack{p\in \mathcal{E}_{k_{n-1},m}:\\ s(p) = v_\ell,\, r(p) = v_j}\right\}\right| \left| \mathfrak{t}_{k_{n-1},\ell}^x(a^{k_{n-1}}_{x,i,\lambda})\right| \\
    &\hspace{.25in}= \left( 1 - c_w\max_{\tau\in \bar\Lambda_x}\| \langle\lambda, \bar\theta_{(k_{n-1}+k_x)}\tau\rangle\|^2_{\mathbb{R}/\mathbb{Z}}\right) \sum_{\ell=1}^M \left(\Theta_x^{(k_{n-1},m)}\right)_{j,\ell} \left| \mathfrak{t}_{k_{n-1},\ell}^x(a^{k_{n-1}}_{x,i,\lambda})\right| .
  \end{split}
  \end{equation*}
The same type of bound applies to $\left| \mathfrak{t}_{k_{n-1},\ell}^x(a^{k_{n-1}}_{x,i,\lambda})\right|$. Thus, recursively, after finitely many bounds, one arrives at (\ref{eqn:twistBnd1}) in the case that $k_{n-1}+k_x\leq m\leq k_{n}$. In the case that $k_n\leq m <k_n+|w^+|$ the same argument holds, the point being that the number of terms in the product is one less than the number of times the word $w^+$ appears in $x$ up to its $m^{th}$ spot.
\end{proof}
\section{Twisted integrals}
\label{sec:twisted}
      The purpose of this section is to understand the behavior of twisted ergodic integrals along the dense $\mathbb{R}^d$ orbits on a compact tiling space $\Omega$. More specifically, given $\mathcal{T}\in\Omega_x$ and $f\in L^2$, recall the twisted ergodic integrals of $f$ by $\lambda$ as the family
      \begin{equation}
        \label{eqn:twisted}
\mathcal{S}_R^{\mathcal{T}}(f,\lambda) = \int_{C_R(0)} e^{-2\pi i \langle \lambda, \tau\rangle} f\circ \varphi_\tau(\mathcal{T})\, d\tau,
      \end{equation}
      where $C_R(0) = [-R,R]^d$ is the $L^\infty$-ball of radius $R$ in $\mathbb{R}^d$ centered at the origin.  Note that if $f\in L^2$ is an eigenfunction with eigenvalue $\lambda$, then $\mathcal{S}_R^\mathcal{T}(f,\lambda) =  C_d f(\mathcal{T})R^d$ for all $R$ and $\mathcal{T}\in\Omega_x$. Moreover, the spectral measure for an eigenfunction consists of a Dirac measure on $\lambda$.   So in order to rule out eigenfunctions it suffices to show either that the lower local dimension of spectral measures for a dense set of functions in $L^2$ is bounded below by a positive quantity, or that the twisted integrals (\ref{eqn:twisted}) for the same dense family of functions has growth of order $R^{d-\alpha}$ for some positive $\alpha$ (see Lemma \ref{lem:dimension}).

      From the time-change equation (\ref{eqn:conj}) and the change of variables formula it is straight-forward to verify that
      \begin{equation}
        \label{eqn:TwistedChange}
        \mathcal{S}_{R}^\mathcal{T}(f,\lambda) =\theta_{(n)_x}^{-d}\mathcal{S}_{R_{(n)_x}}^{\mathcal{T}_{(n)_x}}\left(f_{(n)_x},\lambda_{(n)_x}\right),
      \end{equation}
      for any $n>0$, where
      \begin{equation}
        \label{TwistedChange2}
        \begin{split}
          f_{(n)_x} = \Phi_{\sigma^{-n}(x)}^{(n)*}f,\hspace{.85in} \lambda_{(n)_x} &= \theta^{-1}_{(n)_x}\lambda, \hspace{.85in} R_{(n)_x} = \theta_{(n)_x}R,\\ \mbox{ and } \hspace{.25in} \mathcal{T}_{(n)_x} &= \Phi^{-1}_{\sigma^{-n}(x)}\circ\cdots\circ \Phi^{-1}_{\sigma^{-1}(x)}(\mathcal{T}).
        \end{split}
      \end{equation}
      This will become useful since it allows to convert twisted integrals of level-$n$ TLC functions on $\Omega_x$ into twisted integrals of level-$0$ TLC functions on $\Omega_{\sigma^{-n}(x)}$.

      For $\mathcal{T}\in\Omega_x$, let $g:\mathbb{R}^d\rightarrow \mathbb{R}$ be a (level-0) $\mathcal{T}$-equivariant function which is supported in the union of the interiors of tiles of type $t_\ell$ for some $\ell$. This function can be more concretely described as follows. There is a compactly supported function $\psi^\mathcal{T}_\ell:\mathbb{R}^d\rightarrow \mathbb{R}$ and a Delone set $\Lambda(\mathcal{T},\ell)$ such that
      $$g = \sum_{x\in \Lambda(\mathcal{T},\ell)}\delta_x * \psi_\ell^\mathcal{T},$$
      where $\Lambda(\mathcal{T},\ell)$ describes the position of the tiles of type $\ell$ in $\mathcal{T}$ relative to the origin.
      Recall that there is a (level-0) TLC function $h$ such that $g(t) = h\circ \varphi_t(\mathcal{T})$. Let $t_j^{(k)}\subset \mathcal{T}$ be a level-$k$ supertile of type $j$ which is a patch of $\mathcal{T}$. Then
      \begin{equation}
        \label{eqn:localTwist}
        \begin{split}
          \int_{t_j^{(k)}} e^{-2\pi i \langle \lambda, \tau \rangle} h\circ \varphi_\tau (\mathcal{T})\, d\tau &= \int_{t_j^{(k)}} e^{-2\pi i \langle \lambda, \tau \rangle} g(\tau)\, d\tau \\
       &= e^{2\pi i \langle \lambda , \tau(t^{(k)}_j)\rangle} \mathfrak{t}^x_{k,j}(a^k_{x,\ell,\lambda})\widehat{\psi^\mathcal{T}_\ell}(\lambda),
        \end{split}
      \end{equation}
      where $\tau(t^{(k)}_j)\in\mathbb{R}^d$ is the vector which describes the position of the supertile $t^{(k)}_j$ with respect to the canonical level-$k$ supertile $\mathcal{P}_k(v_j)$, i.e. the translation equivalence. This immediately generalizes to any level-$0$, $\mathcal{T}$-equivariant (equivalently, TLC) functions as follows.

Let $h = \sum h_i:\Omega_x\rightarrow \mathbb{R}$ be a level-$0$ TLC function, where $h_1,\dots, h_M:\Omega_x\rightarrow\mathbb{R}$ are the associated functions for the $M$ different level-$0$ tiles and, for $\mathcal{T}\in\Omega$, $g_\mathcal{T} = \sum g_i$ the associated $\mathcal{T}$-equivariant functions on $\mathbb{R}^d$. Then there exist compactly supported functions $\psi_1^\mathcal{T},\dots, \psi_M^\mathcal{T}$ and Delone sets $\Lambda(\mathcal{T},1),\dots,  \Lambda(\mathcal{T},M) $ such that
$$g_\mathcal{T} = \sum_{i=1}^M\sum_{\tau\in \Lambda(\mathcal{T},i)} \delta_\tau * \psi_i^\mathcal{T},$$
where the support of $\psi^\mathcal{T}_i$ is translation equivalent to the interior of $t_i$. Then as in (\ref{eqn:localTwist}), for a patch $t_j^{(k)}\subset \mathcal{T}$ which is also a level-$k$ supertile:
      \begin{equation}
        \label{eqn:localTwist2}
        \begin{split}
          \int_{t_j^{(k)}} e^{-2\pi i \langle \lambda, \tau \rangle} h\circ \varphi_\tau (\mathcal{T})\, d\tau &= \int_{t_j^{(k)}} \sum_{\ell=1}^Me^{-2\pi i \langle \lambda, \tau \rangle} g_\ell(\tau)\, d\tau \\
       &= e^{2\pi i \langle \lambda , \tau(t^{(k)}_j)\rangle} \sum_{\ell=1}^M\mathfrak{t}^x_{k,j}(a^k_{x,\ell,\lambda})\widehat{\psi^\mathcal{T}_\ell}(\lambda),
        \end{split}
      \end{equation}
      where, again, $g_\mathcal{T}(t) = h\circ \varphi_t(\mathcal{T})$ was used.

      Recall the renormalization maps $\Phi_{\sigma^{-1}(x)}:\Omega_{\sigma^{-1}(x)}\rightarrow\Omega_{x} $ and observe that if $h:\Omega_x\rightarrow \mathbb{R}$ is a level-$1$ TLC function, then $h':=\Phi_{\sigma^{-1}(x)}^*h:\Omega_{\sigma^{-1}(x)}\rightarrow \mathbb{R}$ is a level-$0$ TLC function. In particular, if $h = \sum_{i=1}^M h_i$, where $h_i$ is supported on tilings $\mathcal{T}\in\Omega_x$ whose level-$1$ supertile around the origin is of type $i$, then $h' =\sum_{i=1}^M \Phi_{\sigma^{-1}(x)}^*h_i = \sum_{i=1}^M h'_i$, where $h_i'$ are supported on tilings $\mathcal{T}'\in\Omega_{\sigma^{-1}(x)}$ whose tiles containing the origin is of type $i$. In addition, if $g_\mathcal{T}(t) = h\circ\varphi_t(\mathcal{T})$, $g_{\mathcal{T}'}'(t) = h'\circ\varphi_t(\mathcal{T}')$ and $\Phi_{\sigma^{-1}(x)}(\mathcal{T}') = \mathcal{T}$, using (\ref{eqn:conj}), then
      $$g'_{\mathcal{T}'}(t) = h'\circ \varphi_t(\mathcal{T}') = h\circ\Phi_{\sigma^{-1}(x)}\circ\varphi_t(\mathcal{T}') = h\circ\varphi_{\bar\theta_{x_1}t}\circ\Phi_{\sigma^{-1}(x)}(\mathcal{T}') = h\circ\varphi_{\bar\theta_{x_1}t}(\mathcal{T}) = g_\mathcal{T}(\bar\theta_{x_1}t),$$
      so they can both be written as
$$g_{\mathcal{T}'}'(t)= \sum_{i=1}^M\sum_{\tau\in \Lambda_i^{\mathcal{T}'}}\delta_\tau * \psi^{\mathcal{T}'}_i\hspace{.4in}\mbox{ and }\hspace{.4in} g_{\mathcal{T}}(t)= \sum_{i=1}^M\sum_{\tau\in \Lambda_i^{\mathcal{T}'}}\delta_{\bar{\theta}_{x_1}\tau} * (\bar{\theta}_{x_1}^*\psi^{\mathcal{T}}_i).$$

\begin{comment}

      If $t_j^{(i-1)}\subset \mathcal{T}'\in\Omega_{\sigma^{-1}(x)}$ is a level-$i-1$ supertile, $i>0$, then $\theta_{x_1}^{-1}t_j^{(i-1)}\subset \Phi_{x}(\mathcal{T})\in \Omega_x$ is a level-$i$ supertile of type $j$, so using (\ref{eqn:conj}):
      \begin{equation}
        \label{eqn:localTwist3}
        \begin{split}
          \int_{t_j^{(i-1)}}e^{-2\pi i \langle \lambda, \tau \rangle} h'\circ\varphi_\tau(\mathcal{T}')\, d\tau &= \int_{t_j^{(i-1)}}e^{-2\pi i \langle \lambda, \tau \rangle} h\circ \Phi_{\sigma^{-1}(x)}  \circ\varphi_\tau(\mathcal{T}')\, d\tau \\
          &= \int_{t_j^{(i-1)}}e^{-2\pi i \langle \lambda, \tau \rangle} h\circ\varphi_{\theta_{x_1}^{-1}\tau} \circ \Phi_{\sigma^{-1}(x)}(\mathcal{T}')  \, d\tau \\
          &= \theta_{x_1}^d\int_{\bar\theta_{x_1}t_j^{(i-1)}}e^{-2\pi i \langle \lambda, \theta_{x_1}\tau \rangle} h\circ\varphi_{\tau} (\mathcal{T})  \, d\tau \\
          &= \theta_{x_1}^d\int_{t_j^{(i)}}e^{-2\pi i \langle \theta_{x_1} \lambda, \tau \rangle} h\circ\varphi_{\tau} (\mathcal{T})  \, d\tau \\
        \end{split}
      \end{equation}
      Putting this together with (\ref{eqn:localTwist2}):
\begin{equation}
  \label{eqn:localTwist4}
  \begin{split}
    \int_{t_j^{(i)}}e^{-2\pi i \langle  \lambda, \tau \rangle} h\circ\varphi_{\tau} (\mathcal{T})  \, d\tau  &= \theta^{-d}_{x_1} \int_{t_j^{(i-1)}}e^{-2\pi i \langle \theta^{-1}_{x_1} \lambda, \tau \rangle} h'\circ\varphi_\tau(\mathcal{T}')\, d\tau \\
    &= e^{2\pi i \langle \lambda , \tau(t^{(i-1)}_j)\rangle} \sum_{\ell=1}^M\theta^{-d}_{x_1}\mathfrak{t}^{\sigma^{-1}(x)}_{i-1,j}(a^{i-1}_{\sigma^{-1}(x),\ell,\bar\theta_{x_1}\lambda})\widehat{\psi^\mathcal{T}_\ell}(\bar\theta_{x_1}\lambda) \\
    &= e^{2\pi i \langle \lambda , \tau(t^{(i-1)}_j)\rangle} \sum_{\ell=1}^M\mathfrak{t}^{\sigma^{-1}(x)}_{i-1,j}(a^{i-1}_{\sigma^{-1}(x),\ell,\bar\theta_{x_1}\lambda})\widehat{\psi^{\mathcal{T}'}_\ell}(\lambda),
  \end{split}
  \end{equation}

Let $h = \sum h_i:\Omega_x\rightarrow \mathbb{R}$ be a level-$k$ TLC function, where $h_1,\dots, h_M:\Omega_x\rightarrow\mathbb{R}$ are the associated functions with the $M$ different level-$k$ supertiles and, for $\mathcal{T}\in\Omega_x$, $g_\mathcal{T} = \sum g_i$ the associated $\mathcal{T}$-equivariant functions on $\mathbb{R}^d$. Then there exist compactly supported functions $\psi_1^\mathcal{T},\dots, \psi_M^\mathcal{T}$ and Delone sets $\Lambda^k_1(\mathcal{T}),\dots,  \Lambda^k_M(\mathcal{T}) $ such that
$$g_\mathcal{T} = \sum_{i=1}^M\sum_{\tau\in \Lambda^k_i(\mathcal{T})} \delta_\tau * \psi_i^\mathcal{T},$$
where the support of $\psi^\mathcal{T}_i$ is a level-$k$ supertile of $\mathcal{T}$ conformally equivalent to $t_i$. The Delone sets $\Lambda_i^k(\mathcal{T})$ are the locations of the canonical control points of a level-$k$ supertile of type $i$. Using the maps $\Phi_{\sigma^{-k}(x)}^{(k)}:\Omega_{\sigma^{-k}(x)}\rightarrow\Omega_x$, it is straight-forward to generalize (\ref{eqn:localTwist4}). In particular, if $t^{i}_j\subset \mathcal{T}\in\Omega_x$ is a level $i>k$ supertile
\begin{equation}
  \label{eqn:localTwist5}
    \int_{t_j^{(i)}}e^{-2\pi i \langle  \lambda, \tau \rangle} h\circ\varphi_{\tau} (\mathcal{T})  \, d\tau  = e^{2\pi i \langle \lambda , \tau(t^{(i-k)}_j)\rangle} \sum_{\ell=1}^M\mathfrak{t}^{\sigma^{-k}(x)}_{i-k,j}(a^{i-k}_{\sigma^{-k}(x),\ell,\bar\theta_{(-k)_x}\lambda})\widehat{\psi^{\mathcal{T}'}_\ell}(\lambda)
\end{equation}
\end{comment}
\begin{lemma}[\cite{ST:random}]
\label{lem:packing}
  Let $S_1,\dots, S_N$ be a collection of uniformly expanding compatible substitution rules on $m$ prototiles. For $\mu$ a minimal, $\sigma$-invariant ergodic probability measure on $\Sigma_N$, for almost every $x$ and tiling $\mathcal{T}\in\Omega_x$ and $R>0$ there exists an integer $n = n(R)$ and a decomposition
  \begin{equation}
    \label{eqn:decomp3}
    \mathcal{O}_\mathcal{T}^-(C_R(0)) = \bigcup_{i = 0}^n\bigcup_{j=1}^M \bigcup_{k=1}^{\kappa_j^{(i)}}t^{(i)}_{j,k},
  \end{equation}
  where $t^{(i)}_{j,k}$ is a level-$i$ supertile of the tiling $\mathcal{T}$ of type $j$, such that
  \begin{enumerate}
  \item $\kappa_j^{(n)}\neq 0$ for some $j$ and $\mathrm{Vol}(C_R(0)) = (2R)^d\leq \RU \theta_{(n)_x}^{-d}$,
    \item $\sum_{j=1}^M \kappa_j^{(i)} \leq \RN \mathrm{Vol}(\partial C_R(0))  \theta_{(i)_x}^{d-1}$ for $i = 0,\dots, n-1$
  \end{enumerate}
  for some $\RU, \RN$ which depend only on the substitution rules.
\end{lemma}

\begin{comment}
\begin{corollary}
  \label{cor:packing}
(Assume all the right cocycle conditions for a postal measure and a positive word $w = w^-w^+$). Suppose $\mathcal{O}_\mathcal{T}^-(C_R(0))$ has decomposition (\ref{eqn:decomp3}) for $\mathcal{T}\in\Omega_x$. Let $k_n\rightarrow \infty$ be a maximal sequence of return times to $C([w^-.w^+])$. Then for any $R$ large enough there is a $N(R)\in\mathbb{N}$ such that,
  \begin{equation}
    \label{eqn:decomp3'}
    \mathcal{O}_{\mathcal{T}}^-(C_{R}(0)) = \bigcup_{i = 1}^{N(R)}\bigcup_{j=1}^M \bigcup_{n=1}^{\bar\kappa_j^{(k_i)}}t^{(k_i)}_{j,n},
  \end{equation}
  with
  \begin{equation}
    \label{eqn:telPacking}
    \begin{split}
      \sum_{j=1}^M \bar\kappa_j^{(k_i)}&\leq K\sum_{\ell=0}^{k_{i+1}-k_i-1}\|\Theta_x^{(k_i,k_i+\ell)}\|R^{d-1}\theta^{d-1}_{(k_i+\ell)_x}\hspace{.6in} \mbox{ for all $i=1,\dots, N(R)-1$ and }\\
      \sum_{j=1}^M \bar\kappa_j^{(k_{N(R)})}&\leq K'\sum_{\ell=0}^{n(R)-k_{N(R)}-1}\|\Theta_x^{(k_{N(R)},k_{N(R)}+\ell)}\|R^{d-1}\theta^{d-1}_{(k_{N(R)}+\ell)_x},
    \end{split}
  \end{equation}
  where $n(R)$ is the number given by Lemma \ref{lem:packing}.
\end{corollary}
\begin{proof}
  By repetitivity, there is a $R_0$ big enough so that every ball of radius $R$ contains a level-$k_1$ supertile. This $R_0$ is independent of both $\mathcal{T}$ and locally constant around $x\in\Sigma_N$. Taking the decomposition given by Lemma \ref{lem:packing}, the goal is to express level-$i$ supertiles with $k_m<i<k_{m+1}$, as the union of level $k_m$ supertiles, with bounds on the number of level-$k_m$ supertiles needed for each level-$i$ supertile. This can be bounded since there are no more than $\|\Theta_x^{(k_m,i)}\|$ level-$k_m$ supertiles inside each level-$i$ supertile and using the estimate (ii) from Lemma \ref{lem:packing}. More specifically:
  $$  \sum_{j=1}^M \kappa_j^{(k_i)}\leq K\sum_{\ell=0}^{k_{i+1}-k_i-1}\|\Theta_x^{(k_i,k_i+\ell)}\|\sum_{J=1}^M\kappa_{J}^{(k_i+\ell)} = K'\sum_{\ell=0}^{k_{i+1}-k_i-1}\|\Theta_x^{(k_i,k_i+\ell)}\|R^{d-1}\theta^{d-1}_{(k_i+\ell)_x},$$
  which proves the first inequality of (\ref{eqn:telPacking}); the second follows by the same argument.
\end{proof}
\end{comment}
\begin{proposition}
\label{prop:twistedBound}
  Let $S_1,\dots, S_N$ be a collection of uniformly expanding compatible substitution rules on $M$ prototiles. Let $\mu$ be a positively simple, $\sigma$-invariant ergodic probability measure for which the trace cocycle is integrable. For every positively simple word $w = w^-w^+$ such that $\mu(C([w^-.w^+]))>0$ there exists a finite set of vectors $\Lambda_w\subset \mathbb{R}^d$ and $c_w>0$ such that for any $\epsilon\in(0,1)$, for $\mu$-almost every $x\in\Sigma_N$ there is a subsequence $k_n\rightarrow \infty$, $z>1$ and $\bar{K}_{\epsilon,x}>0$ such that for any level-$y$ bounded TLC function $f:\Omega_x\rightarrow \mathbb{R}$, $\mathcal{T}\in\Omega_x$ and $\lambda\in\mathbb{R}^d$
  $$\left|\mathcal{S}^\mathcal{T}_R(f,\lambda)\right|\leq C_{x,f,\epsilon} R^{d+\epsilon} \prod_{m_0<n<N_\epsilon(R)} \left( 1-c_w \max_{\tau\in \Lambda_w}\left\| \left\langle\lambda, \theta^{-1}_{(k_n+|w^+|)_x}\tau\right\rangle\right\|^2_{\mathbb{R}/\mathbb{Z}}  \right)+ \mathcal{O}(R^{d-1})$$
 for all $R>z^{k_{m_0}}$, and $N_\epsilon(R)$ is the largest return of $x$ to $C([w^-.w^+])$ which is less than $\frac{d\log R - \bar{K}_{\epsilon,x}}{\eta_1+\epsilon}$ and $m_0$ is the smallest integer which satisfies $k_{m_0}\geq y$.
\end{proposition}

\begin{proof}
  Let $w$ be a positively simple word such that the cylinder set $C([w])$ has $\mu(C([w]))>0$. For $\mu$-almost every $x$ which is Furstenberg-Kesten-regular for the trace cocycle, without loss of generality, it can be assumed that $x^+ = w_0w^+y_1w^-w^+y_2w^-w^+y_3\cdots$ as in the proof of Proposition \ref{prop:TracesBound}. Let $\{k_m\}_m$ be defined recursively with $k_1 = |w_0|$ and $k_{m+1} = k_m + |w| + |y_n|$, and $k_x = |w^+|$. This defines the maximal sequence of return times of $x$ to the cylinder set $C([w])$ under $\sigma^{-1}$. Let $x\in\Sigma_N$ be one such $\mu$-generic, Furstenberg-Kesten-regular point.

Recall in the proof of Proposition \ref{prop:TracesBound} a finite set of special vectors denoted as $\bar\Lambda_x$ whose importance comes from the fact that they are good return vectors found inside a level-$|w^+|$ supertile for any tiling $\mathcal{T}\in\Omega_z$ for any $z\in C([w^-.w^+])$. This set will be denoted by $\Lambda_w$, and so scaled versions of $\Lambda_w$ are good return vectors for $\mu$-almost every $x$. In addition, the constant $c$ which appears in Proposition \ref{prop:TracesBound} depends only on $w$, so it will be denoted $c_w$ here for the same reasons. In order to simplify some notation, the product in Proposition \ref{prop:TracesBound} will be shortened to:
\begin{equation}
  \label{eqn:prod}
  \mathfrak{P}(m,n) = \prod_{m\leq i<n}\left( 1 - c_w\max_{\tau\in \Lambda_w} \|\langle\lambda,\theta_{(k_{i}+|w^+|)_x}^{-1}\tau\rangle \|^2_{\mathbb{R}/\mathbb{Z}}\right).
\end{equation}

Since $x$ is Furstenberg-Kesten-regular, for all $\varepsilon>0$ there exists a $C_\varepsilon = C_{\varepsilon,x}$ such that $\|\Theta_x^{(n)}\|\leq C_\varepsilon e^{(\eta_1+\varepsilon)n}$ for all $n>0$, where $\eta_1$ is the largest Lyapunov exponent of the trace cocycle. Given also that $n^{-1}\log \theta^{-d}_{(n)_x}\rightarrow \eta_1$, for any $\varepsilon>0$ there is a $K_\varepsilon = K_{\varepsilon,x}>0$ such that,
$$K_{\varepsilon}^{-1}\exp(\eta_1-\varepsilon)n< \theta^{-d}_{(n)_x}<K_{\varepsilon} \exp(\eta_1+\varepsilon)n$$
for all $n>0$. This implies that
$$\theta^{d-1}_{(n)_x}\leq K^2_{\varepsilon}\exp\left[\left(\frac{1-d}{d}\eta_1+\frac{d+1}{d}\varepsilon\right)n\right]\hspace{.2in}\mbox{ and }\hspace{.2in}\theta^{1-d}_{(n)_x}\leq C_\varepsilon K_\varepsilon \exp\left[\left(\frac{d-1}{d}\eta_1+\frac{d+1}{d}\varepsilon\right)n\right],$$
for all $n>0$, and in particular for any $0<i\leq n$, that
\begin{equation}
  \label{eqn:bdryEffect}
  \theta_{(n)_x}^{1-d}\theta_{(i)_x}^{d-1}\leq C_\varepsilon K^3_\varepsilon \exp\left[\frac{d-1}{d}\eta_1(n-i)+\frac{d+1}{d}\varepsilon (n+i)\right],
\end{equation}
which will be used below. Moreover, for every $\varepsilon>0$ there exists a $\Jh_\varepsilon>1$ such that
\begin{equation}
  \label{eqn:bdryEffect2}
\sum_{i=1}^{n-1} e^{\frac{\eta_1}{d}i} \leq \Jh_\varepsilon \left(1-\frac{c_\mu}{2}\right)\exp\left[\left(\frac{\eta_1+\varepsilon}{d}\right)n\right]
\end{equation}
for all $n>0$. Finally, for every $\varepsilon>0$ there is a $\Ch_\varepsilon>1$ such that for all $n>0$
\begin{equation}
  \label{eqn:bdryEffect3}
  \exp\left[\left( \eta_1+\frac{3(d+1)}{d}\varepsilon \right)n\right]\leq \Ch_\varepsilon \theta_{(n)_x}^{-\left(d+\frac{3d+4}{\eta_1}\varepsilon\right)},
\end{equation}
 which will also be used below.

 Let $h = \sum h_i :\Omega_x\rightarrow \mathbb{R}$ be a level-$y$ TLC function and let $\Rn\geq y$ be the smallest natural number such that $\sigma^{-\Rn}(x)\in C([w^-.w^+])$, so there is an integer $m_0\geq 0$ such that $\Rn = k_{m_0}$. By Lemma \ref{lem:TLCfunctions} $h':= \Phi^*_{\sigma^{-\Rn}(x)}h = \sum \Phi^*_{\sigma^{-\Rn}(x)} h_i = \sum  h_i'$ is a level-$0$ TLC function on $\Omega_{\sigma^{-\Rn}(x)}$. By (\ref{eqn:TwistedChange}), instead of computing the twisted integral $\mathcal{S}_R^\mathcal{T}(h,\lambda)$ one can compute the twisted integral $\theta_{(\Rn)_x}^{-d}\mathcal{S}_{R'}^{\mathcal{T}'}(h',\lambda')$, where $x' = \sigma^{-\Rn}(x)$, $R' = \theta_{(\Rn)_x}R$, $\mathcal{T}' = \Phi^{-1}_{\sigma^{-\Rn}(x)}\circ \cdots \circ \Phi_{\sigma^{-1}(x)}^{-1}(\mathcal{T})$ and $\lambda' = \theta^{-1}_{(\Rn)_x}\lambda$.

Note that in order for the decmposition in (\ref{eqn:decomp3}) to have at least one level-$\Rn$ supertile it is necessary that $R>C_{\varepsilon,x}'z^{\Rn}$ for some $C_{\varepsilon,x}'>0$, where $z:=e^{(\eta_1+\varepsilon)}$. This is a consequence of repetitivity and the fact that $\theta^{-d}_{(n)_x}$ and $e^{\eta_1 n}$ have the same exponential growth, and $\theta^{-d}_{(\Rn)_x}$ is comparable to the volume of level-$\Rn$ supertiles. For such large $R$ the decomposition from Lemma \ref{lem:packing} gives a decomposition for $\mathcal{O}^-_{\mathcal{T}'}(C_{R'}(0))$ as
$$\mathcal{O}_{\mathcal{T}'}^-(C_{R'}(0)) = \bigcup_{i' = 0}^{n'}\bigcup_{j=1}^M \bigcup_{k=1}^{\kappa_j^{(i)}}\bar{t}^{(i)}_{j,k},$$
where $n'(R) = n(R)-\Rn$ is the shifted index and $\bar{t}_{j,n}^{(k_{i'})} =  \theta_{(\Rn)_x} t_{j,n}^{(k_{i'})}$. Using this decomposition for $\mathcal{O}^-_{\mathcal{T}'}(C_{R'}(0))$:
\begin{equation*}
  \begin{split}
    &\theta^{d}_{(\Rn)_x}\mathcal{S}_R^{\mathcal{T}}(h,\lambda) = \mathcal{S}_{R'}^{\mathcal{T}'}(h',\lambda') \\
    &\hspace{.6in}=\int_{\mathcal{O}^-_{\mathcal{T}'}(C_{R'}(0))}e^{-2\pi i \langle\lambda', \tau\rangle}\sum_{\ell=1}^M h'_\ell\circ \varphi_\tau(\mathcal{T}')\, d\tau+ \int_{B_{R'}\backslash \mathcal{O}^-_{\mathcal{T}'}(C_{R'}(0))}e^{-2\pi i \langle\lambda', \tau\rangle} h'\circ \varphi_\tau(\mathcal{T}')\, d\tau \\
    &\hspace{.4in}= \sum_{\ell=1}^M\int_{\mathcal{O}^-_{\mathcal{T}'}(C_{R'})}e^{-2\pi i \langle\lambda', \tau\rangle} h'_\ell\circ \varphi_\tau(\mathcal{T}')\, d\tau + \mathcal{O}(R^{d-1}) \\
    &\hspace{.2in}= \sum_{\ell=1}^M\sum_{i'=0}^{n'(R)}\sum_{j=1}^M\sum_{n=1}^{\kappa^{(k_{i'})}_j}\int_{\bar{t}^{(i')}_{j,n}}e^{-2\pi i \langle\lambda', \tau\rangle} h'_\ell\circ \varphi_\tau(\mathcal{T}')\, d\tau + \mathcal{O}(R^{d-1}) \\
    &= \sum_{i'=0}^{n'(R)}\sum_{j=1}^M\sum_{n=1}^{\kappa^{(i')}_j}  e^{2\pi i \langle \lambda' ,\tau(t^{(i')}_{j,n})\rangle} \sum_{\ell=1}^M   \mathfrak{t}^{x'}_{i',j}\left(a^{i'}_{x',\ell,\lambda'}  \right) \widehat{\psi_\ell^{\mathcal{T}'}}(\lambda')  + \mathcal{O}(R^{d-1}),
  \end{split}
\end{equation*}
where the last line follows from (\ref{eqn:localTwist2}). Note that $|\widehat{\psi_\ell^{\mathcal{T}'}}(\lambda')|$ can be bounded by the $\|h\|_\infty$ times the maximal volume $C$ of a level-$y$ supertile. Using this, Lemma \ref{lem:packing} and Proposition \ref{prop:TracesBound},
\begin{equation}
\label{eqn:twistedBnd}
  \begin{split}
    & \theta^d_{(\Rn)_x} \left|\mathcal{S}_R^\mathcal{T}(h,\lambda)\right|\leq \sum_{i'=0}^{n'(R)}\sum_{j=1}^M\kappa^{(i')}_j \sum_{\ell=1}^M\left|\mathfrak{t}^{x'}_{i',j}\left(a^{i'}_{x',\ell,\lambda'}  \right) \widehat{\psi_\ell^{\mathcal{T}'}}(\lambda')\right| + \mathcal{O}(R^{d-1}) \\
    &\hspace{1in}\leq C\|h\|_\infty \sum_{i'=0}^{n'(R)}\sum_{j=1}^M\kappa^{(i')}_jM \max_\ell\left\{\left|\mathfrak{t}^{x'}_{i',j}\left(a^{i'}_{x',\ell,\lambda'}  \right) \right|\right\} + \mathcal{O}(R^{d-1}) \\
    &\hspace{.75in}\leq C\|h\|_\infty M \sum_{i'=0}^{n'(R)} \RN (2R)^{d-1}\theta^{d-1}_{(i')_x} \max_{\ell,j}\left\{\left|\mathfrak{t}^{x'}_{i',j}\left(a^{i'}_{x',\ell,\lambda'}  \right) \right|\right\} + \mathcal{O}(R^{d-1}) \\
    &\hspace{.5in}\leq C\|h\|_\infty M 2^{d-1} \RN\sum_{i'=0}^{n'(R)}  R^{d-1}\theta^{d-1}_{(i')_x} \max_{\ell,j}\left\{\left|\mathfrak{t}^{x'}_{i',j}\left(a^{i'}_{x',\ell,\lambda'}  \right) \right|\right\} + \mathcal{O}(R^{d-1}) \\
    &\hspace{.25in}\leq C\|h\|_\infty M \RN \RU^{\frac{d-1}{d}} \sum_{i'=0}^{n'(R)}  \theta_{(n)_x}^{1-d}\theta^{d-1}_{(i')_x}\left\|\Theta^{(i')}_x\right\| \mathfrak{P}(0,N(i')) + \mathcal{O}(R^{d-1}),
  \end{split}
\end{equation}
where $N(i')$ is the largest return time to $C([w^-.w^+])$ less than or equal to $i'$, that is, it satisfies $k_{N(i')}\leq i' < k_{N(i')+1}$. Now the estimates (\ref{eqn:bdryEffect})-(\ref{eqn:bdryEffect3}) can be used:
\begin{equation}
  \label{eqn:twistedBnd'}
  \begin{split}
    &\theta^d_{(\Rn)_x}\left|\mathcal{S}_R^\mathcal{T}(h,\lambda)\right|\\
    &\leq C\|h\|_\infty M \RN \RU^{\frac{d-1}{d}} C_\varepsilon K_\varepsilon^4\exp\left[\left(\frac{d-1}{d}\eta_1 +\frac{3d+2}{d}\varepsilon\right)n'\right]\sum_{i'=0}^{n'(R)} \mathfrak{P}(0,N(i')) e^{\frac{\eta_1}{d}i}   + \mathcal{O}(R^{d-1}) \\
    &\hspace{.3in}\leq C\|h\|_\infty M \RN \RU^{\frac{d-1}{d}} C_\varepsilon K_\varepsilon^4\Jh_\varepsilon e^{\frac{\eta_1+\varepsilon}{d}}         \exp\left[\left(\eta_1 +\frac{3(d+1)}{d}\varepsilon\right)n'\right]       \mathfrak{P}(0,N(n'))   + \mathcal{O}(R^{d-1})\\
    &\hspace{.6in}\leq C\|h\|_\infty M \RN \RU^{\frac{d-1}{d}} C_\varepsilon K_\varepsilon^4\Jh_\varepsilon e^{\frac{\eta_1+\varepsilon}{d}}\Ch_\varepsilon \theta_{(n)_x}^{-\left(d+\frac{3d+4}{\eta_1}\varepsilon\right)}\mathfrak{P}(0,N(n'))   + \mathcal{O}(R^{d-1})
  \end{split}
\end{equation}

Since $n(R) = n'+y$ is the largest level of supertile guaranteed to be completely contained in $C_R(0)$, $\min_i\{\mathrm{Vol}(t_i)\}\theta_{n(R)}^{-d} \leq \mathrm{Vol}(C_R(0)) = (2R)^d$. That is, there is a uniform constant $\RU'$ such that $\theta_{(n(R))_x}^{-d}\leq \RU' R^d$ for all $R>1$.
Putting this together with (\ref{eqn:twistedBnd'}),
\begin{equation}
\label{eqn:twistedBnd3}
  \begin{split}
 \left|\mathcal{S}_R^\mathcal{T}(h,\lambda)\right|\leq K_{x,h,\varepsilon} R^{d+\frac{3d+4}{\eta_1}\varepsilon}\mathfrak{P}(m_0,N(n(R))) + \mathcal{O}(R^{d-1})
  \end{split}
\end{equation}
for all $R>z^y$. Now, since $R^d\leq 2^{-d}\RU \theta^{-d}_{(n)_x}\leq 2^{-d}\RU K_\varepsilon e^{(\eta_1+\varepsilon)n}$, setting $\bar{K}_\varepsilon := \log(2^{-d}\RU K_\varepsilon)$ it follows that
$$\frac{d\log R - \bar{K}_\varepsilon}{\eta_1+\varepsilon}\leq n(R)$$
which, combined with (\ref{eqn:twistedBnd3}), finishes the proof.
\end{proof}
\begin{comment}
\begin{proof}
  Let $w$ be a positively simple word such that the cylinder set $C([w])$ has $\mu(C([w]))>0$. For $\mu$-almost every $x$ which is Oseledets-regular for the trace cocycle, without loss of generality, it can be assumed that $x^+ = w_0w^+y_1w^-w^+y_2w^-w^+y_3\cdots$ as in the proof of Proposition \ref{prop:TracesBound}. Let $\{k_m\}_m$ be defined recursively with $k_1 = |w_0|$ and $k_{m+1} = k_m + |w| + |y_n|$, and $k_x = |w^+|$. This defines the maximal sequence of return times of $x$ to the cylinder set $C([w])$ under $\sigma^{-1}$. Let $x\in\Sigma_N$ be one such $\mu$-generic, Oseledets-regular point.

Recall in the proof of Proposition \ref{prop:TracesBound} a finite set of special vectors denoted as $\bar\Lambda_x$: the importance of these vectors is that they are good return vectors found inside a level-$|w^+|$ supertile for any tiling $\mathcal{T}\in\Omega_z$ for any $z\in C([w^-.w^+])$. This set will be denoted by $\Lambda_\mu$ here since the orbit of $\mu$-almost every point visits $C([w^-.w^+])$ infinitely often, and so scaled versions of $\Lambda_\mu$ are good return vectors for $\mu$-almost every $x$. In addition, the constant $c$ which appears in Proposition \ref{prop:TracesBound} depends only on $w$, so it will be denoted $c_\mu$ here for the same reasons. In order to simplify some notation, the product in Proposition \ref{prop:TracesBound} will be shortened to:
\begin{equation}
  \label{eqn:prod}
  \mathfrak{P}(m,n) = \prod_{m\leq i<n}\left( 1 - c_\mu\max_{\tau\in \Lambda_\mu} \|\langle\lambda,\theta_{(k_{i}+|w^+|)_x}^{-1}\tau\rangle \|^2_{\mathbb{R}/\mathbb{Z}}\right).
\end{equation}

Since $x$ is Oseledets regular, for all $\varepsilon>0$ there exists a $C_\varepsilon$ such that $\|\Theta_x^{(n)}\|\leq C_\varepsilon e^{(\eta_1+\varepsilon)n}$, where $\eta_1$ is the largest Lyapunov exponent of the trace cocycle. This implies that for any $0\leq m<n$, $\|\Theta_x^{(m,n)}\|\leq C_\varepsilon e^{((m-n)\eta_1+\varepsilon)n}$

Let $h = \sum h_i :\Omega_x\rightarrow \mathbb{R}$ be a level-$y$ TLC function and let $y'\geq y$ be the smallest natural number such that $\sigma^{-y'}(x)\in C([w^-.w^+])$, so there is an integer $m_0\geq 0$ such that $y' = k_{m_0}$. By Lemma \ref{lem:TLCfunctions} $h':= \Phi^*_{\sigma^{-y'}(x)}h = \sum \Phi^*_{\sigma^{-y'}(x)} h_i = \sum  h_i'$ is a level-$0$ TLC function on $\Omega_{\sigma^{-y'}}$. By (\ref{eqn:TwistedChange}), instead of computing the twisted integral $\mathcal{S}_R^\mathcal{T}(h,\lambda)$ one can compute the twisted integral $\theta_{(y')_x}^{-d}\mathcal{S}_{R'}^{\mathcal{T}'}(h',\lambda')$, where $x' = \sigma^{-y'}(x)$, $R' = \theta_{(y')_x}R$, $\mathcal{T}' = \Phi^{-1}_{\sigma^{-y'}(x)}\circ \cdot \circ \Phi_{\sigma^{-1}(x)}^{-1}(\mathcal{T})$ and $\lambda' = \theta^{-1}_{(-y')_x}\lambda$. Let $n'(R) = n(R)-y'$ be the shifted index in the decompositions for $C_R(0)$ and $C_{R'}(0)$ in (\ref{eqn:decomp3}) and (\ref{eqn:decomp3'}), respectively. Using Corollary \ref{cor:packing} and setting $\bar{t}_{j,n}^{(k_{i'})} =  \theta_{(y')_x} t_{j,n}^{(k_{i'})}$:
\begin{equation*}
  \begin{split}
    &\theta^{d}_{(y')_x}\mathcal{S}_R^{\mathcal{T}}(h,\lambda) = \mathcal{S}_{R'}^{\mathcal{T}'}(h',\lambda') \\
    &\hspace{.6in}=\int_{\mathcal{O}^-_{\mathcal{T}'}(C_{R'}(0))}e^{-2\pi i \langle\lambda', \tau\rangle}\sum_{\ell=1}^M h'_\ell\circ \varphi_\tau(\mathcal{T}')\, d\tau+ \int_{B_{R'}\backslash \mathcal{O}^-_{\mathcal{T}'}(C_{R'}(0))}e^{-2\pi i \langle\lambda', \tau\rangle} h'\circ \varphi_\tau(\mathcal{T}')\, d\tau \\
    &\hspace{.4in}= \sum_{\ell=1}^M\int_{\mathcal{O}^-_{\mathcal{T}'}(C_{R'})}e^{-2\pi i \langle\lambda', \tau\rangle} h'_\ell\circ \varphi_\tau(\mathcal{T}')\, d\tau + \mathcal{O}(R^{d-1}) \\
    &\hspace{.2in}= \sum_{\ell=1}^M\sum_{i'=m_0}^{N(R)}\sum_{j=1}^M\sum_{n=1}^{\bar\kappa^{(k_{i'})}_j}\int_{\bar{t}^{(i')}_{j,n}}e^{-2\pi i \langle\lambda', \tau\rangle} h'_\ell\circ \varphi_\tau(\mathcal{T}')\, d\tau + \mathcal{O}(R^{d-1}) \\
    &= \sum_{i'=m_0}^{N(R)}\sum_{j=1}^M\sum_{n=1}^{\bar\kappa^{(k_{i'})}_j}  e^{2\pi i \langle \lambda' ,\tau(t^{(k_{i'})}_{j,n})\rangle} \sum_{\ell=1}^M   \mathfrak{t}^{x'}_{k_{i'},j}\left(a^{k_{i'}}_{x',\ell,\lambda'}  \right) \widehat{\psi_\ell^{\mathcal{T}'}}(\lambda')  + \mathcal{O}(R^{d-1}),
  \end{split}
\end{equation*}
where the last line follows from (\ref{eqn:localTwist2}). Note that $|\widehat{\psi_\ell^{\mathcal{T}'}}(\lambda')|$ can be bounded by the $\|h\|_\infty$ times the maximal volume of a level-$y$ supertile. Using this and Corollary \ref{cor:packing}:
\begin{equation}
\label{eqn:twistedBnd}
  \begin{split}
    &\theta^d_{(y')_x} \left|\mathcal{S}_R^\mathcal{T}(h,\lambda)\right|\\
    &\hspace{.15in}\leq \sum_{i'=m_0}^{N(R)}\sum_{j=1}^M\bar\kappa^{(k_{i'})}_j \sum_{\ell=1}^M\mathfrak{t}^{x'}_{k_{i'},j}\left(a^{k_{i'}}_{x',\ell,\lambda'}  \right) \widehat{\psi_\ell^{\mathcal{T}'}}(\lambda') + \mathcal{O}(R^{d-1}) \\
    &\hspace{.12in}\leq \sum_{i'=m_0}^{N(R)-1}  K_3   \sum_{n=0}^{k_{i'+1}-k_{i'}-1}\left\|\Theta_x^{(k_{i'},k_{i'}+n)}\right\|R^{d-1}\theta^{d-1}_{(k_{i'}+n)_x} \max_j\sum_{\ell=1}^M\left| \mathfrak{t}^{x'}_{k_{i'},j}\left(a^{k_{i'}}_{x',\ell,\lambda'}  \right)\right| \widehat{\psi_\ell^\mathcal{T}}(\lambda')\\
    & \hspace{.37in}+ K_3  \sum_{n'=0}^{n(R)-k_{N(R)}-1}\left\|\Theta_x^{(k_{N(R)},k_{N(R)}+n')}\right\|R^{d-1}\theta^{d-1}_{(k_{N(R)}+n')_x} \max_j\sum_{\ell=1}^M\left| \mathfrak{t}^{x'}_{k_{N(R)},j}\left(a^{k_{N(R)}}_{x',\ell,\lambda'}  \right)\right| \widehat{\psi_\ell^\mathcal{T}}(\lambda') \\
    &\hspace{.5in} + \mathcal{O}(R^{d-1}) \\
    &\hspace{.1in}\leq \sum_{i'=m_0}^{N(R)-1}  K_4 \max_{j,\ell}\left\{\left| \mathfrak{t}^{x'}_{k_{i'},j}\left(a^{k_{i'}}_{x',\ell,\lambda'}  \right)\right|  \right\} \sum_{n=0}^{k_{i'+1}-k_{i'}-1}\left\|\Theta_x^{(k_{i'},k_{i'}+n)}\right\|R^{d-1}\theta^{d-1}_{(k_{i'}+n)_x}\\
    &\hspace{.3918in} +   K_4 \max_{j,\ell}\left\{\left| \mathfrak{t}^{x'}_{k_{N(R)},j}\left(a^{k_{N(R)}}_{x',\ell,\lambda'}  \right)\right|  \right\} \sum_{n'=0}^{n(R)-k_{N(R)}-1}\left\|\Theta_x^{(k_{N(R)},k_{N(R)}+n')}\right\|R^{d-1}\theta^{d-1}_{(k_{N(R)}+n')_x}  + \mathcal{O}(R^{d-1}).
  \end{split}
\end{equation}
Using Proposition \ref{prop:TracesBound}
\begin{equation}
  \label{eqn:twistedBnd'}
  \begin{split}
  \left|\mathcal{S}_R^\mathcal{T}(h,\lambda)\right| &\leq K_5\sum_{i'=m_0}^{N(R)-1}    \left\| \Theta_x^{(k_{i'})}\right\|
    \mathfrak{P}(m_0,i')
    \sum_{n=0}^{k_{i'+1}-k_{i'}-1}\left\|\Theta_x^{(k_{i'},k_{i'}+n)}\right\|R^{d-1}\theta^{d-1}_{(k_{i'}+n)_x} \\
    &+\hspace{.01in}K_5 \left\|\Theta_x^{(k_{N(R)})}\right\|\mathfrak{P}(m_0,N(R))\sum_{n=0}^{n(R)-k_{N(R)}-1}\left\|\Theta_x^{(k_{N(R)},k_{N(R)}+n)}\right\|R^{d-1}\theta^{d-1}_{(k_{N(R)}+n)_x}  + \mathcal{O}(R^{d-1}).
  \end{split}
\end{equation}

Using the first bound of Lemma \ref{lem:packing}, it follows that $R^{d-1}\leq K' \theta_{(n)_x}^{1-d}$ for some $K'$ independent of $n$. Since $n^{-1}\log\theta_{(n)_x}^{-d}\rightarrow \eta_1$, for all $\varepsilon$ there exists $K_7$ such that
\begin{equation}
  \label{eqn:BdryEstimate}
  R^{d-1}\theta^{d-1}_{(k_{i'}+l)_x}\leq K_6 (\theta_{x_{k_{i'}+l}}\cdots \theta_{x_n})^{1-d} \leq K_7 \exp\left[\left(\frac{d-1}{d}\eta_1+\varepsilon\right)(n(R)-(k_{i'}+l))\right]
\end{equation}
for all $l>0$, so\footnote{We need to address the issue that $\left\|\Theta_x^{(k_{i'},k_{i'}+n)}\right\|\leq C_{\varepsilon,\sigma^{-k_{i'}}(x)}e^{(\eta_1+\varepsilon)n}$; we need a uniform constant.} for $i'<N(R)$
\begin{equation}
  \label{eqn:BdryEstimate2}
  \begin{split}
    &\sum_{n=0}^{k_{i'+1}-k_{i'}-1}\left\|\Theta_x^{(k_{i'},k_{i'}+n)}\right\|R^{d-1}\theta^{d-1}_{(k_{i'}+n)_x}\\
    &\hspace{.2in}\leq K_8\sum_{n=0}^{k_{i'+1}-k_{i'}-1}e^{(\eta_1+\varepsilon)n} \exp\left[\left(\frac{d-1}{d}\eta_1+\varepsilon\right)(n(R)-(k_{i'}+n))\right]\\
    &\hspace{.4in}\leq K_9\exp\left[\left(\frac{d-1}{d}\eta_1+\varepsilon\right)(n(R)-k_{i'})\right]\sum_{n=0}^{k_{i'+1}-k_{i'}-1} \exp\left[\left(\frac{\eta_1}{d}+\varepsilon\right)n \right] \\
    &\hspace{.6in}\leq K_{10} \exp\left[\left(\frac{d-1}{d}\eta_1+\varepsilon\right)(n(R)-k_{i'})\right] \exp\left[\left(\frac{\eta_1}{d}+\varepsilon\right) (k_{i'+1}-k_{i'})\right].
  \end{split}
\end{equation}
and likewise
\begin{equation}
  \label{eqn:BdryEstimate2'}
  \begin{split}
    &  \sum_{n'=0}^{n(R)-k_{N(R)}-1}\left\|\Theta_x^{(k_{N(R)},k_{N(R)}+n')}\right\|R^{d-1}\theta^{d-1}_{(k_{N(R)}+n')_x} \\
    &\hspace{.2in}\leq K_8\sum_{n'=0}^{n(R)-k_{N(R)}-1}e^{(\eta_1+\varepsilon)n'} \exp\left[\left(\frac{d-1}{d}\eta_1+\varepsilon\right)(n(R)-(k_{N(R)}+n'))\right]\\
    &\hspace{.4in}\leq K_9\exp\left[\left(\frac{d-1}{d}\eta_1+\varepsilon\right)(n(R)-k_{N(R)})\right]\sum_{n'=0}^{n(R)-k_{N(R)}-1} \exp\left[\left(\frac{\eta_1}{d}+\varepsilon\right)n' \right] \\
    &\hspace{.6in}\leq K_{10}\exp\left[\left(\eta_1+\varepsilon\right)(n(R)-k_{N(R)})\right] .
  \end{split}
\end{equation}
Putting (\ref{eqn:BdryEstimate2}) and (\ref{eqn:BdryEstimate2'}) into (\ref{eqn:twistedBnd'}),
\begin{equation}
\label{eqn:twistedBnd2}
  \begin{split}
  &  \left|\mathcal{S}_R^\mathcal{T}(h,\lambda)\right|\leq e^{\left(\frac{d-1}{d}\eta_1+\varepsilon\right)n(R)}\sum_{i'=m_0}^{N(R)-1}  K_{10}e^{-\eta_1k_{i'}}   \left\| \Theta_x^{(k_{i'})}\right\|  \mathfrak{P}(m_0,i')  e^{\left(\frac{\eta_1}{d}+\varepsilon\right) k_{i'+1} }    + \mathcal{O}(R^{d-1}) \\
    &\hspace{.55in}+K_{10} \left\| \Theta_x^{(k_{N(R)})}\right\|  \mathfrak{P}(m_0,N(R)) \exp \left[\left(\eta_1+\varepsilon\right)(n(R)-k_{N(R)})\right]\\
    &\hspace{.25in}\leq K_{11}e^{\left(\frac{d-1}{d}\eta_1+\varepsilon\right)n(R)}\sum_{i'=m_0}^{N(R)-1} e^{\varepsilon k_{i'}} \mathfrak{P}(m_0,i') e^{\left(\frac{\eta_1}{d}+\varepsilon\right) k_{i'+1} }\\
    &\hspace{.5in}+K_{11}e^{(\eta_1+\varepsilon)k_{N(R)}} \mathfrak{P}(m_0,N(R))  \exp\left[\left(\eta_1+\varepsilon\right)(n(R)-k_{N(R)})\right]    + \mathcal{O}(R^{d-1})\\
    &\hspace{.25in}\leq K_{12}e^{\left(\frac{d-1}{d}\eta_1+\varepsilon\right)n(R)}\sum_{i'=m_0}^{N(R)-1} e^{\varepsilon k_{i'}} \mathfrak{P}(m_0,i') e^{\left(\frac{\eta_1}{d}+\varepsilon\right) k_{i'+1} }  \\
    &\hspace{.85in}+K_{12} \exp\left[\left(\eta_1+\varepsilon\right)n(R)\right] \mathfrak{P}(m_0,N(R))   + \mathcal{O}(R^{d-1}).
  \end{split}
\end{equation}
By (\ref{eqn:traceExp3}), $c_w\leq 1/4$ in (\ref{eqn:prod}), so the sums in (\ref{eqn:twistedBnd2}) are dominated by the last term, the term involving $N(R)$, so
\begin{equation}
\label{eqn:twistedBnd3}
  \begin{split}
 \left|\mathcal{S}_R^\mathcal{T}(h,\lambda)\right|\leq K_{13} e^{(\eta_1+\varepsilon)n(R)}\mathfrak{P}(m_0,N(R)) + \mathcal{O}(R^{d-1}).
  \end{split}
\end{equation}
Since the $n(R)$ is the largest level of supertile guaranteed to be completely contained in $C_R(0)$, $\min_i\{\mathrm{Vol}(t_i)\}\theta_{n(R)}^{-d} \leq \mathrm{Vol}(C_R(0)) = (2R)^d$. That is, there is a uniform constant $K_{14}$ such that $\theta_{(n(R))_x}^{-d}\leq K_{14} R^d$.
Moreover, since $e^{(\eta_1+\frac{\varepsilon \eta_1}{2d})n}\leq \theta^{-(d+\varepsilon)}_{(n)_x}$, for all $n$ large enough, $e^{(\eta_1+\varepsilon)n}\leq K_{14}R^{d+\frac{2d\varepsilon}{\eta_1}}$ for all $R$ large enough, so putting this together with (\ref{eqn:twistedBnd3}),
\begin{equation}
\label{eqn:twistedBnd3}
  \begin{split}
 \left|\mathcal{S}_R^\mathcal{T}(h,\lambda)\right|\leq K_{15} R^{d+\frac{2d\varepsilon}{\eta_1}}\mathfrak{P}(m_0,N(R)) + \mathcal{O}(R^{d-1})
  \end{split}
\end{equation}
for all $R$ large enough. Finally, since $R^d\leq K_{16} e^{(\eta_1+\varepsilon)n(R)}$ for all $R$ large enough,
\begin{equation}
  \label{eqn:n(R)bnd}
  \frac{d\log R - \log K_{16}}{\eta_1+\varepsilon}\leq n(R),
\end{equation}
so letting $N_\varepsilon(R)$ be the largest return of $x$ which is less than or equal to $\frac{d\log R - \log K_{16}}{\eta_1+\varepsilon}$, i.e., $\sigma^{-k_{N_\varepsilon(R)}}(x)\in C([w])$ and
\begin{equation}
  \label{eqn:n(R)bnd2}
k_{N_\varepsilon(R)}\leq\frac{d\log R - \log K_{16}}{\eta_1+\varepsilon}<k_{N_\varepsilon(R)+1}
\end{equation}
it follows that
$$ \left|\mathcal{S}_R^\mathcal{T}(h,\lambda)\right|\leq K_{15} R^{d+\frac{2d\varepsilon}{\eta_1}}\mathfrak{P}(m_0,N_\varepsilon(R)) + \mathcal{O}(R^{d-1})$$
for all $R$ large enough.
\end{proof}

We can sort this out through the $m\times m$ matrix
 \begin{equation}
   \label{eqn:twistedCocycle}
  \Pi_k^x(\lambda) = \left(\begin{array}{c c c c }
    \Phi_1^x(\mathcal{P}_k(v_1),\lambda ) & \Phi_2^x(\mathcal{P}_k(v_1),\lambda ) & \cdots & \Phi_m^x(\mathcal{P}_k(v_1) ,\lambda )\\
    \vdots & \ddots & \Phi_j^x(\mathcal{P}_k(v_i),\lambda ) & \vdots \\
    \Phi_1^x(\mathcal{P}_k(v_m),\lambda ) & \Phi_2^x(\mathcal{P}_k(v_m),\lambda ) & \cdots & \Phi_m^x(\mathcal{P}_k(v_m) ,\lambda )
  \end{array}\right).
 \end{equation}

 There is a relationship between canonical level-$k$ super tiles and the level-$(k-1)$ supertiles that tile it. More precisely, for $k>0$ and $i,j\in \{1,\dots, m\}$ there is a collection of vectors $\mathcal{U}^x_{i,j,k}$ with $|\mathcal{U}^x_{i,j,k}| = |r^{-1}(v_i)\cap s^{-1}(v_j)| = (A_k^+)_{i,j}$ whose elements are described as follows. Consider the canonical level-$k$ supertile $\mathcal{P}_k(v_i)$. It is tiled by $|r^{-1}(v_i)|$ level-$(k-1)$ supertiles $\mathcal{P}_{k-1}(\bar{p}_e)$, $e\in r^{-1}(v_i)$, all but one of which are not in canonical position. Therefore for each edge $e\in r^{-1}(v_i)$ there is a vector $u_e\in\mathbb{R}^d$ such that $\varphi_{u_e}(\mathcal{P}_{k-1}(\bar{p}_e)) = \mathcal{P}_k(v_j)$ for $e\in r^{-1}(v_i)\cap s^{-1}(v_j)$, that is, so that translating the level-$(k-1)$ supertile $\mathcal{P}_{k-1}(\bar{p}_e)$ is translated into canonical position. Thus $\mathcal{U}^x_{i,j,k}$ is the collection of all such vectors $u_e$ for $e\in r^{-1}(v_i)\cap s^{-1}(v_j)$. It follows that there is a decomposition $\Pi^x_k(\lambda) = M^x_k(\lambda)\Pi^x_{k-1}(\lambda)$ where $M^x_k(\lambda)$ is the matrix
 $$M^x_k(\lambda)_{i,j} = \sum_{u_e\in\, \mathcal{U}^x_{i,j,k}}e^{-2\pi i \langle \lambda , u_e \rangle}.$$
 As such, we have that
\begin{equation}
\label{eqn:twistedCocycle2}
  \Pi^x_k(\lambda) = M^x_k(\lambda )M^x_{k-1}(\lambda)\cdots M^x_1(\lambda).
  \end{equation}
Recall the definition in \S \ref{subsec:retVectors} of the set of return vectors $\Lambda_\mathcal{T}$ for a repetitive tiling $\mathcal{T}$ of finite local complexity. If $\Omega_x$ is the tiling space constructed from the minimal Bratteli diagram $\mathcal{B}_x$, then we denote by $\Lambda_x\subset \mathbb{R}^d$ the set of return vectors, since it is independent of the tiling $\mathcal{T}\in\Omega_x$. The following is the generalization of the concept of good return words in \cite{BS:genus2}.
\begin{definition}
A vector $w\in \mathbb{R}^d$ is a \textbf{good return vector} for a substitution rule $S$ on the prototiles $t_1,\dots, t_m$ if for some $\ell$ and any $i\in\{1,\dots, m\}$, $S(t_i)$ contains\footnote{Maybe formalize the notation $S(t)$ which is substitution and expansion.} two copies of $t_\ell$, $t_{1}$ and $t_{2}$, with $\varphi_w(t_1) = t_2$. The set of good return vectors for a substitution $S$ is denoted by $GV(S)$.
\end{definition}
Note that if $\Omega_x$ is the tiling space constructed from a minimal Bratteli diagram $\mathcal{B}_x$, then there exists a $k>0$ such that the substitution given by composing the first $k$ substitutions given by the positive part of $\mathcal{B}_x$, $S_{(k)}^x = S_{x_1}\circ S_{x_2}\circ\cdots\circ S_{x_{k}}$, has a non-trivial good return vector, and in this case $GV(S_{(k)}^x) \subset \Lambda_x$. In fact, for any $v\in \Lambda_x$, there exists a $k>0$ such that $v\in GV(S_{(k)}^x)$. As such,
$$\Lambda_x = \bigcup_{k>0}GV(S_{(k)}^x).$$
\begin{proposition}
  Let $S_1,\dots, S_N$ be a collection of compatible substitution rules on tiles $t_1,\dots, t_M$. Let $\mu$ be a minimal, contracting $\sigma$-invariant ergodic probability measure. Then for $\mu$-almost every $x\in\Sigma_N$, there exists a $c_1>0$ such that for any $k\in\mathbb{N}$ and finite path $\bar{e}=(e_1,\dots, e_k)\in\mathcal{E}_{0,k}$ and any $i\in\{1,\dots, M\}$, $\lambda\in [0,1)^d$, and $\mathcal{T}\in\Omega_x$:
    $$\left| \Phi_i^x(\mathcal{P}_k(v_j),\lambda)\right|\leq \left\|\Theta^{(k)}_x \right\|_1\prod_{n<k}\left(1-c_1\cdot \max_{w\in GV}\left\|\langle \lambda , \Upsilon_n(w)\rangle\right\|^2_{\mathbb{R}/\mathbb{Z}} \right)$$
\end{proposition}
\end{comment}
  Let
  $$\mathcal{C}_B := \{y\in\mathbb{R}^d: 0< \|\lambda\|\leq B\}.$$
  The following is essentially \cite[Lemma 4.5]{BS:translation}.
  \begin{lemma}
    \label{lem:LipApprox}
  Let $S_1,\dots, S_N$ be $N$ compatible substitution rules on $M$ prototiles $t_1,\dots,t_M$, $\mu$ a minimal $\sigma$-invariant ergodic probability measure and suppose that the trace cocycle is integrable, and $B>1$. For an Oseledets-regular $x\in\Sigma_N$ suppose that for all $y\geq y_0(x,B)$, any bounded level-$y$ TLC function $f_y$ satisfies
\begin{equation}
  \label{eqn:LipApp1}
  \left| \mathcal{S}_R^\mathcal{T}(f_y,\lambda)   \right|\leq C_{x,f_y}R^{d-\gamma/2}
  \end{equation}
 for all $\lambda\in\mathcal{C}_B$ and $R>e^{\gamma^{-1}\eta_1 y}$. Then for any Lipschitz function $f:\Omega_x\rightarrow \mathbb{R}$:
  \begin{equation}
    \label{eqn:LipHold}
    \left| \mathcal{S}_R^\mathcal{T}(f,\lambda)   \right|\leq C_{x,f}R^{d-\gamma/2}
    \end{equation}
 for all $\lambda\in\mathcal{C}_B$ and $R$ large enough.
\end{lemma}
\begin{proof}
  Let $f:\Omega_x\rightarrow \mathbb{R}$ be a Lipschitz function. By Lemma \ref{lem:LipApprox}, for any $y\in\mathbb{N}$ there is a level-$y$ TLC function $f_y$ with
  $$\| f-f_y\|_\infty\leq L_{f}\max_{v\in\mathcal{V}_k}\mu^+_x(v)\hspace{.5in}\mbox{ and }\hspace{.5in}\|f_y\|_\infty\leq \|f\|_\infty.$$
  Note that
  \begin{equation}
    \label{eqn:TransvContr}
    \lim_{n\rightarrow\infty}\frac{\log\mu^+_x(v_i\in\mathcal{V}_n)}{n} = -\eta_1
  \end{equation}
for any $i$. Indeed, since $x$ is Furstenberg-Kesten-regular for the trace cocycle and each renormalization map $\Phi_{\sigma^k(x)}$ along an orbit preserves the unique $\mathbb{R}^d$ invariant measure which is locally $\nu_x\times \mbox{Leb}$, the transverse measure $\nu_x = (\bar\Delta_x)_*\mu^+_x$ is contracted at a rate which is the inverse of that of the expansion of the Lebesgue measure along the leaves, which is $\theta^{-d}_{(n)_x}$. Therefore, since $n^{-1}\log \theta^{-d}_{(n)_x}\rightarrow \eta_1$, (\ref{eqn:TransvContr}) follows. Thus for any $\varepsilon>0$ and $y\geq y_1(x)$,
\begin{equation}
  \label{eqn:LipApp2}
  \|f-f_y\|_\infty \leq K_f e^{-\frac{\eta_1}{2}y}
  \end{equation}
for some $K_f>0$. Let $y_2 = \max\{y_0,y_1\}$, $R_0 = e^{\gamma^{-1}\eta_1y_2}$ and for $R>R_0$, let
$$y_R = \left\lfloor\frac{\gamma\log R}{\eta_1}\right\rfloor$$
so that for $y>y_2$ both (\ref{eqn:LipApp1}) and (\ref{eqn:LipApp2}) hold, and so:
 $$\left| \mathcal{S}_R^\mathcal{T}(f,\lambda)- \mathcal{S}_R^\mathcal{T}(f_y,\lambda)\right|\leq (2R)^dK_fe^{-\frac{\eta_1}{2}y_R} \leq K_f'R^{d-\gamma/2},$$
which combined with (\ref{eqn:LipApp1}) implies (\ref{eqn:LipHold}).

\end{proof}
\section{A quantitative Veech criterion}
\label{sec:criterion}
\begin{definition}
  Let $S_1,\dots, S_N$ be $N$ compatible substitution rules on the prototiles $t_1,\dots, t_M$. Suppose that there is a $\sigma$-invariant probability measure for which there exists a positively simple word $w = w^-w^+$ with the following properties:
  \begin{enumerate}
  \item $\mu(C([w^-.w^+]))>0$,
  \item for any $x_*\in C([w^-.w^+])\subset\Sigma_N$ there is an $\ell\in\{1,\dots,d\}$ such that for any canonical level-$|w^+|$ supertile there are $k$ vectors $\tau_1,\dots, \tau_k\in \Lambda_{x_*,\ell}^{(0)}$, describing translation equivalences of tiles of type $\ell$ in any $\mathcal{P}_{|w^+|}(v)$, with the property that $\alpha_{x_*}(\tau_1),\dots, \alpha_{x_*}(\tau_{k})$ generate $\mathbb{Z}^{r_x}$.
  \end{enumerate}
  Then $w$ is called a \textbf{postal word} and $\mu$ is called a \textbf{postal} measure, since there are return vectors $\tau_1,\dots,\tau_k$ inside any level-$|w^+|$ supertile $\mathcal{P}_{|w^+|}(v)$ which can find any address.
\end{definition}
The following is a consequence of \cite[Lemma 5.1]{BS:translation}.
\begin{lemma}
  \label{lem:Postal}
Let $S_1,\dots, S_N$ be $N$ compatible substitution rules on the prototiles $t_1,\dots, t_M$, and let $w = w^-w^+$ be a positively simple word. Assume that for any $x_*\in U_{i_*}^{r_x}\cap C([w^-.w^+])\subset\Sigma_N$ there is an $\ell\in\{1,\dots,d\}$ such that for any canonical level-$|w^+|$ supertile there are $k$ vectors $\tau_1,\dots, \tau_k\in \Lambda_{x_*,\ell}^{(0)}$, describing translation equivalences of tiles of type $\ell$ in any $\mathcal{P}_{|w^+|}(v)$, with the property that $\alpha_{x_*}(\tau_1),\dots, \alpha_{x_*}(\tau_{k})$ generate $\mathbb{Z}^{r_x}$. Then there exists a $C_{w}>0$ such that
  $$C_w \|z\|_{\mathbb{R}^{r_x}/\mathbb{Z}^{r_x}}\leq \max_{j\leq k} \left\|\left\langle\alpha_{x_*}(\tau_j),z\right\rangle\right\|_{\mathbb{R}/\mathbb{Z}}$$
  for any $z\in\mathbb{R}^{r_x}$.
\end{lemma}

Recall (\ref{eqn:vectorCocycle}), which gives, for $\lambda\in\mathbb{R}^d$ and $\tau\in\Lambda_{\sigma^{-n}(x)}$
\begin{equation}
  \begin{split}
    \left\langle \lambda, \theta_{(n)_x}^{-1}\tau\right\rangle &= \left\langle \lambda, \theta_{(-n)_{\sigma^{-n}(x)}}^{-1}\tau\right\rangle = \left\langle  \lambda,   V_{x} G^{(n)}_{\sigma^{-n}(x)}\alpha_{\sigma^{-n}(x)}(\tau)\right\rangle \\
    &= \left\langle  \alpha_{\sigma^{-n}(x)}(\tau),\bar{G}^{(n)}_x V_{x}^* \lambda \right\rangle
  \end{split}
\end{equation}
for the corresponding some $d\times r_x$ matrix $V_{x}$ and the address vector $\alpha_{\sigma^{-n}(x)}(\tau)\in\mathbb{Z}^{r_x}$. If $k_n\rightarrow \infty$ is a sequence of return times for $x$ to a cylinder set $C([w])$ under $\sigma^{-1}$, then $V_{\sigma^{-k_n}(x)} = V$ for some fixed $V$. In view of the previous Lemma, the goal now is to bound the quantity in Proposition \ref{prop:twistedBound} by something of the form
\begin{equation}
  \label{eqn:oftheForm}
  \begin{split}
    &\prod_{y'<n<N_\varepsilon(R)} \left( 1-c_w \max_{\tau\in \Lambda_w}\left\| \left\langle\lambda, \theta^{-1}_{(k_n+|w^+|)_x}\tau\right\rangle\right\|^2_{\mathbb{R}/\mathbb{Z}}  \right) \\
    &\hspace{2.1in}\leq \prod_{y'<n<N_\varepsilon(R)} \left( 1-c_w' \left\| \bar{G}^{(k_n+|w^+|)}_x V_{x}^*  \lambda    \right\|^2_{\mathbb{R}^{r_x}/\mathbb{Z}^{r_x}}  \right).
  \end{split}
\end{equation}

Suppose $x$ is a $\mu$-generic and Oseledets- and Furstenberg-Kesten-regular point for the respective cocycles, where $\mu$ is a positively simple, postal, ergodic measure. Let $r_\mu$ be the rank of $\Gamma_x$ for $\mu$-almost every $x$. Recall that by \S \ref{subsec:RV} there is a collection of subsets $\{U_i^{r_\mu}\}$ of $\Sigma_N$ with the property that for each $i$ there is a collection of vectors $v_1,\dots, v_{r_\mu}$ such that $\Gamma_x$ is generated by those vectors for any $x\in U_i^{r_\mu}$. By the hypotheses on $\mu$, without loss of generality it can be assumed that there is an $i$ such that $U_i^{r_\mu} = C([w^-.w^+])$, where $w = w^-w^+$ is a positively simple word and postal. As such, $\mu(U_i^{r_\mu})>0$, and set $U_\star = U_i^{r_\mu}$.

Let $k_n\rightarrow \infty$, $n\in\mathbb{N}$, be a maximal sequence of return times to $U_\star$ for $x$ under $\sigma^{-1}$. Since there is a local choice of generators $v_1,\dots, v_{r_\mu}$ for all $\Gamma_x$ with $x\in U_\star$, there is a natural identification of all fibers $\mathfrak{R}_x$ and their lattices $\alpha(\Gamma_x)\subset \mathfrak{R}_x$ for all $x\in U_\star$, and in particular for $\sigma^{k_n}(x)$. Under this identification, the vector space is denoted by $\mathfrak{R}_\star$ and the corresponding lattice is $\Gamma_\star\subset \mathfrak{R}_\star$. This also holds for the dual spaces and lattices $\Gamma_\star^*\subset \mathfrak{R}_\star^*$. As such, for $\rho\in(0,1/2)$, let $B_\rho(\Gamma_\star^*)$ be the $\rho$-neighborhood of the lattice $\Gamma^*_\star\subset\mathfrak{R}^*_\star$.
\begin{proposition}[Quantitative Veech criterion]
  \label{prop:Veech}
  Let $S_1,\dots, S_N$ be a collection of compatible substitution rules on tiles $t_1,\dots, t_M$ and $\mu$ a positively simple, postal, $\sigma$-invariant ergodic probability measure for which the trace and return vector cocycles are integrable. For any positively simple postal word $w = w^-w^+$ with $\mu(C([w]))>0$ there is a $c'_w\in(0,1)$ such that for any Oseledets- and Furstenberg-Kesten-regular $x\in\Sigma_N$, if there exists $\rho,\delta\in (0,\frac{1}{2})$ such that
  \begin{equation}
    \label{eqn:densityVeech}
    \frac{\displaystyle\sum_{j=1}^{N}\mathbbm{1}_{C([w^-.w^+])}\left(\sigma^{-j}(x)\right)\mathbbm{1}_{B_\rho(\Gamma^*_\star)}\left( \bar{G}^{(j)}_x V_{x}^*  \lambda\right)}{\displaystyle\sum_{j=1}^{N}\mathbbm{1}_{C([w^-.w^+])}(\sigma^{-j}(x))}<1-\delta
  \end{equation}

  for some $\lambda \in \mathbb{R}^d\backslash\{0\}$ and $N>N^*$ large enough, then for any Lipschitz function $f:\Omega_x\rightarrow \mathbb{R}$ and $\varepsilon>0$,
  $$\left|\int_{C_R(0)}e^{-2\pi i \langle \lambda, t\rangle} f\circ \varphi_t(\mathcal{T})\, dt\right|\leq C_{x,f,\varepsilon} R^{d+\frac{d\delta\log(1-c'_w\rho)(\mu(C[w])+\varepsilon)}{\eta_1+\varepsilon} +\frac{d}{\eta_1}\varepsilon}+\mathcal{O}(R^{d-1})$$
  for all $R>R_0(\varepsilon, N^*,x)$. Moreover, the lower local dimension of the spectral measure $\mu_f$ satisfies
  $$2\min\left \{1,- \frac{d\delta\log(1-c'_w\rho)\mu(C([w]))}{\eta_1}\right\}\leq d^-_f(\lambda).$$

\end{proposition}
This type of theorem goes by quantitative Veech criterion because it is a quantitative version of Veech's criterion in \cite{Veech:metric1}. There are other qualitative criteria for eigenvalues which are in the spirit to Veech's, e.g \cite{CDHM:eigenfunctions, BDM:eigenvalues}. The criterion here is a generalization to higher rank of the criterion in \cite{BS:translation}.
\begin{proof}
  Let $x\in\Sigma_N$ be an $\mu$-typical, Furstenberg-Kesten-regular point such that the conclusions of Proposition \ref{prop:twistedBound} hold. Since $\mu$ is ergodic, for all $\varepsilon>0$ there exists a $N_*>0$ such that for all $n>N_*$,
  $$\frac{n}{k_n}< \mu(C_w) + \varepsilon$$
  where $C_w:= C([w^-.w^+])$. Denote $\mu_w = \mu(C_w)$. By Lemma \ref{lem:LipApprox} it suffices to check (\ref{eqn:LipApp1}) for all level-$y$ TLC functions for $y>y_0(x,B)$ large enough.

  Let $f:\Omega_x\rightarrow \mathbb{R}$ be a level-$y$ TLC function, $\mathcal{T}\in\Omega_x$ and $\lambda\in\mathbb{R}^d$. If $R>z^{\max\{y,k_{N^*},N_*\}}$, where $z>1$ is given by Proposition \ref{prop:twistedBound}, then given $\varepsilon>0$
  \begin{equation}
    \label{eqn:VeechBnd}
    \begin{split}
      &|\mathcal{S}_R^\mathcal{T}(f,\lambda)|\leq C_{x,f,\varepsilon,y} R^{d+\varepsilon} \prod_{m_0<n<N_\varepsilon(R)} \left( 1-c_w \max_{\tau\in \Lambda_w}\left\| \left\langle\lambda, \theta^{-1}_{(k_n)_x}\tau\right\rangle\right\|^2_{\mathbb{R}/\mathbb{Z}}  \right) +\mathcal{O}(R^{d-1})\\
      &\hspace{1in}\leq C_{x,f,\varepsilon,y} R^{d+\varepsilon}\left(1-c_w'\rho\right)^{\delta N_{\varepsilon}(R)-m_0-2} +\mathcal{O}(R^{d-1}) \\
      &\hspace{.8in}\leq C'_{x,f,\varepsilon,y} R^{d+\varepsilon}\left(1-c_w'\rho\right)^{\delta  (\mu_w+\varepsilon) \left(\frac{d\log R -  \bar{K}_{\varepsilon,x}}{\eta_1+\varepsilon}\right)} +\mathcal{O}(R^{d-1}) \\
      &\hspace{.6in}= C'_{x,f,\varepsilon,y} R^{d+\varepsilon} R^{\frac{\delta d\log(1-c'_w\rho)(\mu_w+\varepsilon)}{\eta_1+\varepsilon}\left(1 - \frac{\bar{K}_{\varepsilon,x}}{d\log R}\right)} +\mathcal{O}(R^{d-1}) \\
      &\hspace{.4in}\leq C'_{x,f,\varepsilon,y} R^{d+\frac{d\delta\log(1-c'_w\rho)(\mu_w+\varepsilon)}{\eta_1+\varepsilon} +\frac{d}{\eta_1}\varepsilon} +\mathcal{O}(R^{d-1})
    \end{split}
  \end{equation}
  for all $R>z^{N^*}$ large enough, which depends only on $\varepsilon$, and where (\ref{eqn:oftheForm}) was used to obtain the second inequality. Now if
  $$-\frac{d\delta\log(1-c'_w\rho)(\mu_w+\varepsilon)}{\eta_1+\varepsilon} > 1,$$
then the $\mathcal{O}(R^{d-1})$ term dominates in (\ref{eqn:VeechBnd}) and so for any $\varepsilon>0$ the growth of the twisted integral is bounded by $D_\varepsilon R^{d-1+\varepsilon}$ for some $D_\varepsilon$. By Lemma \ref{lem:dimension}, since the $L^\infty$ norm dominates the $L^2$ norm, it follows from (\ref{eqn:VeechBnd}) that
  $$2\min\left\{ 1-\varepsilon, -\frac{d\delta\log(1-c'_w\rho)(\mu_w+\varepsilon)}{\eta_1+\varepsilon} - \frac{d}{\eta_1}\varepsilon\right\} \leq d_f^-(\lambda)$$
  for any $\varepsilon>0$.
\end{proof}
\section{Cohomology and deformations of tiling spaces}
\label{sec:cohomology}
Let $\mathcal{T}$ be an aperiodic, repetitive tiling of $\mathbb{R}^d$ of finite local complexity. This section reviews several aspects of cohomology for tiling spaces. See \cite{sadun:book} for a comprehensive introduction.
\subsection{Anderson-Putnam complexes}
For any tile $t$ in the tiling $\mathcal{T}\in\Omega$, the set $\mathcal{T}(t)$ denotes all tiles in $\mathcal{T}$ which intersect $t$. This type of patch is called a \textbf{collared tile}.
\begin{definition}
  \label{def:AP}
  Let $\Omega$ be a tiling space. Consider the space $\Omega\times \mathbb{R}^d$ under the product topology, where $\Omega$ carries the discrete topology and $\mathbb{R}^d$ the usual topology. Let $\sim_1$ be the equivalence relation on $\Omega\times\mathbb{R}^d$ which declares a pair $(\mathcal{T}_1,u_1)\sim_1 (\mathcal{T}_2,u_2)$ if $\mathcal{T}_1(t_1)-u_1 = \mathcal{T}_2(t_2)-u_2$ for some tiles $t_1,t_2$ with $u_1\in t_1\in\mathcal{T}_1$ and $u_2\in t_2\in\mathcal{T}_2$. The space $(\Omega\times\mathbb{R}^d)/\sim_1$ is called the \textbf{Anderson-Putnam (AP) complex} of $\Omega$ and is denoted by $AP(\Omega)$.
\end{definition}
Let us now review the AP-complexes involved in our construction. First, note that for $x  = (x_1,x_2,\dots)\in \Sigma_N$, assuming that $B_x$ is minimal, $AP(\Omega_x)$ only depends on finitely many symbols $x_1,\dots, x_\ell$, since the collaring of tiles in tilings of $\Omega_x$ only depends in the $k^{th}$-approximants $\mathcal{P}_k$ for sufficiently large $k$. Thus for a set of compatible substitution rules $S_1,\dots, S_N$, there exists a partition $U_1,\dots, U_q$ of $\Sigma_N$ by open sets and CW-complexes $\Gamma_1,\dots, \Gamma_q$ such that if $x\in U_i$ and $B_x$ is minimal, then $AP(\Omega_x) = \Gamma_i$.

It will be useful to also consider higher level AP complexes, defined as follows. For $x\in \Sigma_N$, $\mathcal{T}\in\Omega_x$, and denoting by $t^{(1)}$ a level 1 supertile (which are approximants of the form $\mathcal{P}_1(\bar{e})$ as in (\ref{eqn:approximant})), let $\mathcal{T}(t^{(1)})$ be the union of the supertile $t^{(1)}$ and the level 1 supertiles of $\mathcal{T}$ which intersect $t^{(1)}$. Proceeding similarly, we let $\mathcal{T}(t^{(n)})$ denote the \textbf{collared level-$n$ supertile} corresponding to the level-$n$ tupertile $t^{(n)}$. Let $\sim_n$ be the equivalence relation defined by the collared level-$n$ supertiles $\mathcal{T}(t^{(n)})$ in $\Omega$ as in Definition \ref{def:AP}. The quotient $(\Omega\times\mathbb{R}^d)/\sim_n$ is denoted by $AP^n(\Omega)$. By construction, $AP(\Omega_{\sigma^{-n}(x)})$ and $AP^n(\Omega_x)$ are homeomorphic and they are homeomorphic through scaling: $AP^n(\Omega_x)$ is a rescaling of $AP(\Omega_{\sigma^n(x)})$ by $\theta_{x_n}^{-1}\cdots \theta_{x_1}^{-1}$. Denote by
\begin{equation}
  \label{eqn:rescale}
  r_{k,x}:AP^k(\Omega_x)\rightarrow AP(\Omega_{\sigma^{-k}(x)})
\end{equation}
the rescaling homeomorphisms. The following summarizes well-known consequences of the ideas in \cite[\S 4- \S 6]{AP}.
\begin{proposition}
Let $S_1,\dots, S_N$ be a collection of uniformly expanding compatible substitution rules on the set of prototiles $t_1,\dots, t_M$ and suppose $\mathcal{B}_x$ is minimal for some $x\in\Sigma_N$.
  \begin{enumerate}
  \item The substitution rule $S_{x_i}$ induces a continuous map $\gamma_i:AP(\Omega_{\sigma^{-i}(x)})\rightarrow AP(\Omega_{\sigma^{-i+1}(x)})$ defined by $\gamma_i(\mathcal{T},u) = (S_{x_i}(\mathcal{T}),\theta_{x_i}^{-1}u)$ for all $i>0$.
  \item The tiling space $\Omega_x$ is an inverse limit of maps induced by the sequence of substitutions:
    \begin{equation}
      \label{eqn:invLim}
      \Omega_{x} = \varprojlim_{\gamma_i} AP(\Omega_{\sigma^{-i}(x)}).
    \end{equation}
  \item The \v Cech cohomology groups of $\Omega_{x}$ are obtained through the direct limits
    \begin{equation}
      \label{eqn:directLimit}
      \check{H}^i(\Omega_x;\mathbb{Z}) = \varinjlim_{\gamma_k^*}\check{H}^i(AP(\Omega_{\sigma^{-k}(x)});\mathbb{Z}).
    \end{equation}
  \end{enumerate}
\end{proposition}
\subsection{Pattern-equivariant cohomology}
Recall $\mathcal{T}$-equivariant functions from \S \ref{subsubsec:functions}. The set of $C^\infty$ $\mathcal{T}$-equivariant functions is denoted by $\Delta^0_\mathcal{T}$. More generally, a smooth $k$-form $\eta$ on $\mathbb{R}^d$ is $\mathcal{T}$-equivariant if all the functions involved are $\mathcal{T}$-equivariant. Let $\Delta_\mathcal{T}^k$ denote the set of $C^\infty$, $\mathcal{T}$-equivariant $k$-forms on $\mathbb{R}^d$. The usual differential operator of differential forms $d$ takes $\mathcal{T}$-equivariant forms to $\mathcal{T}$-equivariant forms. So $\{\Delta^k_\mathcal{T}\}_k$ is a subcomplex of the de-Rham complex of smooth differential forms in $\mathbb{R}^d$.
\begin{definition}
  The \textbf{pattern-equivariant cohomology} is defined as the quotient
  $$H^k(\Omega_\mathcal{T};\mathbb{R})=\frac{\mathrm{ker}\, d:\Delta^k_\mathcal{T}\rightarrow \Delta^{k+1}_\mathcal{T} }{\mathrm{Im}\, d:\Delta^{k-1}_\mathcal{T}\rightarrow \Delta^{k}_\mathcal{T}}.$$
\end{definition}
\begin{thm}[\cite{KP:RS}]
Let $\mathcal{T}$ be an aperiodic, repetitive tiling of $\mathbb{R}^d$ of finite local complexity. The \v Cech cohomology $\check H^*(\Omega_\mathcal{T};\mathbb{R})$ is isomorphic to the pattern-equivariant cohomology $H^*(\Omega_\mathcal{T};\mathbb{R})$.
\end{thm}
\subsection{Return vectors and cohomology}
\label{subsec:RetVecCoh}
Let $\tau\in\Lambda_x$ be a return vector. Consider a curve $\gamma_\tau:[0,1]\rightarrow \mathbb{R}^d$ which starts and ends on tiles of type $t_i$ such that $\gamma_\tau(0)-\gamma_\tau(1) = \tau$. The projection $\pi_0(\gamma_\tau)$ defines a cycle on $AP(\Omega)$. Note that this cycle is independent of which curve $\gamma_\tau$ was used to realize the return vector $\tau$. As such, a return vector $\tau$ defines a homology class $[\pi_0(\gamma_\tau)]\in H_1(AP(\Gamma);\mathbb{Z})$ whose dual $[\pi_0(\gamma_\tau)]^*\in H^1(AP(\Gamma);\mathbb{Z})$ defines a class in $H^1(\Omega;\mathbb{Z})$ by inclusion. Therefore there is a map
\begin{equation}
  \label{eqn:vectorMap}
  \beta_x:\Gamma_x^*\rightarrow \check H^1(\Omega_x;\mathbb{Z})
\end{equation}
taking $\tau = \sum_ia_i(\tau)v_i^*$ to $\beta(\tau) = \sum_ia_i(\tau)[\pi_0(v_i)]^*$, where the return vectors $\{v_i\}$ form a set of generators of $\Gamma$ and $a_i(\tau)\in\mathbb{Z}$.
\begin{lemma}
  \label{lem:betaMap}
The map (\ref{eqn:vectorMap}) is a surjective homomorphism.
\end{lemma}
\begin{proof}
  Let $c_1,\dots, c_{b_1}$ be a collection of cycles which generate $H^1(AP(\Omega);\mathbb{Z})$. Without loss of generality it can be assumed that each cycle $c_j$ is a concatenation of (oriented) edges $e_1^j,\dots, e_m^j$ of the $1$-skeleton of $AP(\Omega)$. Pick some $\mathcal{T}\in\Omega_x$ and take the starting poing $p_0$ of $e_1^j$, and look at a preimage $p^*\in\pi^{-1}_0(p_0)\subset\mathbb{R}^d$. Starting in $p^*$ there is a set of edges $\bar{e}_i^j$ of tiles in $\mathcal{T}$ which project to $e_i^j$ for all $i=1,\dots, m$. In other words, the path given by the cycle $c_j$ lifts to a path of edges of the tiling $\mathcal{T}$. Were this path to be closed, then it would project to a boundary in $AP(\Omega)$, which cannot happen since the class $c_j$ is nontrivial, and so the path $\bar{e}_1^j,\dots, \bar{e}_m^j$ of edges connects a vertex of a tile with the vertex of another tile \emph{of the same type}. So in fact the cycle $c_j$ lifts to a return vector in a canonical way and so the map $\beta$ is surjective on $H^1(AP(\Omega);\mathbb{Z})$. The same argument allows one to lift cycles which are dual to classes in $H^1(AP^k(\Omega);\mathbb{Z})$ for any $k$. By (\ref{eqn:rescale}) and (\ref{eqn:directLimit}), this means that $\beta$ is surjective, and so $\Gamma_x^*/\mathrm{ker}\, \beta \cong H^1(\Omega_x;\mathbb{Z})$.
\end{proof}
Unfortunately the map $\beta_x$ must not be injective: given (\ref{eqn:directLimit}), since any class in $H^1(AP^k(\Omega_{x});\mathbb{Z})$ is in the image of $\beta_x$ through the construction in the proof above, this means that any class in $\mathrm{ker}\,\gamma_k^*$ is also in the image, and thus there are could be non-zero elements of $\Gamma^*_x$ which are in the kernel of $\beta_x$.
\begin{corollary}
  The map $\beta_x$ induces a surjection from the space of dual return vectors $\mathfrak{R}_x^*$ to the first cohomology space $H^1(\Omega_x;\mathbb{R})$, making the return vector cocycle equivariant with the induced action on cohomology by the homeomorphism $\Phi^*_{\sigma^{-1}(x)}$:
  \begin{equation}
    \label{eqn:betaEquiv}
  \begin{tikzcd}
    \mathfrak{R}_x^*   \arrow{r}{\bar{G}_x} \arrow{d}{\beta_x} & \mathfrak{R}^*_{\sigma^{-1}(x)} \arrow{d}{\beta_{\sigma^{-1}(x)}}\\
    H^1(\Omega_x;\mathbb{R}) \arrow[swap]{r}{\Phi_{\sigma^{-1}(x)}^*}  & H^1(\Omega_{\sigma^{-1}(x)};\mathbb{R}) .
  \end{tikzcd}
  \end{equation}
  As such, any Oseledets decomposition of $\mathfrak{R}_x$ gives a $\Phi_{x}^*$-equivariant decomposition
  $$H^1(\Omega_x;\mathbb{R}) = Z^-_x\oplus Z^+_x := \beta_xY^-_x \oplus \beta_xY^+$$
  where $Z^+_x$ corresponds to classes with positive Lyapunov exponents, and  $Y^\infty_x\subset \mathrm{ker}\, \beta_x$.
\end{corollary}
\subsection{Deformations}
\label{subsec:deform}
Given an polyhedral, repetitive tiling $\mathcal{T}$ of finite local complexity, one can obtain a continuum of tilings $\{\mathcal{T}_{\Rd}\}_{_{\Rd}}$ which are close to $\mathcal{T}$ in some geometric sense. More precisely, the tiling $\mathcal{T}_{\Rd}$ is obtained from $\mathcal{T}$ by a deformation given by $\Rd$. What needs to be established is what the ranges are for the deformation parameter $\Rd$.

The basic idea to understand deformations of tilings $\mathcal{T}$ is to consider the geometric information contained in the geometric object $AP(\Omega_\mathcal{T})$. Not only does it contain information about the topology of the tiling space, it also contains basic geometric information, namely, the lengths and shapes of the edges of the tiles. More precisely, any edge of a tile in $\mathcal{T}$ projects to a 1-chain in $AP(\Omega_\mathcal{T})$, and the vector defined by such edge in $\mathcal{T}$ can be thought of as the value of a $\mathbb{R}^d$-valued 1-cochain evaluated on this edge. One can similarly assign a $\mathbb{R}^d$-valued 1-cochain to every edge in $AP(\Omega_\mathcal{T})$. To be consistent, one must impose that the values of such cochains sum up to zero around the edge of a 2-cell, i.e., that these $\mathbb{R}^d$-valued cochains are in fact $\mathbb{R}^d$-valued cocycles. As such, the geometry of the tiles in a tiling $\mathbb{T}$ is determined by an element $g_0\in H^1(AP(\Omega_\mathcal{T});\mathbb{R}^d)$. Picking other parameters $\Rd\in H^1(AP(\Omega_\mathcal{T});\mathbb{R}^d)$ close enough to $g_0$ gives a continuous family of deformed tilings $\mathcal{T}_{\Rd}$ which are geometrically close to $\mathcal{T}$, all of which have the same combinatorial properties and many of which have tiling spaces $\Omega_{\mathcal{T}_{\Rd}}$ which share similar dynamic properties as those of $\Omega_{\mathcal{T}}$.

One can deform further still by considering $\mathbb{R}^d$-valued 1-cochains in $AP^k(\Omega_\mathcal{T})$. The geometry of level-$k$ supertiles of $\mathcal{T}$ can be deformed by the same method, and so there is an element $g_k\in H^1(AP^k(\Omega_\mathcal{T});\mathbb{R}^d)$ which encodes the geometry of level-$k$ supertiles for all tilings in $\Omega_\mathcal{T}$. Thus, any element $\Rd\in H^1(AP^k(\Omega_\mathcal{T});\mathbb{R}^d)$ close enough to $g_k$ defines a deformation of tilings in $\Omega_{\mathcal{T}}$. The conclusion of all this is that, given (\ref{eqn:rescale}) and (\ref{eqn:directLimit}), the geometry of tilings in $\Omega_\mathcal{T}$ is defined by an element $g_\mathcal{T}\in H^1(\Omega_\mathcal{T};\mathbb{R}^d)=H^1(\Omega_\mathcal{T};\mathbb{Z})\otimes \mathbb{R}^d$ and so any $\Rd\in H^1(\Omega_\mathcal{T};\mathbb{R}^d)$ close enough to $g_\mathcal{T}$ defines a new family of tilings $\Omega^{\Rd}$. Questions about which sort of properties persist under which types of deformations have been studied before in \cite{ClarkSadun:shape, kellendonk:deformation, JulienSadun:deformations}. The goal here will be to deduce under which types of deformations do the systems become weakly mixing.

Going back to the random construction, let $S_1,\dots, S_N$ be a family of uniformly expanding compatible substitution rules on the prototiles $t_1,\dots, t_M$ and suppose $\mathcal{B}_x$ is minimal. Let $\Rd_x\in H^1(\Omega_x;\mathbb{R}^d)$ be the element defining the geometry of tilings in $\Omega_x$. Following \cite[\S 8]{JulienSadun:deformations}, for any $\Rd$ close enough to $\Rd_x$ and $\mathcal{T}\in\Omega_x$, a new deformed tiling $\mathcal{T}'$ is constructed as follows. Let $\alpha$ be a $\mathcal{T}$-equivariant representative of the class $\Rd$, that is, a $\mathcal{T}$-equivariant choice of linear transformations $\alpha(t):\mathbb{R}^d\rightarrow \mathbb{R}^d$.

Define $H_\alpha(s) = \int_{0}^s\alpha $ which, if $\alpha$ is pointwise close enough to the identity (i.e. $\Rd$ is close enough to $\Rd_x$), defines a homeomorphism.
Let $\Lambda_\mathcal{T}$ be the vertex set of $\mathcal{T}$ and consider $\Lambda_\mathcal{T}^{\Rd}$ obtained by applying $H_\alpha$ to each element of $\Lambda_\mathcal{T}$. If $v_1, v_2\in\Lambda_\mathcal{T}$ are two vertices in $\mathcal{T}$ which are joined by an edge, then the vertices $H_\alpha(v_1),H_\alpha(v_2)\in\Lambda^{\Rd}_\mathcal{T}$ are joined by an edge defined by the vector $\int_{v_1}^{v_2}\alpha$. This produces a new tiling $\mathcal{T}_{\Rd}$ whose patches are defined from patches in $\mathcal{T}$. Taking the orbit closure of this tiling one obtains $\Omega_x^{\Rd}$, the tiling space defined by this deformation. Furthermore, by construction, if $\mathcal{C}_\mathcal{T}$ is the set of control points of $\mathcal{T}$, then $\mathcal{C}_\mathcal{T}^{\Rd}:= H_{\alpha}(\mathcal{C}_\mathcal{T})$ is the set of control points of $\mathcal{T}_{\Rd}$.

The set of classes $\Rd\in H^1(\Omega_x;\mathbb{R}^d)$ for which there is a well-defined deformation as above are classified in \cite{JulienSadun:deformations} as follows. Let $C_\mu:H^1(\Omega_x;\mathbb{R}^d)\rightarrow M_{d\times d}$ be the asymptotic cycle defined by averaging representatives of classes:
$$C_\mu:[\alpha]\mapsto C_\mu([\alpha]):=\lim_{R\rightarrow \infty}\frac{1}{\mathrm{Vol}\, B_R(0)}\int_{B_R(0)}\alpha(t)\, dt,$$
for $[\alpha]= \Rd\in H^1(\Omega_x;\mathbb{R}^d)$. This cycle goes also by the name of the Ruelle-Sullivan map. Then $\Omega_x^{\Rd}$ is well-defined through the deformation $H_{\alpha}$ as above if and only if $C_\mu(\alpha)$ is invertible.
\begin{lemma}
  \label{lem:codimension}
The tiling space $\Omega_x^{\Rd}$ is well-defined for $\Rd\in H^1(\Omega_x;\mathbb{R}^d)$ outside a set of codimension 1 and contains tilings of finite local complexity.
\end{lemma}
\begin{proof}
The asymptotic cycle is a linear map on $H^1(\Omega_x;\mathbb{R}^d)$. Thus, the set of $\Rd$ such that $\mathrm{det}\, C_\mu(\Rd) = 0$ is a codimension-1 subset. By \cite[Theorem 8.1]{JulienSadun:deformations} elements $\Rd$ outside this set have a well-defined tiling space $\Omega_x^{\Rd}$ of tilings with finite local complexity.
\end{proof}

The choice of deformation parameter $\Rd$ will be restricted to the open submanifold of $H^1(\Omega_x;\mathbb{R}^d)$ defined by

\begin{equation}
  \label{eqn:moduli}
  \mathcal{M}_x = \left\{ \Rd\in H^1(\Omega_x;\mathbb{R}^d):\mathrm{det}\,C_\mu(\Rd)\neq 0 \right\}
\end{equation}
which carries a natural absolutely continuous measure inherited from the Lebesgue measure on the vector space $H^1(\Omega_x;\mathbb{R}^d)$.

Note immediately that $\mathcal{M}_x$ can be seen to contain a copy of $SL(d,\mathbb{R})$ as follows. Any element $\Rd\in H^1(\Omega_x;\mathbb{R}^d)$ is of the form $\Rd = c\otimes v$ with $c\in H^1(\Omega_x;\mathbb{R})$ and $v\in\mathbb{R}^d$. Let $\mathcal{T}\in\Omega_x$ and $g\in SL(d,\mathbb{R})$. Applying $g$ to the tiling $\mathcal{T}$ as a linear transformation of $\mathbb{R}^d$ preserves the volume of each tile as well as the frequency of the tile types. On individual elements $c\otimes v$, $SL(d,\mathbb{R})$ deforms by $c\otimes v\mapsto c\otimes (gv)$ for any $g\in SL(d,\mathbb{R})$. So the family of deformations given by $SL(d,\mathbb{R})$ can naturally be seen as elements of $\mathcal{M}_x$. It should be noted that the property of unique ergodicity is not altered through any deformations given by $\Rd\in \mathcal{M}_x$ and also that the renormalization maps $\Phi_x$ satisfy $\Phi_x(\Omega_x^{\Rd}) = \Omega_{\sigma(x)}^{\Rd_x^1}$ where $\Rd_x^1 = (\Phi_x^{-1})^*\Rd$ and, more generally,
\begin{equation}
  \label{eqn:equivDeform}
  \Phi_x^{(n)}(\Omega_x^{\Rd}) = \Omega_{\sigma^n(x)}^{\Rd_x^n}
\end{equation}
where $\Rd_x^n = (\Phi_x^{(n)*})^{-1}\Rd$.

Let $v_1,\dots, v_{r_x}\in \Lambda_x$ be a set of generators for $\Gamma_x$, and let $\{w_i\}$ be the dual generators for $\Gamma_x^*$. Putting the $v_i$ as columns, there is a $d\times r_x$ matrix $V_{x}$ such that any return vector $\tau$ satisfies $\tau = V_{x}\alpha_x(\tau)$ for a unique $\alpha_x(\tau)\in\mathbb{Z}^{r_x}$, as well as a dual matrix $V^*_{x}$. As such, $V_{x} \in \Gamma_x\otimes \mathbb{R}^d$, and the deformation $\Rd_x\mapsto \Rd$ induces a deformation of $V_{x}\mapsto V^{\Rd}_{x}$. Note that each matrix $V^{\Rd}_{x}\in \Gamma_x\otimes \mathbb{R}^d\cong \Gamma_x^*\otimes \mathbb{R}^d$. Thus, the map $\beta_x$ in (\ref{eqn:vectorMap}) induces a map $\beta_x:\Gamma^*_x\otimes \mathbb{R}^d \rightarrow H^1(\Omega_x;\mathbb{R}^d)$ which satisfies
\begin{equation}
  \label{eqn:shapeVector}
  \beta_x V^{\Rd *}_{x} = \Rd\in H^1(\Omega_x;\mathbb{R}^d).
\end{equation}
As such, $\Rd = \beta_x V^{\Rd *}_{x}$ can be identified with the element $\Rd \in \mathrm{Hom}(\mathbb{R}^d, H^1(\Omega_x;\mathbb{R}))$ as follows. By definition, an element $\Rd\in H^1(\Omega_x;\mathbb{R}^d)$ can be written as $c\otimes v$ with $c\in H^1(\Omega_x;\mathbb{R})$ and $v\in\mathbb{R}^d$, and this defines a map $\Rd:\lambda\mapsto c\langle v,\lambda\rangle \in H^1(\Omega_x;\mathbb{R})$.

From this point of view the deformation parameter $\Rd$ also parametrizes inclusions\footnote{A dimension count helps: the Grassmanian $G_d(H^1(\Omega_x;\mathbb{R}))$ has dimension $d(\mathrm{dim}\,H^1(\Omega_x;\mathbb{R})- d) = \mathrm{dim}\,H^1(\Omega_x;\mathbb{R}^d)- d^2 $, so an element $H^1(\Omega_x;\mathbb{R}^d)$ defines first a choice of $d$-dimensional subspace of $H^1(\Omega_x;\mathbb{R})$ (parametrized by the Grassmanian), and then a linear transformation inside this subspace, given by a $d\times d$ matrix.} of $\mathbb{R}^d$ into $H^1(\Omega_x;\mathbb{R})$. In \cite{ClarkSadun:shape}, $\Rd$ is called the \emph{shape vector}.

It follows that for $\Rd\in \mathcal{M}_x$ and $\tau\in \Lambda_x^{(0)}$ there is a corresponding vector $\tau^{\Rd}\in\Lambda^{(0)}_{x,\Rd}$ and thus for a generating set $\{\tau_i\}$ of $\Gamma_x$ there is a deformed generating set $\{\tau_i^{\Rd}\}$ of $\Gamma_x^{\Rd}$ and a corresponding matrix $V^{\Rd}_x$ satisfying $\tau^{\Rd} = V^{\Rd}_x \alpha_x(\tau)$ whenever $\tau = V_x \alpha_x(\tau)$. As such, recalling (\ref{eqn:equivDeform}), the deformed version of the return vector cocycle (\ref{eqn:vectorCocycle}) is
\begin{equation}
  \label{eqn:vectorCocycle2}
  \begin{split}
  &\theta_{(-n)_x}^{-1}V_{x}^{\Rd} =  V_{\sigma^n(x)}^{\Rd_x^n}G_{\sigma^{(n-1)}(x)}\cdots G_{\sigma(x)} G_x, \hspace{.5in}\mbox{ and thus } \\
&\theta_{(-n)_x}^{-1}\tau^{\Rd} = \theta_{(-n)_x}^{-1} V_{x}^{\Rd}\alpha_x(\tau) =  V_{\sigma^n(x)}^{\Rd^n_x} G_x^{(n)} \alpha_x(\tau) = V_{\sigma^n(x)}^{\Rd^n_x} \alpha_{\sigma^n(x)}\left(\theta_{(-n)_x}^{-1}\tau^{\Rd}\right).
  \end{split}
\end{equation}

Let $S_1,\dots, S_N$ be a collection of compatible substitution rules on tiles $t_1,\dots, t_M$ and suppose $\mu$ is a $\sigma-$invariant, ergodic probability measure on $\Sigma_N$ such that $\log\|\Phi_x^*\|\in L^1$, where $\Phi_x^*$ is the induced maps on $H^1(\Omega_x;\mathbb{R})$. Oseledets theorem gives a decomposition $H^1(\Omega_x;\mathbb{R}) = Z^+_x\oplus Z^-_x$, where $Z^+_x$ is the Oseledets space corresponding to positive Lyapunov exponents. Consider now the action $\Phi_{\sigma^{-1}(x)}^*:H^1(\Omega_x;\mathbb{R}^d)\rightarrow H^1(\Omega_{\sigma^{-1}(x)};\mathbb{R}^d)$. Since $\Phi_{\sigma^{-1}(x)}^*$ acts on the first component of $[\eta]\otimes v \in H^1(\Omega_x;\mathbb{R}^d) = H^1(\Omega_x;\mathbb{R})\otimes \mathbb{R}^d$, it follows that if
$$\frac{\log\|\Phi_x^*[\eta]\|}{n}\longrightarrow \omega_*\hspace{.6in}\mbox{ then }\hspace{.6in}\frac{\log\|\Phi_x^*([\eta]\otimes e_i)\|}{n}\longrightarrow \omega_*$$
for all basis vectors $e_i\in\mathbb{R}^d$. Thus there is an Oseledets decomposition of $H^1(\Omega_x;\mathbb{R}^d)$ as
$$H^1(\Omega_x;\mathbb{R}^d) = \hat{Z}^-_x\oplus \hat{Z}^+_x = Z^-_x \otimes \mathbb{R}^d\oplus \hat{Z}^+_x\otimes \mathbb{R}^d $$
obtained from the Oseledets decomposition of $H^1(\Omega_x;\mathbb{R})$. Note that $\mathrm{dim}\, \hat{Z}^+_x \geq d^2$, since $Z^+_x$ is always at least $d$-dimensional given that the constant forms $dx_1,\dots, dx_d$ represent $d$-linearly independent elements of $Z^+_x$. Finally, note that the inclusions $\Rd:\mathbb{R}^d\rightarrow H^1(\Omega_x;\mathbb{R})$ defined by $\lambda\mapsto \Rd(\lambda)$ as above are equivariant with respect to the induced maps on cohomology: if $\Rd = c\otimes v$, then
$$(\Phi_x^*\Rd)(\lambda) = [\Phi_x^*(c\otimes v)](\lambda)  = [\Phi_x^*(c)\otimes v](\lambda) = (\Phi_x^*c)\langle v,\lambda \rangle .$$
\section{A cohomological quantitative Veech criterion}
\label{sec:CQVC}

Prior to this section, especially in \S \ref{sec:renorm}-\ref{sec:criterion}, results related to twisted ergodic integrals were obtained for undeformed tiling spaces $\Omega_x$ given a tower structure defined by $x\in \Sigma_N$ through the random substitution tiling construction from \S \ref{sec:RandSub}. It would be sensible now to revisit some of those results, especially Propositions \ref{prop:TracesBound}, \ref{prop:twistedBound} and \ref{prop:Veech}, in light of the results on deformations from the previous section.
\subsection{Propositions \ref{prop:TracesBound} and \ref{prop:twistedBound} revisited}
Let $S_1,\dots, S_N$ be a collection of uniformly expanding compatible substitution rules on $M$ prototiles, $x\in\Sigma_N$, $\Omega_x$ be a tiling space (assuming $\mathcal{B}_x$ is minimal), and $\Rd\in\mathcal{M}_x$. The combinatorics of the tilings in $\Omega_x$ and those in $\Omega_x^{\Rd}$ are the same: for any level-$k$ supertile $\mathcal{P}$ for a tiling in $\Omega$ there is a corresponding deformed level-$k$ supertile $\mathcal{P}_{\Rd}$ for tilings in $\Omega_x^{\Rd}$ which is tiled by the same number of tiles which tile $\mathcal{P}$. In short, the combinatorics of $\Omega_x$ and $\Omega_x^{\Rd}$ are the same; what has changed is the geometry of the return vectors. That is, if $\Lambda^{(0)}_x$ is the set of return vectors for tilings in $\Omega_x$ (see \S \ref{subsec:retVectors}) and $\Lambda^{(0)}_{x,\Rd}$ is the set of return vectors for tilings in $\Omega_x^{\Rd}$, then $\Lambda^{(0)}_{x}\neq \Lambda^{(0)}_{x,\Rd}$, and so $\Gamma_x \neq \Gamma_x^{\Rd} := \mathbb{Z}[\Lambda^{(0)}_{x,\Rd}]$ and $\Gamma_x^{\Rd}$ is still finitely generated by finite local complexity (Lemma \ref{lem:codimension}).

Recall the family of elements $\{ a^k_{x,i,\lambda}\}$ in (\ref{eqn:diagonal}) of the LF algebra $LF(\mathcal{B}^+_x)$, where $k\geq 0$, $1\leq i \leq M$ and $\lambda\in\mathbb{R}^d$. As seen in the previous section, a deformation of $\Omega_x$ by $\Rd\in\mathcal{M}_x$ sends the control points $\mathcal{C}_\mathcal{T}$ of a tiling $\mathcal{T}\in\Omega_x$ to control points $H_{\alpha}(\mathcal{C}_\mathcal{T})$ of the tiling $\mathcal{T}_{\Rd}\in\Omega_x^{\Rd}$, where $[\alpha] = \Rd$. As such, since $\{ a^k_{x,i,\lambda}\}$ depended on control points, this gives a family of elements $\{ a^{k,\Rd}_{x,i,\lambda}\}$ for the deformed tiling space. Since Proposition \ref{prop:TracesBound} concerns bounding traces applied to the elements $\{ a^k_{x,i,\lambda}\}$, the deformed version of this will consider bounding traces applied to the elements $\{ a^{k,\Rd}_{x,i,\lambda}\}$. Note that the traces are independent of deformations.

Recall that $\tau\in \mathcal{W}^{x_k}_{i,j}\subset \Lambda^{(0)}_{\sigma^{-k}(x)}$ if and only if $\theta_{x_{k-1}}^{-1}\cdots \theta_{x_{1}}^{-1}\tau \in \mathcal{U}_{i,j,k}^x\subset \Lambda^{(k)}$. That is, $\tau$ describes how tiles are placed in level-$1$ supertiles of $\Omega_{\sigma^{-k}(x)}$ if and only if $\theta_{x_{k-1}}^{-1}\cdots \theta_{x_{1}}^{-1} \tau$ describes how level-$(k-1)$ supertiles lie in level-$k$ supertiles of $\Omega_x$. In the proof of Proposition \ref{prop:TracesBound}, the sets $\mathcal{U}_{i,j,k}^x$ were used in products $\langle\lambda, u \rangle$ for $u\in \mathcal{U}_{i,j,k}^x$ and $\lambda\in\mathbb{R}^d$. By (\ref{eqn:vectorCocycle2}), if $u = \theta_{(-k)_{\sigma^{-k}(x)}}^{-1}\tau $, in the undeformed space, then in the deformed space the relevant product is
\begin{equation}
  \label{eqn:DeformedProduct}
  \left\langle   \lambda, V_x^{\Rd} G^{(n)}_{\sigma^{-k}(x)} \alpha_{\sigma^{-k}(x)}(\tau) \right\rangle,
  \end{equation}
 where $\alpha_{\sigma^{-k}(x)}(\tau)$ is the address of $\tau$. With all of this in mind, the analogue of Proposition \ref{prop:TracesBound} in the deformed setting is the following proposition; the proof follows the proof of Proposition \ref{prop:TracesBound} line by line except that one needs to use the deformed products (\ref{eqn:DeformedProduct}) in the relevant places, e.g. (\ref{eqn:traceExp2}) - (\ref{eqn:traceExp3}).
\begin{prop*}

  Let $S_1,\dots, S_N$ be a family of uniformly expanding compatible substitution rules on the prototiles $t_1,\dots, t_M$ and a positively simple word $w = w^-w^+$. Suppose  $x\in\Sigma_N$ has infinitely many occurences of the word $w$, both in the future and the past. Denote by $k_n\rightarrow \infty$ the maximal sequence of return times of $x$ to $C([w^-.w^+])$ under $\sigma^{-1}$. Pick $\Rd\in\mathcal{M}_x$. Then there exists a $\ell\in\{1,\dots, M\}$, a finite set of vectors $\bar{\Lambda}_x\subset \mathbb{Z}^{r_x}$, $c>0$, and $k_x>0$ such that for any $m\in\mathbb{N}$
  \begin{equation}
    \label{eqn:twistBnd1}
    \begin{split}
      &\left|\mathfrak{t}^x_{m,j}(a^{m,\Rd}_{x,i,\lambda})\right|\\
      &\hspace{.4in}\leq \left\{
      \begin{array}{ll}
        \displaystyle\left\| \Theta_x^{(m)}\right\|\prod_{n<N}\left( 1 - c\max_{\tau\in \bar\Lambda_x} \| \left\langle\tau, \bar{G}^{(k_n+k_x)}V_x^{\Rd *}\lambda\right\rangle \|^2_{\mathbb{R}/\mathbb{Z}}\right) & \mbox{ if }\,k_{N}\leq m < k_N+k_x  \\
        \displaystyle\left\| \Theta_x^{(m)}\right\|\prod_{n \leq N}\left( 1 - c\max_{\tau\in \bar\Lambda_x} \|\left\langle\tau, \bar{G}^{(k_n+k_x)}V_x^{\Rd *}\lambda\right\rangle \|^2_{\mathbb{R}/\mathbb{Z}}\right) & \mbox{ if }\, k_N+k_x\leq m \leq k_{N+1}
      \end{array} \right.
    \end{split}
  \end{equation}
  for any $i,j$.
\end{prop*}
Given this, the following proposition follows in the same way that Proposition \ref{prop:twistedBound} follows from Proposition \ref{prop:TracesBound}.
\begin{proposition}
\label{prop:twistedBound2}
  Let $S_1,\dots, S_N$ be a collection of uniformly expanding compatible substitution rules on $M$ prototiles. Let $\mu$ be a positively simple, $\sigma$-invariant ergodic probability measure for which the trace cocycle is integrable. For every positively simple word $w = w^-w^+$ such that $\mu(C([w^-.w^+]))>0$ there exists a finite set of vectors $\Lambda_w\subset \mathbb{Z}^{r_x}$ and $c_w>0$ such that for any $\epsilon\in(0,1)$, for $\mu$-almost every $x\in\Sigma_N$ there is a subsequence $k_n\rightarrow \infty$, $z>1$ and $\bar{K}_{\epsilon,x}>0$ such that for any $\Rd\in\mathcal{M}_x$, any level-$y$ bounded TLC function $f:\Omega_x^{\Rd}\rightarrow \mathbb{R}$, $\mathcal{T}\in\Omega_x^{\Rd}$ and $\lambda\in\mathbb{R}^d$
  $$\left|\mathcal{S}^\mathcal{T}_R(f,\lambda)\right|\leq C_{x,f,\epsilon} R^{d+\epsilon} \prod_{m_0<n<N_\epsilon(R)} \left( 1-c_w \max_{\tau\in \Lambda_w}\left\| \left\langle\tau, \bar{G}^{(k_n+|w^+|)}V_x^{\Rd *}\lambda\right\rangle\right\|^2_{\mathbb{R}/\mathbb{Z}}  \right)+ \mathcal{O}(R^{d-1})$$
 for all $R>z^{k_{m_0}}$, and $N_\epsilon(R)$ is the largest return of $x$ to $C([w^-.w^+])$ which is less than $\frac{d\log R - \bar{K}_{\epsilon,x}}{\eta_1+\epsilon}$ and $m_0$ is the smallest integer which satisfies $k_{m_0}\geq y$.
\end{proposition}
Now Lemma \ref{lem:Postal} can be used as in the undeformed version, that is, the deformed analogue of (\ref{eqn:oftheForm}) is
\begin{equation}
  \label{eqn:oftheForm2}
  \begin{split}
    &\prod_{y'<n<N_\varepsilon(R)} \left( 1-c_w \max_{\tau\in \Lambda_w}\left\| \left\langle\tau, \bar{G}^{(k_n+|w^+|)}V_x^{\Rd *}\lambda\right\rangle\right\|^2_{\mathbb{R}/\mathbb{Z}}  \right) \\
    &\hspace{2.1in}\leq \prod_{y'<n<N_\varepsilon(R)} \left( 1-c_w' \left\| \bar{G}^{(k_n+|w^+|)}_x V_{x}^{\Rd *}  \lambda    \right\|^2_{\mathbb{R}^{r_x}/\mathbb{Z}^{r_x}}  \right)
  \end{split}
\end{equation}
provided that $\Lambda_w$ generates $\mathbb{Z}^{r_x}$.
\subsection{The criterion for deformed spaces}

For any positively simple postal word $w = w^-w^+$ with $\mu(C([w]))>0$, any $\rho,\delta\in(0,1/2)$, $\Rd\in\mathcal{M}_x$ and $\lambda\in\mathbb{R}^d$, define
\begin{equation}
  \label{eqn:density}
  \mathcal{D}_N^x(w,\rho,\Rd,\lambda):=  \frac{\displaystyle\sum_{j=1}^{N}\mathbbm{1}_{C([w^-.w^+])}\left(\sigma^{-j}(x)\right)\mathbbm{1}_{B_\rho\left(H^1(\Omega_{\sigma^{-j}(x)};\mathbb{Z})\right)}\left(\Phi_x^{(j)*}\Rd (\lambda)  \right)}{\displaystyle\sum_{j=1}^{N}\mathbbm{1}_{C([w^-.w^+])}(\sigma^{-j}(x))}.
\end{equation}
The quantity $\mathcal{D}_N^x(w,\rho,\Rd,\lambda)$ measures the density of the iterates of $\sigma^{-j}(x)$ up to $N$ which satisfy two simultaneous properties:
\begin{enumerate}
\item that the iterate on the base dynamics lands in the special compact set $C([w^-.w^+])$ for the positively simple word $w = w^-w^+$; and
\item the the fiber dynamics $\Phi^{(n)*}_x\Rd(\lambda)$ land at most $\rho$ from a point in the lattice $H^1(\Omega_{\sigma^{-j}(x)};\mathbb{Z})$.
\end{enumerate}
Since $x$ is taken to be $\mu$-generic and $\mu(C(w^-.w^+))>0$, the iterates $\sigma^{-j}(x)$ will visit the special compact set with asymptotic frequency $\mu(C(w^-.w^+))$. Thus the density quantity $\mathcal{D}_N^x(w,\rho,\Rd,\lambda)$ is really focusing on (ii) above, the behavior of the fiber dynamics.
\begin{lemma}
\label{lemma:OFK}
  Let $\mu$ be a $\sigma$-invariant ergodic probability measure such that $\log \|\Phi_x^*\|\in L^1$, where $\Phi_x^*$ is the induced action on the first cohomology. Suppose $x$ is Oseledets regular for the cohomology cocycle. Then it is Furstenberg-Kesten regular for the trace cocycle.
\end{lemma}
\begin{proof}
  The largest exponents for the cocycle on the first cohomology space correspond to the classes represented by the $d$ constant, pattern equivariant 1-forms $dx_1,\dots, dx_d$. The Lyapunov exponents are
  $$\lambda_{top} = \lim_{n\rightarrow\infty} \frac{\log |\theta_{x_{1}}^{-1}\cdots \theta_{x_{n}}^{-1}|}{n}$$
  which exists by assumption. The trace cocycle by definition tracks the number of prototiles in level-$n$ supertiles as $n$ grows, which is proportional to the volume of level-$n$ supertiles. Thus the top exponent of the trace cocycle is given by
  $$\lim_{n\rightarrow\infty} \frac{\log \left\|\Theta_x^{(n)}\right\|}{n} = \lim_{n\rightarrow \infty}\frac{\log |\theta_{x_{1}}^{-d}\cdots \theta_{x_{n}}^{-d}|}{n} = d \lambda_{top},$$
  so $x$ is Furstenberg-Kesten regular, and $\eta_1 = d\lambda_{top}$.
\end{proof}

The following is a quantitative, non-stationary version of \cite[Theorem 4.1]{ClarkSadun:shape}.
\begin{proposition}[Cohomological quantitative Veech criterion]
  \label{prop:CQVC}
  Let $S_1,\dots, S_N$ be a collection of compatible substitution rules on tiles $t_1,\dots, t_M$ and $\mu$ a positively simple, postal, $\sigma$-invariant ergodic probability measure for which the cocycle on $H^1$ is integrable. For any positively simple postal word $w = w^-w^+$ with $\mu(C([w]))>0$ there is a $c'_w\in(0,1)$ such that for any Oseledets-regular $x\in\Sigma_N$, if there exists $\rho,\delta\in (0,\frac{1}{2})$ and $\Rd\in H^1(\Omega_x;\mathbb{R}^d)$ such that $ \mathcal{D}_N^x(w,\rho,\Rd,\lambda) <1-\delta$
  for some $\lambda \in \mathbb{R}^d\backslash\{0\}$ and $N>N^*$ large enough, then there is a $\delta'>0$ such that for any Lipschitz function $f:\Omega_x^{\Rd}\rightarrow \mathbb{R}$, $\varepsilon>0$, and $\mathcal{T}\in \Omega_x^{\Rd}$,
  $$\left|\int_{C_R(0)}e^{-2\pi i \langle \lambda, t\rangle} f\circ \varphi_t(\mathcal{T})\, dt\right|\leq C_{x,f,\varepsilon} R^{d+\frac{d\delta'\log(1-c'_w\rho)(\mu(C[w])+\varepsilon)}{\eta_1+\varepsilon} +\frac{d}{\eta_1}\varepsilon}+\mathcal{O}(R^{d-1})$$
  for all $R>R_0(\varepsilon, N^*,x)$. Moreover, the lower local dimension of the spectral measure $\mu_f$ satisfies
  \begin{equation}
    \label{eqn:dimBnd}
    2\min\left \{1,- \frac{d\delta'\log(1-c'_w\rho)\mu(C([w]))}{\eta_1}\right\}\leq d^-_f(\lambda).
  \end{equation}

\end{proposition}
\begin{remark}
  By the proof of Lemma \ref{lemma:OFK}, the $\eta_1$ in the bound (\ref{eqn:dimBnd}) in Proposition \ref{prop:CQVC} can be replaced by $d\lambda_{top}$, leaving the bound entirely in terms of data from the cocycle on $H^1(\Omega_x;\mathbb{R})$.
\end{remark}
\begin{proof}
First, consider the density quantity in the deformed situation at the level of return vectors which is analogous to (\ref{eqn:densityVeech}):
  \begin{equation}
    \label{eqn:densityVeech2}
    \frac{\displaystyle\sum_{j=1}^{N}\mathbbm{1}_{C([w^-.w^+])}\left(\sigma^{-j}(x)\right)\mathbbm{1}_{B_\rho(\Gamma^*_\star)}\left( \bar{G}^{(j)}_x V_{x}^{\Rd *}  \lambda\right)}{\displaystyle\sum_{j=1}^{N}\mathbbm{1}_{C([w^-.w^+])}(\sigma^{-j}(x))}<1-\delta'.
  \end{equation}
Given Proposition \ref{prop:twistedBound2} and Lemma \ref{lemma:OFK}, the same calculation as in (\ref{eqn:VeechBnd}) implies that if the density condition (\ref{eqn:densityVeech2}) is satisfied for all $N$ large enough, then the quantitative weak mixing conclusion of Proposition \ref{prop:Veech} holds for the deformed tiling space $\Omega_x^{\Rd}$.

  By compactness, there exists a $C_{w,\mu}>1$ such that
  $$C_{w,\mu}^{-1}\|v\|_{\mathfrak{R}_\star^*}\leq \|\beta_x v\|_{H^1(\Omega_x;\mathbb{R})}\leq C_{w,\mu} \|v\|_{\mathfrak{R}_\star^*} $$
  for all $v\in \mathfrak{R}^*_x\backslash \mathrm{ker}\, \beta_x$ and $x\in C([w^-.w^+])\cap \mathrm{supp}\, \mu$. The equivariance in (\ref{eqn:betaEquiv}) establishes that
  $$\Phi_x^{(j)*}\Rd(\lambda) = \Phi_x^{(j)*}\beta_xV_x^{\Rd *}\lambda = \beta_x \bar{G}_x^{(j)}V_x^{\Rd *}\lambda,$$
from which it follows that the density condition $ \mathcal{D}_N^x(w,\rho,\Rd,\lambda) <1-\delta$ for all $N$ large enough implies the density condition in (\ref{eqn:densityVeech2}) for all $N$ large enough with $\delta' = \delta/C_{w,\mu}$, from which the conclusion follows.
\end{proof}
\section{The Erd\H{o}s-Kahane method \`a la Bufetov-Solomyak}
\label{sec:dimension}
In this section it is shown that the typical deformation parameter $\Rd$ satisfies the conditions in the cohomological quantitative Veech criterion (Proposition \ref{prop:CQVC}). The argument very closely follows that of Bufetov-Solomyak in \cite[\S 6]{BS:translation}, which they call the \emph{vector form of the Erd\H{o}s-Kahane method}, and it is a tool to bound the Hausdorff dimension of a set of bad values of $\Rd\in\mathcal{M}_x$. It should be noted that the case when $d=1$ is already done in \cite[\S 6]{BS:translation}, so what is relevant here is the case $d>1$.

In this section it is assumed that $\mu$ is a positively simple, postal, $\sigma$-invariant ergodic probability measure for which the return vector cocycle and the trace cocycle are integrable. Throughout $x$ will denote a $\mu$-generic and Oseledets-regular point (for both cocycles). Let $r_\mu$ be the rank of $\Gamma_x$ for $\mu$-almost every $x$. Recall that by \S \ref{subsec:RV} there is a collection of subsets $\{U_i^{r_\mu}\}$ of $\Sigma_N$ with the property that for each $i$ there is a collection of vectors $v_1,\dots, v_{r_\mu}$ such that $\Gamma_x$ is generated by those vectors for any $x\in U_i^{r_\mu}$. By the hypotheses on $\mu$, without loss of generality it can be assumed that there is an $i$ such that $U_i^{r_\mu} = C([w^-.w^+])$, where $w = w^-w^+$ is a positively simple word and postal. As such, $\mu(U_i^{r_\mu})>0$, and set $U_\star = U_i^{r_\mu}$.

Let $k_n\rightarrow \infty$, $n\in\mathbb{N}$, be a maximal sequence of return times to $U_\star$ for $x$ under $\sigma^{-1}$. Since there is a local choice of generators $v_1,\dots, v_{r_\mu}$ for all $\Gamma_x$ with $x\in U_\star$, there is a natural identification of all fibers $\mathfrak{R}_x$ and their lattices $\alpha(\Gamma_x)\subset \mathfrak{R}_x$ for all $x\in U_\star$, and in particular for $\sigma^{k_n}(x)$. Under this identification, the vector space is denoted by $\mathfrak{R}_\star$ and the corresponding lattice is $\Gamma_\star\subset \mathfrak{R}_\star$. This also holds for the dual spaces and lattices $\Gamma_\star^*\subset \mathfrak{R}_\star^*$.

Let $V_\star$ be the $ d\times r_\mu$ matrix associated with $U_\star$ as in (\ref{eqn:vectorCocycle}). That is, for any $x\in U_\star$ and $\tau\in \Lambda_x$ there is a unique $\alpha_\star(\tau)\in\Gamma_\star$ such that $\tau = V_\star \alpha_\star(\tau)$. For $\lambda\in\mathbb{R}^d$, let $K_x^n(\lambda)\in H^1(\Omega_x;\mathbb{Z})$ be the closest lattice point to $\Phi_x^{(n)*}\Rd(\lambda)$ and set
\begin{equation}
  \label{eqn:Kepsilon}
\Phi_x^{(n)*} \Rd( \lambda) = K_x^n(\lambda) + \varepsilon_x^n(\lambda),\;\;\;\; \|\varepsilon_x^n(\lambda)\| := \left\|\Phi_x^{(n)*}\Rd( \lambda) \right\|_{H^1(\Omega_x;\mathbb{R})/H^1(\Omega_x;\mathbb{Z})},
\end{equation}
where $\|\cdot\|_{V/G}$ is the shortest distance of an element $v$ in a vector space $V$ to a point in a lattice $G\subset V$. For any $B>1$, let
$$\mathcal{M}_x^B:= \{\Rd = c\otimes v \in\mathcal{M}_x: \|c\|\leq B\;\; \mbox{ and } \;\;   \|v\|\leq B\}$$
and furthermore, given $\rho,\delta>0, B>1$ and $x\in\Sigma_N$, let
\begin{equation}
  \begin{split}
    E_N^x(\rho,\delta, B) &:= \left\{\Rd(\lambda) \in H^1(\Omega_x;\mathbb{R}): \Rd\in \mathcal{M}_x^B,\;\; \lambda\in\mathcal{C}_B ,\; \mbox{ and }\;\mathcal{D}_N^x(w,\rho,\delta,\Rd,\lambda) <1-\delta\right\} \\
    \mathcal{E}_N(\rho,\delta, B) &:= \left\{\Rd\in\mathcal{M}_x^B:\, \mbox{ there is }\lambda\in\mathcal{C}_B\mbox{ such that } \Rd(\lambda) \in E^x_N(\rho,\delta, B)\right\}
  \end{split}
\end{equation}
  and set
  $$\mathfrak{B} = \mathfrak{B}(\rho,\delta,B):= \limsup \mathcal{E}_N(\rho,\delta,B) = \bigcap_{N_0>0}\bigcup_{N \geq N_0}\mathcal{E}_N(\rho,\delta, B).$$

  Denote by $K_n = K_x^n(\lambda)$ and $\varepsilon_n = \varepsilon_x^n(\lambda)$. Let $\Pi_x^+:H^1(\Omega_x;\mathbb{R})\rightarrow Z_x^+$ be the projection onto the unstable space of the fiber over $x$. Then
  $$\Phi_x^{(n)*}\Pi_{x}^+\Rd(\lambda) = \Pi_{\sigma^{-n}(x)}^+\Phi_x^{(n)*}\Rd(\lambda)  = \Pi^+_{\sigma^{-n}(x)}K_n + \Pi^+_{\sigma^{-n}(x)}\varepsilon_n $$
  and thus
  \begin{equation}
    \label{eqn:EK1}
    \Pi_{x}^+\Rd(\lambda) = \Phi_x^{(n)*-1}\left(\Pi^+_{\sigma^{-n}(x)}K_n\right) + \Phi_x^{(n)*-1}\left( \Pi^+_{\sigma^{-n}(x)}\varepsilon_n \right)
  \end{equation}
 Since $x$ is Oseledets regular, for any $\epsilon>0$ and all $n$ sufficiently large,
  $$\left\| \Phi_x^{(n)*-1} \cdot \Pi^+_{\sigma^{-n}(x)}\right\|\leq e^{-(\eta_*-\epsilon)n},$$
  where $\eta_* = \omega_{\sh_\mu^+}^+$ is the smallest positive Lyapunov exponent for the return vector cocycle (equivalently, the cocycle defined by the induced action on $H^1(\Omega_x;\mathbb{R})$ by the maps $\{\Phi_x^*\}$). Therefore, from (\ref{eqn:EK1}), since $\varepsilon_n<1$,
  \begin{equation}
    \label{eqn:EK2}
    \left\| \Pi_{x}^+\Rd(\lambda) - \Phi_x^{(n)*-1}\left(\Pi^+_{\sigma^{-n}(x)}K_n\right) \right\|< e^{-(\eta_*-\epsilon)n}
  \end{equation}
for all $n$ large enough.

Set
\begin{equation}
  \label{eqn:EK3}
  W_n = W_n(x):= \log \left\|  \Phi_{\sigma^{-n}(x)}^{*} \right\|,\;\;\;\;M_n:= (2+\exp(W_{n+1}))^{\beta_\mu} \;\;\mbox{ and }\;\; \rho_n:= \frac{1}{2(1+\exp(W_{n+1}))},
\end{equation}
where $\beta_\mu = \mathrm{dim}\,H^1(\Omega_x;\mathbb{R})$, which is constant for $\mu$-almost every $x$.

Define
$$ \bar{E}_N^x(\delta,B) := \left\{ (\Rd,\lambda) \in \mathcal{M}_x^B\times \mathcal{C}_B : \mathrm{card}\{n\leq N :\mbox{max}\{\|\varepsilon_n\|,\|\varepsilon_{n+1}\|\}\geq \rho_n\}<\delta N \right\},$$
and let $\mathbb{E}:\bar{E}_N^x(\delta,B) \rightarrow H^1(\Omega_x;\mathbb{R})$ be the evaluation map $(\Rd,\lambda)\mapsto \Rd(\lambda)\in H^1(\Omega_x;\mathbb{R}).$
\begin{lemma}
  \label{lem:badSets}
  For $\mu$-almost every $x$, given $\delta>0$
  $$E_N^x(\varrho,\delta/4,B)\subset \mathbb{E}( \bar{E}_N^x(\delta,B))$$
  for all $N$ large enough, where $\varrho$ depends on $\delta$.
\end{lemma}
\begin{proof}
  The following result, which combines Proposition 6.1 in \cite{BS:genus2} and Corollary 4.3 in \cite{BS:translation}, will be used.
  \begin{proposition}
    \label{prop:BSlargeDev}
  There exists a positive constant $L$ such that for $\mu$-almost every $x$, for any $\delta>0$ there is an $N_*$ such that
  $$\max\left\{ \sum_{n\in \Psi}W_{n+1}: \Psi\subset \{1,\dots, N\},\, |\Psi|\leq \delta N \right\}\leq L \log(1/\delta)\cdot \delta N.$$
for all $N>N_*$. Moreover, for any $C>0$
  $$\mbox{card}\{n\leq N:W_{n+1}>CL\cdot\log(1/\delta)\}\leq \frac{\delta N}{C}.$$
  \end{proposition}

  Let $K = 2L\log (1/ \delta)$, where $L$ is obtained from Proposition \ref{prop:BSlargeDev}, define $\varrho = (2+2e^K)^{-1}$, and suppose $\Rd(\lambda)\not\in\mathbb{E}( \bar{E}_N^x(\delta, B))$. Then there is a subset $\Psi$ of $\{1,\dots, N\}$ of cardinality greater than or equal to $\delta N$ such that $\max\{\|\varepsilon_n\|,\|\varepsilon_{n+1}\|\}\geq \rho_n$
  for all $n\in\Psi$. Since $\rho_n<\varrho$ is equivalent to $W_{n+1}>K$, by the second bound in Proposition \ref{prop:BSlargeDev}, there are no more than $\delta N /2$ integers $n\leq N$ for which $W_{n+1}>K$ for all $N$ large enough. Thus
  $$\mathrm{card}\{n\leq N: \max\{   \|\varepsilon_n\|,\|\varepsilon_{n+1}\|\}\geq \varrho \}\geq \frac{\delta N}{2},$$
  so there are no less than $\delta N /4$ integers $n\leq N$ with $\|\varepsilon_n\|>\varrho$, and so it follows that $\Rd(\lambda) \not\in E_N^x(\varrho,\delta/4,B)$.
\end{proof}
\begin{proposition}
  \label{prop:dimension}
  There exists a $\varrho>0$ such that for $\mu$-almost every $x$ and any $\epsilon_*>0$, for all $\delta>0$ small enough and any $B>1$:
  $$\mathrm{dim}(\mathfrak{B}_x(\varrho, \delta, B)) \leq \mathrm{dim}\,\mathcal{M}_x^B - \mathrm{dim}\, Z^+_x+d+\epsilon_*.$$
\end{proposition}
\begin{proof}
  In light of Lemma \ref{lem:badSets} it suffices to show that for all $\delta>0$ small enough, setting $\beta_{\epsilon_*} := \mathrm{dim}\,\mathcal{M}_x^B-\mathrm{dim}\,Z^+_x+d+\epsilon_* = d\cdot\mathrm{dim}\,Z^-_x + (d-1)\mathrm{dim}\,Z^+_x +d+\epsilon_*$, it follows that
  $$\mathrm{dim}(\mathfrak{B}'_x(\delta, B)\cap \mathcal{M}^B_x) \leq \beta_{\epsilon_*},\,\, \mbox{ where }\,\, \mathfrak{B}'_x(\delta,B):= \limsup\bar{\mathcal{E}}_N^x(\delta,B) = \bigcap_{N_0>0}\bigcup_{N \geq N_0}\bar{\mathcal{E}}_N^x(\delta,B)$$
  and
  \begin{equation}
    \label{eqn:projN}
    \begin{split}
      \bar{\mathcal{E}}_N^x(\delta,B):&= \left\{\Rd\in\mathcal{M}_x^B: \mbox{ there is a $\lambda\in\mathcal{C}_B$ such that $(\Rd,\lambda)\in\bar{E}_N^x(\delta,B)$} \right\}\\
      &=\pi_x(\bar{E}_N^x(\delta,B)),
    \end{split}
  \end{equation}
where $\pi_x$ is the natural projection map $\pi_x:\bar{E}_N^x(\delta,B)\rightarrow \bar{\mathcal{E}}_N^x(\delta,B)$ onto the $\mathcal{M}_x$-coordinate. In order to estimate the dimension of $\mathfrak{B}'_x$, it suffices to construct a cover of $\bar{E}^x_N(\delta,B)$ since the projection map $\pi_x:\bar{E}_N^x(\delta,B)\rightarrow \bar{\mathcal{E}}_N^x(\delta,B)$ is Lipschitz \cite[Corollary 2.4]{falconer:book}. By definition,
\begin{equation}
  \label{eqn:localProduct}
  \bar{E}_N^x(\delta,B)\subset  \Pi^-_x(\mathcal{M}_x^B) \times \Pi^+_x(\pi_x(\bar{E}_N^x(\delta,B)))\times \mathcal{C}_B.

\end{equation}
In slight abuse of notation, the set $\Pi^\pm_x(\bar{E}_N^x(\delta,B))$ will be short for $\Pi^\pm_x(\pi_x(E_N^x(\delta,B)))\times \mathcal{C}_B$. To produce the desired dimension estimates, one needs to find a cover of the set $\Pi^+_x(\bar{E}_N^x(\delta,B))$.

To this end, for any $c\in E^+_x \subset H^1(\Omega_x;\mathbb{R})$, $B>1, \epsilon>0$ and $A\subset \Pi^+_x \mathcal{M}_x^B\times\mathcal{C}_B$, define
\begin{equation}
  \label{eqn:locus}
  \begin{split}
    \mathbb{E}^{-1}_B(c) &:= \left\{ (\Rd,\lambda)\in \Pi^+_x\mathcal{M}_x^B\times \mathcal{C}_B: \mathbb{E}(\Rd,\lambda) = c\right\} \hspace{.2in}\mbox{ and } \\ \partial_\epsilon^B A &:= \left\{(\Rd,\lambda)\in \Pi^+_x\mathcal{M}_x^B\times\mathcal{C}_B :\mathrm{dist}\, ((\Rd,\lambda), A)\leq \epsilon \right\}
  \end{split}
\end{equation}
and note that $\mathrm{dim}\, \mathbb{E}^{-1}_B(c) \leq (d-1)d_\mu^+ +d $, where $d^+_\mu = \mathrm{dim}\, E^+_x$. Moreover, by compactness, for any $\eta>0$ there exists a $F^B_x>0$ such that
$$\Pi^+_xB_\eta(c)\subset \mathbb{E}(\partial_{F_x^B\eta}^B(\mathbb{E}^{-1}_B(c))).$$

\begin{comment}
  For $c\in E^+_x\subset H^1(\Omega_x;\mathbb{R})$ and $\lambda\in \mathbb{R}^d$ define
$$\mathcal{N}_x^B(c,\lambda) := \{\Rd\in \Pi_x^+ \mathcal{M}_x^B: \Rd(\lambda)=c\}$$
and note that since $\mathrm{dim}\, \Pi^+_x\mathcal{M}^B_x = \mathrm{dim}\, \hat{E}_x^+ - 1 = d\cdot  \mathrm{dim}\, E_x^+-1$,
\begin{equation}
  \label{eqn:dimensionNull}
  \mathrm{dim}\, \mathcal{N}_x^B(c,\lambda) \leq (d-1) \mathrm{dim}\, E_x^+-1 = (d-1)d_\mu^+-1.
\end{equation}
Therefore, since $\mathcal{M}_x^B$ is a bounded subset of $H^1(\Omega_x;\mathbb{R}^d)$, there exists a $K(B,x)$ such that for any $\eta>0$ the set $\mathcal{N}_x^B(c,\lambda)$ can be covered by $K(B,x)e^{\eta ((d-1)d_\mu^+) N}$ balls of radius $e^{-\eta  N}$.

For any $\lambda\in\mathbb{R}^d$, denote by $\mathbb{E}_\lambda:\mathcal{M}^B_x\rightarrow H^1(\Omega_x;\mathbb{R})$ the function defined by $\Rd\mapsto \mathbb{E}(\Rd,\lambda)\in H^1(\Omega_x;\mathbb{R})$ for any $\Rd\in\mathcal{M}_x^B$. For any $\epsilon>0$ and $c\in E^+_x$ set
$$\partial_\epsilon \mathcal{N}_x^B(c,\lambda) := \left\{ \Rd\in \Pi^+_x\mathcal{M}_x^B: \mathrm{dist}\, (\Rd,\mathcal{N}_x^B(c,\lambda))\leq \epsilon \right\}.$$
The map $\mathbb{E}_\lambda$ is linear and as such there is a $F_\lambda^B\geq 1$ such that for any $\eta>0$
$$B_\eta(c)\subset \mathbb{E}_\lambda(\partial_{F^B_\lambda\eta}\mathcal{N}_x^B(c,\lambda)).$$
In words, the image under $\mathbb{E}_\lambda$ of the $F_\lambda^B\eta$-neighborhood of $\mathcal{N}_x^B(c,\lambda)$ contains the $\eta$-neighborhood around $c$.

For $v,\lambda\in \mathbb{R}^d$ and $B,\epsilon>1$ define
$$\mathcal{Y}_v^B(\lambda):= \left\{\lambda'\in\mathcal{C}_B: \langle v,\lambda'\rangle = \langle v,\lambda\rangle\right\} \hspace{.2in}\mbox{ and }\hspace{.2in}\partial_\epsilon \mathcal{Y}_v^B(\lambda) := \{\lambda'\in\mathbb{R}^d: \mathrm{dist}\, (\lambda',\mathcal{Y}_v^B(\lambda))\leq \epsilon\}$$
and note that $\mathrm{dim} \mathcal{Y}_v^B(\lambda)\, = d-1$. For any $\Rd = c\otimes v\in\mathcal{M}_x^B$, denote by $\mathbb{E}_{\Rd}:\mathbb{R}^d\rightarrow H^1(\Omega_x;\mathbb{R})$ the function defined by $\lambda\mapsto \mathbb{E}(\Rd,\lambda)\in H^1(\Omega_x;\mathbb{R})$ for any $\Rd\in\mathcal{M}_x^B$. As such there is a $F_{v}^B>0$ such that for any $\eta>0$
$$B_\eta(c\langle v,\lambda\rangle)\subset \mathbb{E}_{\Rd}(\partial_{F_{v}^B\eta} \mathcal{Y}_v^B(\lambda)).$$

\end{comment}
Suppose $(\Rd,\lambda) = (c\otimes v,\lambda)\in \bar{E}_N^x(\delta,B)$, and let $K_n$ and $\varepsilon_n$ be the corresponding sequences as defined in (\ref{eqn:Kepsilon}). From (\ref{eqn:EK2}) it follows that $\Pi_x^+ \Rd(\lambda) = \Pi_x^+c\langle v,\lambda\rangle $ is in the ball of radius $e^{-(\eta_*-\epsilon)N}$ centered at $P_N(\Rd,\lambda):= \Phi_x^{(N)*-1}\left(\Pi^+_{\sigma^{-N}(x)}K_N\right)$, and so
\begin{equation}
  \label{eqn:coverCovers}
  \mathrm{dist}\left(\left(\Pi^+_x\Rd,\lambda\right),\mathbb{E}^{-1}_B(\{P_N(\Rd,\lambda)\}\right)\leq F_x^B e^{-(\eta_*-\epsilon)N}.
\end{equation}
Therefore $(\Rd,\lambda)$ is in one of at most $K_{B,x} \left(F_x^B e^{-(\eta_*-\epsilon)N}\right)^{-[(d-1)d_\mu^++d]}$ balls of radius $F_x^B e^{-(\eta_*-\epsilon)N}$ which cover $\mathbb{E}^{-1}_B(\{P_N(\Rd,\lambda)\})\subset \Pi^+_x\mathcal{M}^B_x\times\mathcal{C}_B$. The goal is now to consider all possible sequences of $\{K_n\}$ that can appear for any such $(\Rd,\lambda)\in \bar{E}_N^x(\delta,B)$, which will give all the possible sets of the form $\partial_{\epsilon_N}\mathbb{E}_B^{-1}(\{P_N(\Rd,\lambda)\})$ which will cover $\Pi^+_x(\bar{E}_N^x(\delta,B))$, for $\epsilon_N = e^{-(\eta_*-\epsilon)N}$.

Let $\Psi_N$ be the set of subsequences of $n\in \{1,\dots, N\}$ for which $\max\{\|\varepsilon_n\|,\|\varepsilon_{n+1}\| \}\geq \rho_n$, and note that each such subsequence has at most $\delta N$ elements in it and moreover that there are at most $\displaystyle \sum_{i<\delta N}\left(\begin{array}{c} N \\ i\end{array} \right)$ such sets. By \cite[Lemma 6.1]{BS:translation}, for a fixed $\Psi_N$ the number of subsequences $\{K_n\}$ is
  at most $\displaystyle \mathbb{M}_N:= \prod_{n\in\Psi_N}M_n$ times the number of possible starting values for $K_0$, which is bounded independently of $N$ by compactness since $\Rd\in\mathcal{M}_x^B$ and $\lambda\in\mathcal{C}_B$. By (\ref{eqn:EK3}) and Proposition \ref{prop:BSlargeDev}, since $M_n\leq 3^{\beta_\mu}e^{\beta_\mu W_{n+1}}$, for all $N$ sufficiently large,
  $$\mathbb{M}_N\leq K_B 3^{\beta_\mu \delta N}\exp\left( \beta_\mu \sum_{n\in\Psi_N}W_{n+1}\right)\leq K_B 3^{\beta_\mu\delta N}\exp(\beta_\mu L \log(1/\delta)(\delta N)).$$
  It follows from (\ref{eqn:EK2}) that the number of balls of radius $F_x^B e^{-(\eta_*-\epsilon)N}$ needed to cover $\Pi^+_x(\pi_x\bar{E}_N^x(\delta,B))$ is no more than
  \begin{equation*}
    \begin{split}
      K_B  \sum_{i<\delta N}\left(\begin{array}{c} N \\ i\end{array} \right)3^{\beta_\mu \delta N} &\exp(\beta_\mu L \log(1/\delta)(\delta N)) \\
        &\hspace{1in}\leq K_B\mathrm{exp}\, \left(\left( H(\delta)\log 2 + \delta\beta_\mu\log 3 + \beta_\mu L \delta \log(1/\delta)\right)N\right),
    \end{split}
  \end{equation*}
 where the right-hand side follows from the bound of the sum of binomial coefficients up to $\delta N$ being bounded by $2^{H(\delta)N}$, where $H(\delta)$ is the binary entropy function. Therefore, since $H(\delta)$ and $\delta\log(1/\delta)$ go to zero as $\delta\rightarrow 0^+$, there exists a $\delta_0$ such that $\delta\in(0,\delta_0)$ implies that that
    \begin{equation}
      \label{eqn:EK4}
      H(\delta)\log 2 + \delta\beta_\mu\log 3 + \beta_\mu L \delta \log(1/\delta)\leq \epsilon_*(\eta_*-\epsilon).
    \end{equation}
    The set $ \Pi^-_x(\mathcal{M}^B_x)$ can be covered with roughly $\exp( (\eta_*-\epsilon)(\beta_{\epsilon_*}-d-\epsilon_* - (d-1)d^+_\mu)N)$ balls of radius $e^{-(\eta_*-\epsilon)N}$, whereas by (\ref{eqn:locus})-(\ref{eqn:EK4}) the set $\Pi^+_x(E_N^x(\delta,B))$ can be covered by roughly $ \exp( (\eta_*-\epsilon)N[(d-1)d^+_\mu+d+\epsilon_*])$ balls of radius $e^{-(\eta_*-\epsilon)N}$. By (\ref{eqn:localProduct}) then, it follows that for all $N$ large enough, $\bar{E}_N^x(\delta,B)$ can be covered by roughly
    $$\exp(\eta_*-\epsilon)\beta_{\epsilon_*}N = \exp\left[(\eta_*-\epsilon)N( \mathrm{dim}\,\mathcal{M}_x^B +d- d^+_\mu +\epsilon_* ) \right]$$
    balls of radius $e^{-(\eta_*-\epsilon)N}$ for all large $N$. So the $\beta_{\epsilon_*}$-Hausdorff measure of the set $\limsup_N \bar{E}_N^x(\delta,B)$ is finite, which implies that $\mathrm{dim}(\mathfrak{B}'_x(\delta, B)) \leq \beta_{\epsilon_*}$, from which the result follows.
\end{proof}
\section{Some proofs}
\label{sec:proofs}
\begin{proof}[Proof of Theorem \ref{thm:main}]
Suppose for a Oseledets regular $x$ the deformation parameter $\Rd\not\in \mathfrak{B}_x(\rho,\delta,B)$ for some $\delta,\rho,B$. This implies that for all $N$ large enough $\Rd\not \in \mathcal{E}_N^x(\rho,\delta, B)$, which means that for all $\lambda\in \mathcal{C}_B$, $\Rd(\lambda)\not\in E_N^x(\rho,\delta,B)$, which means that for all $N$ large enough, $\mathcal{D}_N^x(w,\rho, \Rd, \lambda)<1-\delta$. If $\mathrm{dim}\, Z^+_x>d$ then the dimension of $\mathfrak{B}_x(\rho,\delta,B)$ is strictly less than $\mathcal{M}_x$, and so the density condition $\mathcal{D}_N^x(w,\rho, \Rd, \lambda)<1-\delta$ holds for almost every $\Rd$, for all $N$ large enough. The result then follows from the cohomological quantitative Veech criterion, Proposition \ref{prop:CQVC}.
\end{proof}
\begin{proof}[Proof of Corollary \ref{cor:main}]
  By considering a high power of a single substitution, it can be assumed by primitivity that each tile is subdivided into enough tiles so that there are enough translation vectors between tiles of the same type which generate the group of return vectors, and so postal words are trivial. Thus if $x\in\Sigma_N$ is fixed for the shift map and represents this substitution, the measure $\delta_x$ is a positively simple, postal ergodic probability measure. Thus, if the unstable space of $H^1(\Omega_x;\mathbb{R})$ with respect to the induced action by the substitution has dimension greater than $d$, Theorem \ref{thm:main} applies.
  \end{proof}

\begin{proof}[Proof of Lemma \ref{lem:dimension}]
  By definition:
  \begin{equation}
    \label{eqn:L2bound}
    \begin{split}
      \left\| \mathcal{S}_R(f,\lambda) \right\|^2 & = \left\langle\int_{C_R(0)} e^{-2\pi i \langle \lambda,\tau \rangle}f\circ\varphi_\tau(\mathcal{T})\, d\tau, \int_{C_R(0)} e^{-2\pi i \langle \lambda,s \rangle}f\circ\varphi_s(\mathcal{T})\, ds \right\rangle \\
      &= \int_\Omega  \int_{C_R(0)}\int_{C_R(0)} e^{-2\pi i \langle \lambda,\tau-s \rangle}f\circ\varphi_\tau(\mathcal{T})\, d\tau  f\circ\varphi_s(\mathcal{T})\, ds\,    d\mu(\mathcal{T}) \\
      &=  \int_{C_R(0)}  \int_{C_R(0)} e^{-2\pi i \langle \lambda,\tau-s \rangle}\langle f\circ\varphi_{\tau-s},f\rangle\, d\tau \, ds \\
      &=  \int_{C_R(0)}  \int_{C_R(0)} e^{-2\pi i \langle \lambda,\tau-s \rangle}\left(\int_{\mathbb{R}^d}e^{2\pi i \langle\tau-s,z\rangle}\, d\mu_f(z) \right)\, d\tau\, ds \\
      &=  \int_{\mathbb{R}^d} \int_{C_R(0)}  \int_{C_R(0)} e^{2\pi i \langle z-\lambda,s-\tau \rangle} \, d\tau\, ds\,  d\mu_f(z)\\
      &= \int_{\mathbb{R}^d}\prod_{i=1}^d\frac{\sin^2(2\pi (z_i-\lambda_i)R)}{\pi^2(z_i-\lambda_i)^2} \,  d\mu_f(z).
    \end{split}
  \end{equation}
  There exists a $C_2$ such that for $r = \frac{1}{2R}$,
  $$C_2R^2\leq \frac{\sin^2(2\pi w R)}{\pi^2w^2}$$
  for $w\in[-r,r]$. Since $B_r(\lambda)\subset C_r(\lambda)$, by (\ref{eqn:L2bound}), the hypothesis implies that
  \begin{equation}
    \begin{split}
      \mu_f(B_r(\lambda)) &\leq \mu_f(C_r(\lambda)) = \int_{C_r(\lambda)}\, d\mu_f(z) \leq\frac{1}{(C_2R)^{d}} \int_{C_r(\lambda)} \prod_{i=1}^d\frac{\sin^2(2\pi (z_i-\lambda_i)R)}{\pi^2(z_i-\lambda_i)^2} \,  d\mu_f(z)\\
      &\leq \frac{1}{(C_2R^2)^{d}} \int_{\mathbb{R}^{d}}\prod_{i=1}^d\frac{\sin^2(2\pi (z_i-\lambda_i)R)}{\pi^2(z_i-\lambda_i)^2} \, d\mu_f(z) = \frac{1}{(C_2R^2)^{d}} \|\mathcal{S}_R(f,\lambda)\|^2 \\
      & \leq \frac{C_1^2}{C_2^dR^{2d}} R^{2(d-\alpha)} \leq C r^{2\alpha}
    \end{split}
  \end{equation}
  for any $r<(2R_0)^{-1}$.
\end{proof}
\section{Uniform rates of weak mixing}
\label{sec:uniform}
Let $\Omega$ be a tiling space corresponding to repetitive aperiodic tilings of finite local complexity with a uniquely ergodic $\mathbb{R}^d$ action. For $f\in L^2$ of zero average, H\"older's inequality gives the bound
\begin{equation}
  \label{eqn:Holder}
  \left| \int_{C_R(0)}e^{2\pi i \langle\lambda,t\rangle}\langle f\circ \varphi_t, f\rangle \, dt \right|\leq \|f\|_{L^2}\left\|\mathcal{S}_R(f,-\lambda)\right\|_{L^2}.
\end{equation}
By the polarization identity, in order to bound correlations of the form $\langle f\circ \varphi_t,g\rangle$, it suffices to bound self-correlations of the form $\langle f\circ \varphi_t,f\rangle$. For such correllations, assuming $f$ has zero average and using that $\langle f\circ \varphi_t,f\rangle = \hat{\mu}_f(t)$ (Bochner's theorem),
\begin{equation}
  \label{eqn:correllation}
  \begin{split}
    \int_{C_R(0)}|\langle f\circ \varphi_t,f\rangle|^2\, dt &= \int_{\mathbb{R}^d}\mathbbm{1}_{C_R(0)} \langle f\circ \varphi_t,f\rangle \overline{\langle f\circ \varphi_t,f\rangle}\, dt \\
    &= \int_{\mathbb{R}^d}\int_{C_R(0)}e^{2\pi i \langle\lambda,t\rangle}\langle f\circ \varphi_t, f\rangle \, dt\, d\bar{\mu}_f(\lambda).
  \end{split}
\end{equation}
It follows that in order to obtain a rate of growth of the correllations (\ref{eqn:correllation}) then by (\ref{eqn:Holder}) one has to bound the twisted integrals of $f$ for all spectral parameters $\lambda\in\mathbb{R}^d$. For any $B>1$ this problem will be subdivided in several parts:
\begin{enumerate}
\item for $0< \|\lambda\|\leq B$, the control of the twisted integrals is obtained from the results of Theorem \ref{thm:main},
\item for $\|\lambda\| >B$, the twisted integrals will be bounded using integration by parts, assuming that the functions are sufficiently smooth in the leaf direction,
\item for $\lambda=0$, the twisted integrals will be bounded using the bounds on the growth of ergodic integrals of $f$, assuming that such behavior is known.
\end{enumerate}
More precisely, let
\begin{equation*}

    \mathcal{C}_B= \{\lambda \in \mathbb{R}^d: 0< \|\lambda\|\leq B\},\hspace{1in}  O_{B} = \{\lambda\in\mathbb{R}^d: \|\lambda\|> B \},

\end{equation*}
so by (\ref{eqn:correllation}) and (\ref{eqn:Holder}) it follows that
\begin{equation}
  \label{eqn:breakdown}
  \int_{C_R(0)}|\langle f\circ \varphi_t,f\rangle|^2\, dt \leq \|f\|_{L^2} \sum_{S\in\{\mathcal{C}_B, O_B,\{0\}\} }\int_{S} \left\|\mathcal{S}_R(f,-\lambda)\right\|_{L^2}\, d\bar{\mu}_f(\lambda).
  \end{equation}
Again, the bound for $\mathcal{C}_B$ is given by the results of Theorem \ref{thm:main}, so it remains to establish bounds for the other subsets of $\mathbb{R}^d$
\subsection{Bounds on $O_B$}
\label{subsec:O_B}
Let $f\in \RI_r(\Omega)$ for some $r>1$ and let $\lambda\in O_B$ for some $B>r$. Assume that $\ell$ is an index such that $|\lambda_\ell|>B$. The twisted integral can be written as
\begin{equation}
  \label{eqn:IBP}
  \begin{split}
    \mathcal{S}_R^{\mathcal{T}}(f,-\lambda) &= \int_{-R}^R\cdots \int_{-R}^R e^{2\pi i \langle\lambda,t\rangle}f\circ\varphi_{t}(\mathcal{T})\, dt \\
    &= \frac{1}{2\pi i \lambda_\ell}\int_{-R}^R\cdots \int_{-R}^R\left( \frac{\partial}{\partial t_\ell} e^{2\pi i \langle \lambda, t\rangle }\right)f\circ\varphi_{(t_1,\dots, t_d)}(\mathcal{T})\, dt_\ell\wedge \star dt_\ell ,

  \end{split}
\end{equation}
where $\star$ is the Hodge-$\star$ operator taking $1$-forms to $d-1$-forms. Doing integration by parts, it follows that
\begin{equation}
  \label{eqn:1iteration}
  \begin{split}
    \mathcal{S}_R^{\mathcal{T}}(f,-\lambda) & = \frac{1}{2\pi i \lambda_\ell}\left[ \int_{\partial_\ell^- C_R(0)} e^{2\pi i \langle \lambda, t\rangle}f\circ \varphi_{t}(\mathcal{T})\, \star dt_\ell - \int_{\partial_\ell^+ C_R(0)} e^{2\pi i \langle \lambda, t\rangle}f\circ \varphi_{t}(\mathcal{T}) \star dt_\ell \right]\\
  &  \hspace{1in}+ \frac{1}{2\pi i \lambda_\ell}\mathcal{S}_R(X_\ell f,-\lambda),
  \end{split}
\end{equation}
where $\partial^\pm_\ell C_R(0)$ are the two $d-1$ dimensional faces of $C_R(0)$ given when $\lambda_\ell = \pm R$. Therefore the twisted integral can be bounded by
$$\|\mathcal{S}_R(f,-\lambda)\|_{L^2} \leq \frac{1}{2\pi i |\lambda_\ell|}\left( 2(2R)^{d-1}\|f\|_{L^2} + \|\mathcal{S}_R(X_\ell f,-\lambda)\|_{L^2}\right).$$
Doing integration by parts $k$ times as in (\ref{eqn:IBP}) and (\ref{eqn:1iteration}),
$$\|\mathcal{S}_R(f,-\lambda)\|_{L^2} \leq \frac{1}{(2\pi  |\lambda_\ell|)^k} \|\mathcal{S}_R(X_\ell^k f,-\lambda)\|_{L^2} + (2R)^{d-1} \sum_{j=1}^k\frac{\|X_\ell^{j-1}f\|_{L^2} }{(\pi |\lambda_\ell|)^j} .$$
Thus, for $f\in \RI_r(\Omega)$, by taking $k\rightarrow \infty$ there exists a constant $C_{f,r}$ such that
$$\|\mathcal{S}_R(f,-\lambda)\|_{L^2} \leq C_{f,r} R^{d-1}$$
for all $R>1$ and any $\lambda\in\mathbb{R}^d$ with $\|\lambda\| > B>r$.
\subsection{Bounds for $\lambda=0$}
\label{subsec:0}
When $\lambda=0$ the twisted ergodic integrals $\mathcal{S}_R(f,\lambda)$ reduce to regular ergodic integrals. By \cite[Corollary 2.19]{horan:lyapunov}, the trace cocycle has a spectral gap, that is, for the largest two Lyapunov exponents $\lambda_1, \lambda_2$ it holds that $\lambda_1>\lambda_2$ and so by the main result of \cite{T:TTT}, for any $\varepsilon>0$ there exists a $C_\varepsilon$ such that ergodic integrals $\mathcal{S}_R^\mathcal{T}(f,0)$ for functions of zero average are bounded by
$$|\mathcal{S}_R^\mathcal{T}(f,0)|\leq C_{\varepsilon,f} \max\left\{ R^{d\frac{\lambda_2}{\lambda_1}+\varepsilon},R^{d-1+\varepsilon}\right\}$$
for all $R>1$.
\subsection{Proof of (\ref{eqn:avgCorr}) in Theorem \ref{thm:main2}}
Let $S_1,\dots, S_N$ be a collection of $N$ uniformly expanding compatible substitution rules on the $M$ prototiles $t_1,\dots, t_M$ of dimension $d$ and $\mu$ a $\sigma$-invariant positively simple, postal ergodic probability measure on $\Sigma_N$ such that $\log \|\Phi_x^*\|\in L^1_\mu$, where $\Phi^*_x$ is the map on the first cohomology of $\Omega_x$ induced by the shift, and assume that $\mathrm{dim}(E^+_x)>d$. Pick an Oseledets regular $x$ and $\Rd\in \mathcal{M}_x$ such that the conclusions of Theorem \ref{thm:main} hold. Let $f,g\in \RI_r(\Omega_x^{\Rd})$ and let $B>r$.

Theorem \ref{thm:main} gives bounds for $\lambda\in \mathcal{C}_B$ in (\ref{eqn:breakdown}). The bounds for $\lambda\in O_B$ in (\ref{eqn:breakdown}) follow from \S \ref{subsec:O_B}, whereas the bounds for $\lambda=0$ are given in \S \ref{subsec:0}. Therefore it has been established that
    \begin{equation}
      \label{eqn:avgCorr2}
  \int_{C_R(0)}|\langle f\circ \varphi_t,g\rangle|^2\, dt \leq C_{f,g,r,\varepsilon} R^{d-\alpha'_\mu+\varepsilon}
    \end{equation}
    for all $R>1$, where
    $$\alpha'_\mu = \min\left\{\alpha_\mu,d\left(1-\frac{\lambda_2}{\lambda_1}\right),1\right\},$$
    $\alpha_\mu$ is the constant given in Theorem \ref{thm:main}, and $\lambda_1>\lambda_2$ are the top two Lyapunov exponents for the trace cocycle. Finally, by using the Cauchy-Schwartz inequality in (\ref{eqn:avgCorr2}), (\ref{eqn:avgCorr}) follows.
\subsection{Proof of uniform bounds for twisted integrals in Theorem \ref{thm:main2}}Let $S_1,\dots, S_N$ be a collection of $N$ uniformly expanding compatible substitution rules on the $M$ prototiles $t_1,\dots, t_M$ of dimension $d$ and $\mu$ a $\sigma$-invariant positively simple, postal ergodic probability measure on $\Sigma_N$ such that $\log \|\Phi_x^*\|\in L^1_\mu$, where $\Phi^*_x$ is the map on the first cohomology of $\Omega_x$ induced by the shift, and assume that $\mathrm{dim}(E^+_x)>d$. Pick an Oseledets regular $x$ and $\Rd\in \mathcal{M}_x$ such that the conclusions of Theorem \ref{thm:main} hold, and so by the previous subsection, (\ref{eqn:avgCorr}) also holds. The argument here follows the argument of Venkatesh in \cite[Lemma 3.1]{venkatesh:sparse}.

First some estimates need to be worked out. For a function $f\in \RI_r(\Omega_x)$, $\mathcal{T}\in\Omega_x$, $t\in\mathbb{R}^d$ and $H>0$ define
$$D_H^\mathcal{T}(f,t) := \int_{C_H(0)}e^{-2\pi i \langle \lambda,x  \rangle}f\circ\varphi_{x+t}(\mathcal{T})\, dx,$$
and note that
$$D_H^\mathcal{T}(f,t)^2 = \int_{C_H(0)^2}e^{-2\pi i \langle \lambda,x+y  \rangle}f\circ\varphi_{x+t}(\mathcal{T}) f\circ\varphi_{y+t}(\mathcal{T})\, dx \, dy,$$
so
$$\int_{C_R(0)} D_H^\mathcal{T}(f,t)^2\, dt \leq \int_{C_H(0)^2} \int_{C_R(0)} (f\circ\varphi_{x} f\circ\varphi_{y})\circ\varphi_t(\mathcal{T})\, dt\, dx\, dy.$$
In addition, by \cite{T:TTT} and \cite[Corollary 2.19]{horan:lyapunov},
\begin{equation}
  \label{eqn:corBnd1}
  \begin{split}
    \int_{C_R(0)} (f\circ\varphi_x\cdot f\circ\varphi_y)\circ\varphi_t(\mathcal{T})\, dt &= (2R)^d \mu(f\circ\varphi_x\cdot f\circ\varphi_y) + \mathcal{O}\left(\max\left\{R^{d\frac{\lambda_2}{\lambda_1}},R^{d-1}\right\}\right) \\
   &= (2R)^d\langle f\circ\varphi_{x-y},f\rangle  + \mathcal{O}\left(\max\left\{R^{d\frac{\lambda_2}{\lambda_1}},R^{d-1}\right\}\right),
  \end{split}
\end{equation}
where $\lambda_1>\lambda_2$ are the top two Lyapunov exponents of the trace cocycle.
Now, the twisted integral of $D_H^{\mathcal{T}}(f,t)$ is
\begin{equation}
  \label{eqn:CS1}
  \begin{split}
    \int_{C_R(0)} e^{-2\pi i \langle \lambda,t \rangle} D_H^{\mathcal{T}}(f,t) \, dt &= \int_{C_H(0)}\int_{C_R(0)}e^{-2\pi i \langle\lambda, s+t\rangle}f\circ \varphi_{s+t}(\mathcal{T})\, dt\, ds \\
    &= \int_{C_H(0)}\int_{\varphi_{s}(C_R(0))} e^{-2\pi i \langle \lambda, z\rangle} f\circ \varphi_z(\mathcal{T})\, dz\, ds
  \end{split}
\end{equation}
after using Fubini and changing variables $z = s+t$. So the difference between the twisted integral in (\ref{eqn:CS1}) and $(2H)^d\mathcal{S}_R^\mathcal{T}(f,\lambda)$ is bounded as
\begin{equation}
  \label{eqn:CS2}
  \begin{split}
    &\left|  (2H)^{d}\int_{C_R(0)} e^{-2\pi i \langle\lambda, t\rangle }f\circ \varphi_t(\mathcal{T})\, dt -  \int_{C_R(0)} e^{-2\pi i \langle \lambda,t \rangle} D_H^{\mathcal{T}}(f,t) \, dt\right| \\
    & \hspace{.4in}= \left| \int_{C_H(0)}\int_{C_R(0)} e^{-2\pi i \langle \lambda, t\rangle} f\circ \varphi_t(\mathcal{T})\, dt - \int_{\varphi_s(C_R(0))}e^{-2\pi i \langle \lambda, t\rangle}f\circ \varphi_t(\mathcal{T})\, dt\, ds\right|\\
    &\hspace{.4in}\leq  (2H)^d \|f\|_\infty^d
  \end{split}
\end{equation}
since there are cancellations in the difference of the integrals. The twisted integral $\mathcal{S}_R^{\mathcal{T}}(f,\lambda)$ can now be bounded:
\begin{equation}
  \label{eqn:final1}
  \begin{split}
    \left|\mathcal{S}_R^{\mathcal{T}}(f,\lambda)\right| &\leq \left|\mathcal{S}_R^{\mathcal{T}}(f,\lambda) - \frac{1}{(2H)^d}\int_{C_R(0)} e^{-2\pi i \langle \lambda,t \rangle} D_H^{\mathcal{T}}(f,t) \, dt\right| + \frac{1}{(2H)^d}\left|\int_{C_R(0)} e^{-2\pi i \langle \lambda,t \rangle} D_H^{\mathcal{T}}(f,t) \, dt\right| \\
    &\leq \|f\|_\infty^d + \frac{1}{(2H)^d}\left|\int_{C_R(0)} e^{-2\pi i \langle \lambda,t \rangle} D_H^{\mathcal{T}}(f,t) \, dt\right|.
    \end{split}
\end{equation}
  Finally, to bound the last term in (\ref{eqn:final1}), let $\lambda^* = \max\left\{d\frac{\lambda_2}{\lambda_1},d-1\right\}$. Then:
\begin{equation}
  \label{eqn:CS3}
  \begin{split}
    &\left|\int_{C_R(0)} e^{-2\pi i \langle \lambda,t \rangle} D_H^{\mathcal{T}}(f,t) \, dt\right|^2 = \left|\left\langle e^{-2\pi i \langle\lambda, \cdot\rangle} D_H^{\mathcal{T}}(f,\cdot),1\right\rangle_{L^2(C_R(0))} \right|^2 \leq  (2R)^d\int_{C_R(0)}D_H^\mathcal{T}(f,t)^2\, dt \\
    &\hspace{1.8in}= (2R)^d \int_{C_R(0)}\int_{C_H(0)^2}e^{-2\pi i \langle\lambda,x+y\rangle}f\circ\varphi_{t+x}(\mathcal{T})\cdot f\circ\varphi_{t+y}(\mathcal{T}) \, dx\, dy\, dt \\
    &\hspace{1.6in}\leq (2R)^d \int_{C_H(0)^2} \left| \int_{C_R(0)}f\circ\varphi_{t+x}(\mathcal{T})\cdot f\circ\varphi_{t+y}(\mathcal{T}) \, dt\right| \, dx\, dy \\
    &\hspace{1.4in}\leq (2R)^d \int_{C_H(0)^2} \left| (2R)^d\langle f\circ\varphi_{x-y},f\rangle  + \mathcal{O}\left( R^{\lambda^*}\right)\right| \, dx\, dy \\
    &\hspace{1.2in}\leq (2R)^{2d} \int_{C_H(0)^2} \left| \langle f\circ\varphi_{x-y},f\rangle\right|\, dx\, dy  +C_\varepsilon  R^{d+\lambda^*+\varepsilon}(2H)^{2d} \\
    &\hspace{1in}\leq (2R)^{2d}C_{f,r,\varepsilon}^2 H^{2d-\alpha_\mu'+2\varepsilon} +C_\varepsilon  R^{d+\lambda^*+\varepsilon}(2H)^{2d} \\
    &\hspace{.8in}\leq C'_{f,r,\varepsilon}\left(R^{2d}H^{2d-\alpha_\mu'+2\varepsilon}+R^{d+\lambda^*+\varepsilon}H^{2d}\right),
  \end{split}
\end{equation}
where the first inequality follows from the Cauchy-Schwarz inequality, (\ref{eqn:corBnd1}) was used in the third inequality and (\ref{eqn:avgCorr}) in the fifth. Chosing $H = R^{\frac{d-\lambda^*-\varepsilon}{\alpha_\mu'-2\varepsilon}}$ it follows that $R^{2d}H^{-\alpha_\mu'+2\varepsilon}=R^{d+\lambda^*+\varepsilon}$, so dividing (\ref{eqn:CS3}) by $(2H)^{2d}$ and using in (\ref{eqn:final1}) with this choice of $H$:
\begin{equation*}
  \label{eqn:final2}
    \left|\mathcal{S}_R^{\mathcal{T}}(f,\lambda)\right| \leq \|f\|_\infty^d + C''_{f,r,\varepsilon}R^{\frac{d+\lambda^*+\varepsilon}{2}}
\end{equation*}
which completes the proof.
\begin{comment}
\section{Absence of absolute continuity for spectral measures}
This section is devoted to the proof of Theorem \ref{thm:main3}. The arguments essentially follow the ones of Bufetov-Solomyak adapted to our setting.

Recall the spectral cocycle $\mathbb{M}_{x}^k(\lambda)$ in (\ref{eqn:SpecCocycle}). For each $v\in\mathbb{R}^d$ define the upper Lyapunov exponent
$$\chi^+_{x,\lambda,v} := \limsup_{n\rightarrow \infty}\frac{\log\|\mathbb{M}_x^{(n)}(\lambda) v\|}{n}.$$
\begin{lemma}
  \label{lem:HofConverse}
  Suppose that for a fixed $\lambda\in\mathbb{R}^d$, $\alpha\in(0,2)$, $r_0\in(0,1)$ and $f\in L^2(\Omega_x)$,
  $$\mu_f(B_r(\lambda))<C r^\alpha$$
  for all $r\in(0,r_0)$. Then
  $$\|\mathcal{S}_R(f,\lambda)\|\leq C_3 R^{2(1-\alpha)}$$
  for all $R> r_0^{-\frac{2}{2-\alpha}}$.
\end{lemma}
\begin{proof}
  Recall (\ref{eqn:L2bound}) and for $R>0$ define $K_R:\mathbb{R}^d\rightarrow \mathbb{R}$ by
  $$K_R(z) = \prod_{i=1}^d\frac{\sin^2(2\pi z_iR)}{\pi^2 z_i^2}$$
  for any $z\in\mathbb{R}^d$ and note that
  $$K_R(z) \leq \left\{\begin{array}{ll} R^{2d} & \mbox{ if $|z_i|\leq \frac{1}{R}$ for all $i$,}\\ \frac{1}{\pi z}&\mbox{ if $|z_i|> \frac{1}{R}$ for all $i$,} \end{array}\right. .$$
  It follows that
  $$\|\mathcal{S}_R(f,\lambda)\|\leq R^{2d}\mu_f(C_{1/R}(\lambda))+\frac{1}{\pi^2}\int_{\mathbb{R}^d\backslash C_{1/R}(\lambda)}\prod_{i=1}^d\frac{1}{(\tau_i-\lambda_i)^2}\, d\mu_f(\tau).$$
  For the second term:
  \begin{equation}
    \label{eqn:weirdterm}
    \begin{split}
      \int_{\mathbb{R}^d\backslash C_{1/R}(\lambda)}\prod_{i=1}^d\frac{1}{(\tau_i-\lambda_i)^2}\, d\mu_f(\tau) &=  \int_{C_{R^2}(\lambda)}\\
    \end{split}
  \end{equation}
\end{proof}
It remains to show that for every $\epsilon>0$
$$\limsup_{r\rightarrow 0}\frac{\mu_f(B_R(\lambda))}{r^{2-\gamma+\epsilon}},$$
where $\gamma = 2\chi^+_{x,\lambda, \bar{\psi}(\lambda)}/\lambda_1$. If this does not hold then there exists an $\epsilon_0>0$ such that $\mu_f(B_r(\lambda))\leq r^{2-\gamma+\epsilon_0}$ for all $r$ small enough. Therefore, by Lemma \ref{lem:HofConverse},
\begin{equation}
  \|\mathcal{S}_R(f,\lambda)\|^2\leq R^{\gamma-\epsilon_0}
\end{equation}
for all $R>0$ large enough.
\vspace{3in}
Let $\mathcal{T}\in\Omega_x$ and $g = \sum_i g_i:\mathbb{R}^d\rightarrow \mathbb{R}$ be a level-0 $\mathcal{T}$-equivariant function, so there exists a function $f:\Omega_x\rightarrow \mathbb{R}$ such that $g(t) = f\circ\varphi_t(\mathcal{T}) = \sum_i f_i\circ\varphi_t(\mathcal{T})$. Assume that for every $i$ the function $g_i$ has support in the interior of the union of tiles of type $i$ in $\mathcal{T}$. The correlation function $\langle f,f\circ \varphi_t\rangle $ can be expressed as
$$\langle f,f\circ \varphi_t\rangle  = \sum_{i,j} \langle f_i,f_j\circ \varphi_t\rangle =\sum_{i,j} \hat{\mu}^f_{ij}(t),$$
where the $\mu_{ij}^f$ are the spectral measures associated to the correlations $\langle f_i,f_j\circ \varphi_t\rangle$. Because each $f_i$ has support in the interior of tiles of type $i$, there exists a collection of functions $h_{ij}$ of compact support and countable sets $\Lambda_{ij}$ such that
$$\hat{\mu}^f_{ij} = \sum_{\tau\in \Lambda_{ij}} h_{ij}* \delta_\tau.$$
The collections $\Lambda_{ij}$ are the set of all vectors $\tau$ which connects the control point of a tile of type $i$ with a control point of a tile of type $j$.
\end{comment}
\appendix
\section{Twisted cohomology for tiling spaces}
\label{sec:plotTwist}
Let $\bar{\Delta}^k_\mathcal{T}$ the set of smooth ($C^\infty$), $\mathbb{C}$-valued $\mathcal{T}$-equivariant $k$-forms. Let $\eta\in\Delta^1_\mathcal{T}$ be a representative of a class $[\eta] \in H^1(\Omega_\mathcal{T};\mathbb{R})$ and define the operator $d_\eta:\bar{\Delta}^k_\mathcal{T}\rightarrow \bar{\Delta}^{k+1}_\mathcal{T}$
$$d_\eta:\alpha\mapsto d_\eta\alpha:= d\alpha - 2\pi i \eta\wedge \alpha. $$
It is immediate to check that
$$d^2_\eta\alpha = (d - 2\pi i \eta\wedge )(d\alpha - 2\pi i \eta\wedge \alpha) = -2\pi i d\eta\wedge \alpha = 0,$$
making $(\bar{\Delta}^k, d^2_\eta)$ a cochain complex with well-defined cohomology.
\begin{definition}
  For any real, closed $\eta\in\Delta^1$, the \textbf{twisted cohomology spaces of $\Omega$} are defined as the quotients
  $$H_\eta^k(\Omega;\mathbb{C}) := \frac{\mathrm{ker}\, d_\eta:\bar{\Delta}^k\rightarrow \bar{\Delta}^{k+1}}{\mathrm{Im}\, d_\eta:\bar{\Delta}^{k-1}\rightarrow \bar{\Delta}^k }.$$
\end{definition}
Any $\lambda\in\mathbb{R}^d$ defines a natural constant, closed, $\mathcal{T}$-equivariant $1$-form $\sum \lambda_i\, dx_i$. For any such constant form the twisted cohomology is denoted by $H_\lambda^*(\Omega;\mathbb{C})$.
\begin{proposition}
  \label{prop:Fact1}
  \begin{enumerate}
  \item The first twisted cohomology $H^1_\eta(\Omega;\mathbb{C})$ is defined, up to unitary equivalence, by the cohomology class $[\eta]\in H^1(\Omega;\mathbb{R})$.

  \item For a constant 1-form $\lambda\in \mathbb{R}^d$, the twisted differential $d_\lambda$ satisfies $d_\lambda = U_\lambda^{-1} d U_\lambda$, where $U_\lambda:\Delta^k_\mathcal{T}\rightarrow \Delta_\mathcal{T}^k$ is the twisted multiplication operator $U_\lambda:\omega(x)\mapsto e^{-2\pi i \langle\lambda, x\rangle}\omega(x)$ \textbf{which does not preserve pattern-equivariance}.

  \end{enumerate}
\end{proposition}
The first item in the proposition is essentially due to Forni \cite[Lemma 4.2]{forni:twist}.
\begin{proof}
  For the first claim, consider two closed 1-forms $\eta, \eta'\in\Delta^1_\mathcal{T}$ such that $\eta-\eta' = df$ for some $f\in\bar\Delta_\mathcal{T}^0$. Note that $e^{-2\pi i f}\in\bar\Delta^1_\mathcal{T}$, and so for any form $\omega\in\bar{\Delta}_\mathcal{T}^k$:
  $$d_\eta(e^{-2\pi i f}\omega) = e^{-2\pi i f}(-2\pi i df \wedge \omega  + d\omega - 2\pi i \eta \wedge \omega) = e^{-2\pi i f} d_{\eta'}\omega.$$
  This means that if $\omega$ is $d_{\eta'}$-closed, then $e^{-2\pi i f}\omega$ is $d_\eta$-closed. Thus multiplication by $e^{-2\pi i f}$ sends the $d_{\eta'}$-closed, pattern equivariant forms to the $d_{\eta}$-closed pattern equivariant forms. Similarly, multiplication by $e^{-2\pi i f}$ gives a bijection between $d_{\eta'}$-exact pattern equivariant forms and $d_\eta$-exact pattern equivariant forms.

  For the second claim, a straight-forward computation shows that $d(U_\lambda\omega) = U_\lambda d_\lambda \omega$ for any $\omega\in \Delta_\mathcal{T}^k$, from which the claim follows.

\end{proof}

\begin{proposition}
 Let $\eta\in\Delta^1$ be a real, closed 1-form. Then $H^0_\eta(\Omega;\mathbb{C})$ is non-trivial if and only if $\eta$ is a constant 1-form $\sum_i\lambda_i \,dx_i$ for some $\lambda\in\mathbb{R}^d$ which is an eigenvalue for the $\mathbb{R}^d$ action on $\Omega$, in which case it is one-dimensional.
\end{proposition}
\begin{proof}
  For $\lambda\in\mathbb{R}^d$, let $\eta$ be the constant form given by $\lambda$. As such for $f\in\bar\Delta^0_\mathcal{T}$,
  $$d_\eta f = \sum_{k=1}^d (\partial_k f - 2\pi i f\lambda_k)\, dx_k = 0$$
  implies that $\partial_k f = 2\pi i\lambda_k f $ for all $k$. For $k=j$, this differential equation has solution
  $$f(x_1,\dots, x_k) = C_j \exp\left(2\pi i \lambda_jx_j  \right),$$
  where $C_j$ does not depend on $x_j$. This implies that
  $$f(x) = C e^{2\pi i \langle \lambda,x\rangle},$$
which makes it an eigenfunction of the $\mathbb{R}^d$ action on $\Omega$ by the correspondence (\ref{eqn:imap}). Since the multiplicity of any dynamical eigenvalue is one, $H^0_\eta(\Omega;\mathbb{C})$ is one-dimensional.

  Alternatively, suppose $\eta  = \sum_k\eta_k\,dx_k$ is not constant and suppose $\eta_j$ is not constant. Let $f = R+iI$ be a smooth function. If $f$ is $d_\eta$-closed:
  $$d_\eta f = dR - 2\pi i R\eta + i(dI-2\pi i I \eta) = 0,$$
  then $dR = -2\pi i I\eta$ and $dI = 2\pi i R\eta$, from which it follows that
  \begin{equation}
    \begin{split}
      d_\eta\bar{f} &= dR - 2\pi i R\eta - i(dI-2\pi i I \eta)\\
      &= -2\pi i I\eta - 2\pi i R\eta - i(2\pi i R\eta-2\pi i I \eta) \\
      &= -2\pi i (I+R)\eta + 2\pi( R \eta -  I \eta) \\
      &= 2\pi (R-iI - I - iR)\eta = 2\pi (\bar{f}-i\bar{f})\eta \\
      &= 2\pi (1-i)\eta\bar{f}.
    \end{split}
  \end{equation}
Putting this together with the definition of $d_\eta$, it follows that $d\bar{f} = 2\pi \eta \bar{f} $, and therefore also that $\partial_k \bar{f} = 2\pi \bar{f} \eta_k$ for all $k$. Also, by definition,
  $$d_\eta f = \sum_{k=1}^d (\partial_k f - 2\pi i f\eta_k)\, dx_k = 0$$
  implies that $\partial_k f = 2\pi i f \eta_k$ for all $k$. For $k=j$, this differential equation has solution
  \begin{equation}
    \label{eqn:difEq}
    f(x_1,\dots, x_k) = C_j \exp\left(2\pi i \int_0^{x_j} \eta_j(x_1,\dots, z_j, \dots, x_d)\, dz_j\right),
  \end{equation}
  where $C_j$ does not depend on $x_j$. Similarly, the differential equation $\partial_j \bar{f} = 2\pi \bar{f} \eta_j$ has solution
  $$\bar{f}(x_1,\dots, x_k) = C'_j \exp\left(2\pi \int_0^{x_j} \eta_j(x_1,\dots, z_j, \dots, x_d)\, dz_j\right),$$
  where $C'_j$ does not depend on $x_j$. Taking the complex conjugate in (\ref{eqn:difEq}), it follows that
  $$C'_j \exp\left(2\pi \int_0^{x_j} \eta_j(x_1,\dots, z_j, \dots, x_d)\, dz_j\right)  = C_j \exp\left(-2\pi i \int_0^{x_j} \eta_j(x_1,\dots, z_j, \dots, x_d)\, dz_j\right) .$$

  It follows that, as a function of $x_j$,
  $$\mathrm{Const.} = \exp\left(2\pi (1+i) \int_0^{x_j} \eta_j(x_1,\dots, z_j, \dots, x_d)\, dz_j\right), $$
which contradicts that $\eta_j$ is not constant. So there is no solution to $d_\eta f = 0$ and $H^0_\eta(\Omega;\mathbb{C})$ is trivial.
\end{proof}
 Given a pattern equivariant $d$-form $\omega\in\bar\Delta^d_\mathcal{T}$, one would like to know when the cohomological equation
\begin{equation}
  \label{eqn:CEsolution}
  d_\lambda u = \omega,
\end{equation}
can be solved, that is, when we can find a $u\in\bar\Delta^{d-1}_\mathcal{T}$ which satisfies $d_\lambda u = \omega$.

Let $\sum_i \lambda_i \,dx_i \in \Delta^1$ be a constant 1-form and, for $R>0$, $\Upsilon^\lambda_R:\bar\Delta^d_\mathcal{T}\rightarrow \mathbb{C}$ the one-parameter family of currents defined by
\begin{equation}
  \label{eqn:current}
  \Upsilon^\lambda_R (\omega) := \mathcal{S}^\mathcal{T}_R(i^{-1}_\mathcal{T}\star \omega,\lambda) = \int_{C_R(0)}U_\lambda \omega.
\end{equation}
\begin{proposition}
\label{prop:CEsolution}
Let $u\in\bar\Delta^{d-1}_\mathcal{T}$ be a pattern-equivariant solution of (\ref{eqn:CEsolution}). There exists $C'>0$ such that the spectral measure $\mu_f$ of $f = i_\mathcal{T}^{-1}\star \omega$ satisfies
  $$\mu_f(B_r(\lambda))\leq C'r^{2}$$
  for all $r<1/2$. In particular, the lower local dimension satisfies $2\leq d^-_{i^{-1}_\mathcal{T}\star \omega}(\lambda)$. Therefore, if (\ref{eqn:CEsolution}) has a solution and $\lambda$ is an eigenvalue for the system, then $f=\star \omega$ lies in the orthogonal complement of $h_\lambda$, where $h_\lambda$ is an eigenfunction with eigenvalue $\lambda$:
  $$\star d_\lambda\Delta^{d-1}_\mathcal{T}\subset \langle h_\lambda\rangle^\perp.$$
\end{proposition}
\begin{proof}
  Recall by Proposition \ref{prop:Fact1} that $d_\lambda = U^{-1}_\lambda dU_\lambda$, so supposing that $d_\lambda\eta = \omega$,
  \begin{equation}
    \begin{split}
      \Upsilon_R^\lambda(\omega) &= \Upsilon_R^\lambda(d_\lambda\eta) = \Upsilon_R^\lambda(U^{-1}_\lambda d U_\lambda\eta) = \int_{C_R(0)}U_\lambda U_\lambda^{-1} d U_\lambda \eta \\
      &= \int_{C_R(0)} d U_\lambda \eta = \int_{\partial C_R(0)} U_\lambda \eta.
    \end{split}
  \end{equation}
  It follows that
  $\left| \Upsilon_R^\lambda(\omega) \right|\leq \|\eta\|_\infty (2R)^{d-1}$ for all $R>0$. The pointwise dimension estimates follow from (\ref{eqn:current}) and Lemma \ref{lem:dimension}.
\end{proof}

\begin{question}
What are the twisted cohomology groups $H^*_\eta(\Omega_x;\mathbb{C})$ of a given tiling space $\Omega_x$ constructed from substitutions?
\end{question}
\section{On the structure of return vectors}
\label{app:1}
The goal here is to relate the return vector cocycle to the cocycle given by the substitution matrices. The following is due to Kenyon; see \cite[Theorem 5.1]{SolomyakEigenfunctions}.
\begin{thm}
  Let $\mathcal{T}$ be a self-similar tiling of $\mathbb{R}^d$ with uniform expansion $\theta$, which is an algebraic integer. Then there exists a basis $\{b_1,\dots, b_d\}$ of $\mathbb{R}^d$ such that
  \begin{equation}
    \label{eqn:kenyon}
    \Lambda_\mathcal{T}\subset \mathbb{Z}[\theta]b_1+\cdots + \mathbb{Z}[\theta]b_d.
    \end{equation}

\end{thm}
In this context, $\theta>1$ is the Perron-Frobenius eigenvalue of the substitution matrix $F$.

If $\tau = (\tau_1,\dots, \tau_d)\in\Lambda_\mathcal{T}\subset\mathbb{R}^d$, then $\theta\tau\in \Lambda_\mathcal{T}$, so by (\ref{eqn:kenyon}) it follows that if $\tau_i = p(\theta)b_i$ for some polynomial $p(\theta)\in \mathbb{Z}[\theta]$, then
$$(\theta\tau)_i =\theta\tau_i= \theta p(\theta)b_i,$$
which can be represented as the action of the companion matrix $C_S$ on the minimal polynomial $p_S$ of $S$. If the substitution matrix is derogatory\footnote{That is, its minimal and characteristic polynomials are the same.}, the companion matrix $C_S$ is in fact similar to $S$ \cite[\S 3.3]{HJ:Matrix}, so they have the same spectral properties. In general, a matrix is similar to a matrix made up of blocks of companion matrices, each of which is a companion matrix to a polynomial which divides the characteristic polynomial of $S$.

To summarize, the action of the substitution on the set of return vectors is given by the action of the companion matrix on each of the coordinates and which, after a change of coordinates, is described by the substitution matrix (whenever the substitution matrix is derogatory). Moreover the substitution matrix and the action on the group of return vectors described by the companion matrix have the same spectrum.

The goal now is to prove a statement in the non-stationary setting analogous to Kenyon's theorem. In what follows, if $\theta_i$ is the contraction constant for the substitution rule $S_i$, let $\bar{\theta}_i = \theta_i^{-1}$ be its expansion.
\begin{theorem}
\label{thm:Kenyon2}
  Let $S_1,\dots, S_N$ be $N$ uniformly expanding compatible substitutions on the prototiles $t_1,\dots, t_M\subset \mathbb{R}^d$ with expansion constants $\bar{\theta}_1,\dots, \bar{\theta}_N > 1$. Let $\mu$ be a minimal, $\sigma$-invariant ergodic probability measure on $\Sigma_N$. Then for $\mu$-almost every $x\in\Sigma_N$ there exists a basis $\{b_1,\dots, b_d\}$ of $\mathbb{R}^d$ such that
  \begin{equation}
    \label{eqn:kenyonExpanded}
    \Lambda_x\subset \mathbb{Z}[\bar\theta_1,\bar\theta_2,\dots, \bar\theta_N]b_1+\cdots + \mathbb{Z}[\bar\theta_1,\bar\theta_2,\dots, \bar\theta_N]b_d.
    \end{equation}
\end{theorem}
Note that
$$\mathrm{rank}\, \mathbb{Z}[\bar\theta_1,\bar\theta_2,\dots, \bar\theta_N] \leq \prod_{i=1}^N \mathrm{rank}\, \mathbb{Z}[\bar\theta_i].$$

\begin{proof}
  The proof here follows Solomyak's proof in \cite[Theorem 5.1]{SolomyakEigenfunctions} for the stationary case. The hypotheses imply that the set of points $x$ which are recurrent, have $r_x = r_\mu$ and have $\mathcal{B}_x$ minimal is a set of full $\mu$ measure. Let $x\in\Sigma$ be one such point. By (\ref{eqn:CtrlPt1}), minimality, and (\ref{eqn:CtrlPt3}), it suffices to assume that $\mathcal{T} = \Delta_x(\bar{e})$ for any maximal choice path $\bar{e}\in X_c^x$ and it suffices to show that
  \begin{equation}
    \label{eqn:sufficient}
    \mathcal{C}_\mathcal{T}\subset \mathbb{Q}(\bar\theta_1,\dots, \bar\theta_N)e_1 + \cdots + \mathbb{Q}(\bar\theta_1,\dots, \bar\theta_N)e_d
    \end{equation}
  for some basis $\{e_1,\dots, e_d\}$ of $\mathbb{R}^d$. Indeed, since $\mathbb{Z}(\mathcal{C}_\mathcal{T}) = \Gamma_x = \mathbb{Z}(\Lambda_x)$, (\ref{eqn:kenyonExpanded}) follows from (\ref{eqn:sufficient}).

  Let $k_n\rightarrow \infty$ be a maximal sequence of recurrent times such that $\Gamma_{\sigma^{k_n}(x)} = \Gamma_x$ for all $n\geq 0$ as guaranteed by Corollary \ref{cor:2}.  Let $\mathcal{T}\in\mho_x$ and pick a set $\{e_1,\dots, e_d\}\subset \mathcal{C}_\mathcal{T}$ which spans $\mathbb{R}^d$. Define the vector space
  $$V_{\mathcal{C}_\mathcal{T}} := \mbox{span}_{\mathbb{Q}(\bar\theta_1,\dots, \bar\theta_N)}\,\mathcal{C}_\mathcal{T}$$
  over the field $\mathbb{Q}(\bar\theta_1,\dots, \bar\theta_N)$. The goal now is to show that $\{e_1,\dots, e_d\}$ is a basis for $V_{\mathcal{C}_\mathcal{T}}$.

  Let $\pi$ be a projection from $V_{\mathcal{C}_\mathcal{T}}$ to  $\mbox{span}_{\mathbb{Q}(\bar\theta_1,\dots, \bar\theta_N)}\, \{e_1,\dots ,e_d\}$. Let $\kappa=(\kappa_1,\dots, \kappa_N)\in\mathbb{Z}^N$ be a multi-index and define $\bar\theta^{\kappa} = \bar\theta_1^{\kappa_1}\cdots \bar\theta_N^{\kappa_N}$. Since $V_{\mathcal{C}_\mathcal{T}}$ is a vector space over $\mathbb{Q}(\bar\theta_1,\dots, \bar\theta_N)$,
  \begin{equation}
    \label{eqn:Sol1}
    \pi(\bar\theta^\kappa x) = \bar\theta^\kappa\pi(x)\hspace{.4in}\mbox{ for any }\hspace{.4in}x\in V_{\mathcal{C}_\mathcal{T}},\;\; \kappa\in\mathbb{Z}^N.
  \end{equation}
  Define
  $$\mathcal{C}_\mathcal{T}^{\infty} := \bigcup_{n\geq 0} \theta_{(-k_n)_x}\mathcal{C}_{\Phi_x^{(k_n)}(\mathcal{T})}.$$
which is a dense subset of $\mathbb{R}^d$.
  \begin{lemma}
$\mathcal{C}_\mathcal{T}^{\infty} \subset V_{\mathcal{C}_\mathcal{T}}$
  \end{lemma}
  \begin{proof}
    Since $\Phi^{\pm(\pm k)}_x(\mathcal{T})\in\mho_{\sigma^{\pm k}(x)}$ for all $k$, it follows that $ \mathcal{C}_{\Phi_x^{(k_n)}(\mathcal{T})}\subset \Gamma_{\sigma^{k_n}(x)} = \Gamma_x$  for all $n\geq 0$. Thus $\theta_{(-k_n)_x} \mathcal{C}_{\Phi_x^{(k_n)}(\mathcal{T})}\subset  \theta_{(-k_n)_x} \Gamma_x \subset V_{\mathcal{C}_\mathcal{T}}$.
  \end{proof}
Define
  \begin{equation}
    \label{Sol2}
f(x) = \pi(x) \hspace{.8in}\mbox{ for all }\hspace{.8in} x\in\mathcal{C}_\mathcal{T}^{\infty}
    \end{equation}
and let $\|\cdot \|$ be any norm on $\mathbb{R}^d$.
  \begin{lemma}[\cite{Lagarias:GeomMod1}]
    There is a $L_1>0$ such that
    $$ \|\pi(\tau_1)-\pi(\tau_2)\|\leq L_1\|\tau_1-\tau_2\|$$
    for all $\tau_1,\tau_2\in\mathcal{C}_\mathcal{T}$.
  \end{lemma}
  By the Lemma above and (\ref{eqn:Sol1}), it follows that $f$ is Lipschitz on $\mathcal{C}^\infty_\mathcal{T}$ with constant $L_1$. It is extended by continuity to all of $\mathbb{R}^d$ and note that it satisfies
  \begin{equation}
    \label{eqn:Sol2}
    f(\theta^\kappa x) = \theta^\kappa f(x)
  \end{equation}
  for any $x\in\mathbb{R}^d$.
  \begin{lemma}
    \label{lem:Sol1}
    Suppose that $t,\varphi_\tau(t)\in\mathcal{T}$ for some $\tau\in\Lambda_i^\mathcal{T}$. If $\omega\in t$, then
    \begin{equation}
      \label{eqn:Sol3}
f(\omega - \tau) = f(\omega) - \pi(\tau).
    \end{equation}
  \end{lemma}
  \begin{proof}
    Note that $\pi(\tau)$ is defined since $\Lambda_i^\mathcal{T} \subset \mathcal{C}_\mathcal{T} - \mathcal{C}_\mathcal{T}\subset V_{\mathcal{C}_\mathcal{T}}$. The goal now is to show that (\ref{eqn:Sol3}) holds on a dense set.    Let $\omega = \theta_{(-k_n)_x}\omega'$ with $\omega'\in\mathcal{C}_{\Phi_x^{(k_n)}}(\mathcal{T})\subset \Gamma_{\sigma^{k_n}(x)}=\Gamma_x = \langle \mathcal{C}_{\mathcal{T}}\rangle$. So $\omega'$ is the linear combination of elements of $\mathcal{C}_\mathcal{T}$. Then as in \cite[Lemma 5.4]{SolomyakEigenfunctions}:
    \begin{equation*}
      \begin{split}
        f(\omega-\tau) &= f(\theta_{(-k_n)_x}\omega'  - \tau )= f(\theta_{(-k_n)_x}(\omega'  - \bar\theta_{(-k_n)_x} \tau) )= \theta_{(-k_n)_x}\pi(\omega'  - \bar\theta_{(-k_n)_x} \tau )\\
        &= \theta_{(-k_n)_x} \pi(\omega')  - \theta_{(-k_n)_x}\pi(\bar\theta_{(-k_n)_x} \tau )= \theta_{(-k_n)_x}f(\omega')  - \pi( \tau ) = f(\omega) - \pi(\tau).
      \end{split}
    \end{equation*}
  \end{proof}
  The function $f$ is differentiable almost everywhere by virtue of being Lipschitz. Let $x\in\mathbb{R}^d$ de such that $H = Df(x)$ exists. Then
  $$f(x+u) = f(x)+Hu+ \Psi(u)$$
  for all $u\in\mathbb{R}^d$ with $\|\Psi(u)\|/\|u\|\rightarrow 0$ as $u\rightarrow 0$. Multiplying by $\bar\theta_{(-k_n)_x}$ and substituting $v = \bar\theta_{(-k_n)_x} u$,
  $$f(\bar\theta_{(-k_n)_x}x + v) = f(\bar\theta_{(-k_n)_x} x)+ Hv +\bar\theta_{(-k_n)_x}\Psi(\theta_{(-k_n)_x}v) $$
  for all $v\in\mathbb{R}^d$. By repetitivity, there exists $R>0$ such that $B_R(0)$ contains a translate of the patch in $\mathcal{T}$ which is the smallest patch which intersects $B_1(\bar\theta_{(-k_n)_x}  x)$ for any $n\in\mathbb{N}$. The vectors corresponding to translates are $\tau_n$ and they are all elements of $\Lambda_\mathcal{T}$. So by Lemma \ref{lem:Sol1} there exists a sequence $z_n\in B_R(0)$ such that
  $$f(z_n+v) = f(z_n)+Hv + \bar\theta_{(-k_n)_x}\Psi(\theta_{(-k_n)_x} v)$$
  for all $v\in B_1(0)$. Let $z'$ be a limit point of $z_n$. By continuity of $f$ and the properties of $\Psi$ near 0 it follows that $f(z'+v) = f(z') + Hv$ for all $v\in B_1(0)$, and so $f$ is flat in some neighborhood. Moreover by (\ref{eqn:Sol2}) it follows that $f$ is flat in an arbitrary large neighborhood, and by repetitivity it follows that $f$ is flat everywhere. Since by definition $f(0) = 0$, $f$ is linear. Now, $f(e_i) = \pi(e_i)$ for all $i$, so $f$ is the identity, $f(\omega) = \omega$ for all $\omega\in\mathcal{C}_\mathcal{T}$, so (\ref{eqn:sufficient}) holds.
\end{proof}
\bibliographystyle{amsalpha}
\bibliography{biblio}
\end{document}

config: arxiv