text:
\begin{document}
\title{Conjugacies of Model Sets}
\author{Johannes Kellendonk and  Lorenzo Sadun}

\address{Johannes Kellendonk\\ Universit\'e de Lyon,
Universit\'e Claude Bernard Lyon 1\\
Institute Camille Jordan, CNRS UMR 5208\\  69622 Villeurbanne, France}\email{kellendonk@math.univ-lyon1.fr}
\address{Lorenzo Sadun\\Department of Mathematics\\The University of
 Texas at Austin\\ Austin, TX 78712 USA}
\email{sadun@math.utexas.edu}
\thanks{
The work of the first author is partially supported by the ANR SubTile
NT09 564112.
The work of the second author is partially supported by NSF
  grant  DMS-1101326}
\date{June 13, 2014}

\keywords{Tilings, Dynamical Systems}

\subjclass{37B50, 52C22}

\begin{abstract}
  Let $M$ be a model set meeting two simple conditions: (1) the
  internal space $H$ is a product of $\R^n$ and a finite
  group, and (2) the window $W$ is a finite union of disjoint
  polyhedra.  Then any point pattern with finite local complexity
  (FLC) that is topologically conjugate to $M$ is mutually locally
  derivable (MLD) to a model set $M'$ that has the same internal group
  and window as $M$, but has a different projection from $H \times
  \R^d$ to $\R^d$.  In cohomological terms, this means that the group
  $H^1_{an}(M,\R)$ of asymptotically negligible classes has dimension $n$.
  We also exhibit a
  counterexample when the second hypothesis is removed, constructing
  two topologically conjugate FLC Delone sets, one a model set and the
  other not even a Meyer set.
\end{abstract}

\maketitle

\setlength{\baselineskip}{.6cm}

\section{Introduction and statement of results.}

A substantial part of the analysis of Delone sets (or tilings) is
based on the study of their associated dynamical systems.
This includes characterizing certain classes of
Delone sets by ergodic and topological properties of their dynamical systems.
For instance, the dynamical system of a
repetitive finite local complexity (FLC)
Delone set of $\R^d$ is topologically conjugate
  to that of a repetitive Meyer set if and only if it has
  $d$ independent topological eigenvalues \cite{KS13}. Similar
  characterizations are known for model sets (see \cite{ABKL} for a review).

  This suggest a natural definition and a question. Two Delone sets
  $\Lambda$ and $\Lambda'$ are called {\em topologically conjugate\/}
  whenever their associated dynamical systems are topologically
  conjugate. We call them {\em pointed topologically conjugate\/} if
  the conjugacy maps $\Lambda$ to $\Lambda'$.  Which properties of FLC
  Delone sets are preserved under topological conjugacy or pointed
  topological conjugacy?

In \cite{KS13} we showed
that the Meyer property is {\em not\/} always
preserved under topological conjugacy. In
view of this, we call a Meyer set {\em \brillig\/}
if every FLC Delone set topologically conjugate to $\Lambda$ is a
Meyer set. Our aim in this article is to study rigity for
model sets.

Recall that a {Meyer set} is a Delone set $\Lambda$ such that the set
of difference vectors $\Lambda-\Lambda$ is uniformly discrete, and
that model sets are Meyer sets arising from a particular
construction. This construction involves a cut \& project scheme,
that is, an internal (locally compact abelian) group $H$, a lattice
$\Gamma\subset H\times\R^d$ and a strip $S=W\times \R^d$ where
$W\subset H$, the so-called {window}, is a compact subset that is the
closure of its interior.  The projection set for these data is the set
of points arising by projecting the points of $S\cap \Gamma$ onto
$\R^d$ along $H$.  Generically (but not always) the projection set is
repetitive in which case we call any element of its hull, that is, any
point set which has the same local patches up to translation, a model
set.  In particular, all model sets are repetitive according to this definition.

We say that a model set $\mo'$ is a {\em reprojection\/} of a model set
$\mo$ if it arises from the same setup, except that the projection of
$S\cap\Gamma$ onto $\R^d$ is not along $H$, but is along $H' \subset
H\times \R^d$ with $H' = \{(h,g(h))| h\in H\}$ where $g: H\to\R^d$ is
a continuous group homomorphism.

We pay particular attention to {\em polyhedral\/} model sets, by which
we mean model sets satisfying two additional assumptions:
\begin{enumerate}
\item[H1.] The internal space $H$ is the product of the vector space
$\R^n$ with a finite (discrete, pure torsion) group $C$.
\item[H2.] The window $W$ is a finite union of polyhedra.
\end{enumerate}
Our main result states that such model sets are extremely rigid:

\begin{thm}\label{NewMainTheorem}
If $\mo$ is a polyhedral model set

and $\Lambda$ is a Delone set of finite local complexity that is pointed topologically conjugate to
$\mo$, then $\Lambda$ is mutually locally derivable (MLD) to a reprojection of $\mo$.
\end{thm}
Without hypothesis H2, this theorem is false.  In Section
\ref{examples} we exhibit a one dimensional model set $\mo$
satisfying H1 but not H2, and a Delone set $\Lambda$ of finite local complexity that is pointed
topologically conjugate to $\mo$, with $\Lambda$ not being a
Meyer set, much less a model set or a reprojection of $\mo$.
\medskip

A first result in the direction of studying topological conjugacies
can be found in \cite{KS13}. It says that any pointed topological
conjugacy between repetitive FLC Delone sets is the composition of an
MLD transformation followed by a {\em shape conjugacy\/} (defined below)
which can be chosen arbitrarily close to the identity.  Given that MLD
transformations are well-understood, this reduces the task of
understanding topological conjugacies to a study of shape conjugacies.
Shape conjugacies modulo MLD transformations are parametrized (at
least infinitesimally) by a subgroup of the first tiling cohomology
with $\R^d$-coefficients $H^1(\Lambda,\R^d)$, called the {\em
  asymptotically negligible group\/} in \cite{CS2}, which we denote by
$H^1_{an}(\Lambda,\R^d)$.

If $\Lambda$ is a Meyer set, then within $H^1_{an}(\Lambda, \R^d)$
there is a subgroup of shape deformations that preserves the Meyer
property. We call these {\em \ns\/}, and denote the subgroup
$H^1_{ns}(\Lambda, \R^d)$. If $\Lambda$ is a model set, there is a
further subgroup, denoted $H^1_{repr}(\Lambda, \R^d)$, corresponding
to reprojections. One can similarly define $H^1_{an}(\Lambda, \R)$,
$H^1_{ns}(\Lambda, \R)$, and $H^1_{repr}(\Lambda, \R)$. (See Sections
\ref{sec-define} and \ref{sec-tove}, below.)  In cohomological terms,
Theorem \ref{NewMainTheorem} can be restated as follows:

\begin{thm} \label{MainTheorem2} If $\mo$ is a polyhedral model set

  then $H^1_{repr}(\mo,\R)=H^1_{ns}(\mo, \R)=H^1_{an}(\mo, \R)$.
\end{thm}

Another cohomological restatement is as follows: There is a natural map from the cohomology of
$H\times\R^d/\Gamma$ to the cohomology of a model set constructed from the data
$(\Gamma,H,\R^d)$. Let us denote its image by $H^1_{max}(\mo,\R)$
\begin{thm} \label{MainTheorem3} If $\mo$ is a polyhedral model set

  then $H^1_{an}(\mo,\R)$ is $n$-dimensional and is contained in $H^1_{max}(\mo,\R)$.
\end{thm}

The organization of the paper is as follows. In Section
\ref{sec-define} we review the machinery of Delone dynamical systems
that is needed in the remainder of the paper.
In Section \ref{sec-model} we review the theory of model sets, identifying how
different model sets with the same parameter can differ.
In Section
\ref{sec-tove} we introduce the notion of \ns{} generators of shape conjugacies
and show
(Theorem \ref{mainlemma}) that all \ns{} generators are, up to local
deformation, generators of reprojections. In Section \ref{brillig} we
show (Theorem \ref{PolyhedralTheorem}) that asymptotically negligible
classes are represented by coboundaries of \ns{} generators. Taken
together, this proves Theorem \ref{NewMainTheorem}.  In Section
\ref{sec-cohomology} we interpret these results in terms of cohomology
and prove Theorems \ref{MainTheorem2} and \ref{MainTheorem3}.

Nonslip generators are introduced as a means of proving Theorem
\ref{NewMainTheorem}, but we believe that they have independent
interest. In Section \ref{sec-Meyer} we explore the significance of
the \ns{} property for model sets that do not necessarily satisfy
hypotheses H1 and H2, and for more general Meyer sets.  We prove
\begin{thm}\label{MainTheorem4}
Let $\Lambda$ be a repetitive Meyer set and $F$ a generator of
a shape deformation. If $F$ is not \ns{}, then the deformed set $\Lambda^F$
is not Meyer.
\end{thm}
In Section \ref{examples} we exhibit a model set that does not satisfy
H2 and a generator of shape conjugacies that is not \ns{}, and hence a
deformation of a model set that is not Meyer. We do not
know whether it is ever possible to construct a \ns{} class that is
not a reprojection.

\section{Preliminaries on point sets and their dynamical
  systems}\label{sec-define}
In this section we review some of the necessary background on Delone
sets and their dynamical systems.  Apart from the notion of
reprojection for model sets, this is all well-established in the
literature.

\subsection{Dynamical system of a Delone set}
A {\em Delone set\/} is a set $\Lambda \subset \R^d$ that is uniformly
discrete and relatively dense. That is, there exists an $r>0$ such
that every ball of radius $r$ contains at most one point of $\Lambda$,
and there exists an $R>0$ such that every ball of radius $R$ contains
at least one point of $\Lambda$.

A {\em Meyer set\/} is a Delone set $\Lambda$ for which
$\Lambda-\Lambda$ (i.e., the set of displacement vectors between
points of $\Lambda$) is uniformly discrete.
Equivalently, $\Lambda$
is Meyer if 0 is not a limit point of $(\Lambda-\Lambda) - (\Lambda -
\Lambda)$.

Let $B$ be a compact subset of $\R^d$.
The $B$-{\em patch\/} of a point set $\Lambda\subset \R^d$ is the intersection $P=\Lambda\cap B$
of $\Lambda$ with $B$.
We denote it by $(P,B)$ or simply by $P$. An $R$-patch of $\Lambda$ at $x$ is
the intersection of $\Lambda$ with $B=B_R(x)$, the ball of radius $R$ at $x$.

A Delone set has {\em finite local complexity\/}, or FLC, if for each
$R>0$ the set $\{B_R(0)\cap(\Lambda-x)|x\in\Lambda\}$ is finite, or
stated differently, the number of $R$-patches occuring at points of
$\Lambda$ and counted up to translation is finite.  A Delone set is
{\em repetitive\/} if for every patch $P$ of $\Lambda$, there exists an
$R$ such that every $R$-patch of $\Lambda$ contains at least one
translated copy of $P$. {\em Henceforth all Delone sets in this paper
  will be assumed to have FLC and to be repetitive.}

Delone sets are associated with dynamical systems as follows. We pick
a metric on the space of Delone sets with given inner and outer radii
$r$ and $R$ such that two Delone sets are close if their restriction
to a large ball around the origin are close in the Hausdorff
metric. If the Delone sets have FLC, this means that they agree
exactly on a large ball, up to a small translation. $\R^d$ acts on the
space of Delone sets by translation. The closure of the orbit of a
Delone set $\Lambda$ is called the {\em continuous hull\/} of
$\Lambda$, and is denoted $\Omega_\Lambda$, or just $\Omega$ when
there is no ambiguity about which Delone set is being considered. If
$\Lambda$ is a repetitive FLC Delone set, then $(\Omega_\Lambda,
\R^d)$ is a minimal dynamical system. We will also consider the
{\em canonical transversal\/} $\Xi_\Lambda$ (or simply $\Xi$) of
$\Omega_\Lambda$ which is given by the closure of the set
$\{\Lambda-x:x\in\Lambda\}$.  $\Xi_\Lambda$ consists of all point
patterns of $\Omega_\Lambda$ that contain the origin.

A Delone set $\Lambda'$ is {\em locally derived\/} from $\Lambda$ if
there exists a radius $R>0$ such that, whenever $\Lambda-x_1$ and
$\Lambda-x_2$ agree to radius $R$ around the origin, $\Lambda'-x_1$
and $\Lambda'-x_2$ agree to radius 1 around the origin. If $\Lambda'$
is locally derived from $\Lambda$ and $\Lambda$ is locally derived
from $\Lambda'$, we say that $\Lambda$ and $\Lambda'$ are {\em
  mutually locally derivable\/}, or MLD.

A local derivation of $\Lambda'$ from $\Lambda$ extends to a factor
map from $\Omega_\Lambda$ to $\Omega_{\Lambda'}$. If $\Lambda$ and
$\Lambda'$ are MLD, then this factor map is a topological conjugacy
called an {\em MLD map\/}.

\subsection{Maximal equicontinuous factor}
An important factor of the dynamical system $(\Omega,\R^d)$ is the
largest factor (up to conjugacy) on which the action is
equicontinuous. We denote this so-called {\em maximal equicontinuous
  factor\/} by $\Omega_{max}$ and the factor map by $\pi_{max}$.

The equivalence relation
$$\Rmax = \{(\Lambda_1,\Lambda_2)\in \Omega_\Lambda\times\Omega_\Lambda:
\pi_{max}(\Lambda_1) = \pi_{max}(\Lambda_2)$$ can be alternatively
described using the (regional) proximality relation and its strong
version. This description can be used to show the following two
results results needed later on:
\begin{itemize}
\item Any topological conjugacy preserves $\Rmax$.
\item If two elements $\Lambda_1$, $\Lambda_2$ in the hull of a Meyer
set satisfy
$\pi_{max}(\Lambda_1) = \pi_{max}(\Lambda_2)$, then they share a point.
\end{itemize}
Proofs can be found in \cite{BargeKellendonk}.

\section{Model sets}\label{sec-model}

A model set (or cut \& project set) is a Meyer set that is
obtained by a particular construction.

\subsection{Cut  and project scheme}
To construct a model set one needs a {\em cut
  and project scheme\/}
$(\Gamma,H,\R^d)$ and a subset $W\subset H$ called the {\em window}.

The cut \& project scheme consists of the space $\R^d$ in which the
model set lives (the {\em parallel\/} or {\em physical\/} space), a
locally compact abelian group $H$ (called the {\em internal group\/} or
{\em perpendicular space\/}) and a lattice (a cocompact discrete
subgroup) $\Gamma\subset H\times\R^d$. The set $S:=W\times \R^d$ is
called the {\em strip\/}.  As usual, we require three further
assumptions:
\begin{enumerate}
\item The projection onto the second factor $\pi^\|:H\times \R^d\to \R^d$ is
injective when restricted to the lattice $\Gamma$,
\item Projection onto the first factor $\pi^\perp:H\times \R^d\to H$ maps
the lattice $\Gamma$ densely into $H$,
\item If $W + h = W$ for $h\in H$ then $h=0$.
\end{enumerate}
We also use the notation $\Gamma^\|=\pi^\|(\Gamma)$,
$\Gamma^\perp=\pi^\perp(\Gamma)$, $x^\|=\pi^\|(x)$ and $x^\perp=\pi^\perp(x)$.
We write $\pi_\Gamma^\perp$ for the restriction of $\pi^\perp$ to $\Gamma$. The point set
$$\m_\xi(W) := \{\pi^\|(\gamma): \gamma \in S\cap(\Gamma+\xi)\}$$
is called the projection set of the cut \& project scheme with window
$W$ and parameter $\xi\in H\times \R^d/\Gamma$.
Note that any element of $\ker\pi_\Gamma^\perp$ is a period of the projection set $\m_\xi(W)$.

We are interested in windows $W$ that are compact and the closure of
their interiors. In this case, the projection set $\m_\xi(W)$ is repetitive
if the parameter $\xi$ is such that
$\pi^\perp(\Gamma + \xi) \cap \partial W$ is empty. We call such
parameters {\em non-singular\/} and denote the
set of non-singular parameters by $NS$.

\begin{dfn}
  A {\em model set\/} (with window $W$) is an element in the hull of a
  projection set $\m_\xi(W)$ whose parameter is non-singular. We
  always assume that $W$ is compact and the closure of its interior.
  We call the model set {\em polyhedral\/} if $H$ is the product of $\R^n$
  with a finite group and the window is a finite disjoint
  union of polyhedra.
\end{dfn}
Model sets are repetitive Meyer sets.
It is well-known that MLD maps send model sets
to model sets, and send Meyer sets to Meyer sets.

\begin{remarks}
  Our use of the term {\em model set\/} is slightly different than
  that of some other authors.  First, a model set in our sense need
  not to be a projection set with closed window, but might be what
  elsewhere is called an {\em inter-model set\/}, containing some but
  not all points $x^\|$ for which $x^\perp$ lies on the boundary of
  $W$. Second, the requirement that the parameter be non-singular
  automatically makes our model sets repetitive. So model sets in our
  sense might elsewhere be called repetitive inter-model sets (or
  complete Meyer sets).  We should also mention that the window being
  the closure of its interior is not required by all authors. For
  instance \cite{BaakeGrimm13} demand only that the window be
  relatively compact and have non-empty interior.
\end{remarks}

\subsection{Reprojection}
The projection $\pi^\|$ is along $H$ (onto $\R^d$).  If we change the
direction along which we project, that is, if we tilt the space $H$,
but otherwise leave the strip $S$ and the parameter $\xi$ fixed, then
this affects the projection set $\m_\xi(W) = \{\pi^\|(x):x\in S\cap
(\Gamma+\xi)\}$ rather mildly.  Let $\pi':H\times\R^d\to \R^d$ be the
projection onto $\R^d$ along another group $H'\subset H\times \R^d$
transversal to $\R^d$.  We call
$$ \m_\xi'(W) = \{\pi'(x):x\in S\cap (\Gamma+\xi)\}$$
the {\em reprojection\/} of $\m_\xi(W)$ along $H'$. More generally, if $\Lambda$ is a subset of
$\pi^\|(\Gamma+\xi)$

we call
$$\Lambda' = \{\pi'\circ{\pi^\|_\xi}^{-1}(\lambda):\lambda\in\Lambda\}$$
its {\em reprojection\/} along $H'$. Here $\pi^\|_\xi$ is the
restriction of $\pi^\|$ to $\Gamma+\xi$ which is injective by our
assumption. We can think of $H'$ as being the image of $H$ under a
group isomorphism which corresponds to a shear transformation in the
case $H$ is a vector space: there exists a continuous group
homomorphism $g:H\to \R^d$ such that $H' = \{(h,g(h)) : h\in H\}$.

\subsection{Non-singular parameters and model sets}
A model set is called {\em non-singular\/} if it is a projection set
$\m_\xi(W)$ with non-singular parameter $\xi$.
Since the window is
the closure of its interior, this occurs for a dense $G_\delta$-set of
$\xi$. Therefore these model sets are also
called {\em generic\/}.

For a fixed window $W$, the hull of a non-singular model set $\m_{\xi_0}(W)$ contains
all other model sets $\m_\xi(W)$ with $\xi \in NS$. In other words,
the hull of a non-singular model set with window $W$ depends not on
the choice of the non-singular parameter $\xi$ but only on the
window. We therefore denote the hull with $\Omega(W)$. Hence a model
set with window $W$ is an element of $\Omega(W)$.

It is not difficult to see that $\m_{\xi}(W)=\m_{\xi'}(W)$ if
and only if $\xi-\xi'\in\Gamma$.
The map $\m_\xi(W)\mapsto \xi$ (for $\xi\in NS$)
 is thus surjective. It is called the torus parametrization map.

It turns out that this torus is also the maximal equicontinuous
factor, $ \Omega_{max} = (H\times \R^d)/\Gamma$, the action being given
by left translation on the second factor $\R^d$
\cite{BargeKellendonk}.
The factor map $\pim:\Omega(W)\to (H\times \R^d)/\Gamma$ is the continuous
extension of the map $\m_\xi(W)\mapsto \xi$ (for $\xi\in NS$)
which associates to a non-singular model set
$\m_\xi(W)$ its parameter $\xi$. This map is injective precisely on
its pre-image of $NS$.
The elements of the hull that are mapped
by $\pim$ to singular points are called {\em singular model sets\/}.

\subsection{The star map}
The star map sends points in $\R^d$ to points in $H$. There are several related star maps to be considered:
\begin{itemize}
\item The {\em general star map\/} $\sigma$ is a group homomorphism
  $\Gamma^\| \to H$, $\sigma(x) = \pi^\perp(\gamma)$ where
  $\gamma\in\Gamma$ is the unique lift of $x\in\Gamma^\|$ in $\Gamma$
  under $\pi^\|$. In particular, $[(\sigma(x),x)]_\Gamma =
  [(0,0)]_\Gamma$, where $[\cdot]_\Gamma$ denotes an equivalence class
  mod $\Gamma$. We denote the general star map also simply with a
  star, $\sigma(x)=x^*$.  The general star map is not continuous if
  one gives $\pi^\|(\Gamma)$ the relative topology induced by $\R^d$.

\item The {\em general star map $\sigma_\xi$ with parameter $\xi$\/}
  sends $\pi^\| (\Gamma + \xi)$ to $H$, where $(\sigma_\xi(x), x)$ is
  the unique lift of $ x \in \pi^\| (\Gamma + \xi)$ to $\Gamma+\xi$.
  Like the general star map without parameter (or equivalently, with
  parameter 0), this has a dense domain in $\R^d$, dense range in $H$,
  and is not continuous.

\item If $\Lambda$ is a Delone subset of $\pi^\| (\Gamma + \xi)$ we
  denote the restriction of $\sigma_\xi$ to $\Lambda$ also by
  $\sigma_\Lambda$ and call it the {\em star map of $\Lambda$\/}.  Note
  that the support of $\sigma_\Lambda$ is uniformly discrete. This
  will allow us later to talk about weak pattern equivariance of
  $\sigma_\Lambda$. If $\Lambda$ is a model set with window $W$ then
  the image of $\sigma_\Lambda$ is a dense subset of $W$.
\end{itemize}
Note that if $x_1$ and $x_2$ are points of $M$ and $\pim(M) = \xi$,
then both $(\sigma_M(x_1), x_1)$ and $(\sigma_M(x_2),x_2)$ are in
$\Gamma + \xi$, so $x_2-x_1\in\Gamma^\|$ and
$$\sigma_M(x_2) - \sigma_M(x_1) =\sigma_\xi(x_2) - \sigma_\xi(x_1) = (x_2 - x_1)^*.$$
The factor map $\pim$ is related to the star map of a pattern as
follows. If $M$ is a model set and $\xi = \pim(M)$ then
$(\sigma_M(x),x) \in \Gamma + \xi$ for all points $x\in M$. Thus
\begin{equation*}
\pim(M)= [\sigma_M(x),x]_\Gamma.
\end{equation*}
In particular, if $0 \in M$, then $\pim(M)= [\sigma_M(0),0]_\Gamma$.
Let $\imath_{H}:H\to H\times\R^d/\Gamma$ be given by $\imath_{H}(h) =
[h,0]_\Gamma$. By assumption $\imath_{H}$ is injective. Furthermore,
by equivariance of $\pim$ we have $\pim(M-x)= [\sigma_M(x),0]_\Gamma$
for all $x\in M$. Thus
\begin{equation}\label{eq-pistar}
\sigma_M(x) = \imath_H^{-1}\circ \pim(M-x)
\end{equation}
for all $x\in M$.

\subsection{Acceptance domains of patches}
Let $M$ be a model set with window $W$

let $B$ be a compact set

and let $P= M \cap B$. We call $P$ the ``$B$-patch of $M$''.  More
generally we say that a finite set $P\in\R^d$ is a patch for a model set with window $W$
if there is $M\in\Omega(W)$ and a compact set $B\in\R^d$ such that $P
= M\cap B$.

We wish to determine a condition for another model set $M'$ in the
hull of $M$ to have $M' \cap B = P$.  It will be sufficient for our applications to consider the case that
$P$ contains the origin which we will assume troughout.

Let $D$ be the preimage $\sigma^{-1}((W-W)\cap\Gamma^\perp)$.
This gives the set of all possible displacements between points in the same pattern, since if
$x_1, x_2 \in M$, then $\sigma_M(x_2) - \sigma_M(x_1)=\sigma(x_2-x_1) \in W-W$.

Let $P' = (D \cap B) \backslash P$. $P'$
the set of points that can appear in a $B$-patch of some model set which contains $0$
but are not in the $B$-patch $P$. Note that $P'$ is a finite set. Let
$$W_P^o = {\bigcap_{x \in P} (Int(W) - {x}^*) \cap \bigcap_{x' \in P'} (W^c - {x'}^*)}.$$
$W_P:=\overline{W_P^o}$
is called the {\em acceptance domain\/} of $P$. By construction, it is
the closure of an open set and hence the closure of its interior.

\begin{prop}\label{prop-acc}
Let $P$ be a $B$-patch for a model set with window $W$.
We assume that $P$ contains the origin. Let
$M'\in\Omega(W)$ be a possibly different model set which contains the origin.

If $M'$ is non-singular then, for all $x\in M$,  $\sigma_{M'} (x)
\in Int(W_P)$ if and only if $P = (M-x)\cap B$.
\end{prop}

\begin{proof}
Let $\xi=\pim(M')$. Assuming that $M'$ is
non-singular this means that $M' =\m_\xi(W)$.
The condition $M' \cap B= P$ has two parts:
\begin{enumerate}
\item All the points of $P$ should be in $M'$. This is equivalent to
  having $\sigma_{\xi}(x) \in W$ for all $x\in P$. But
  $\sigma_{\xi}(x)-\sigma_{\xi}(0) = x^*$, so this is in turn
  equivalent to $\sigma_{M'}(0)\in W - x^*$.
\item All the points of $P'$ should not be in $M'$. That is, for each
  $x' \in P'$, $\sigma_{\xi}(x') \in W^c$, so $\sigma_{M'}(0) \in W^c
  - {x'}^*$, where $W^c$ denotes the complement of $W$.
\end{enumerate}

Thus $P \in M'$ if and only if
\begin{equation} \label{accept}
\sigma_{M'}(0) \in \bigcap_{x \in P} (W - {x}^*) \cap \bigcap_{x' \in P'} (W^c - {x'}^*).
\end{equation}
Recall that $\pim(M') = [\sigma_{M'}(0)]_\Gamma$. Therefore and
since $M$ was assumed non-singular,
$\sigma_{M'}(0)$ cannot lie on the boundary of $W_P^o$. Thus
we can replace the right-hand side with the closed set $W_P$.
\end{proof}

\begin{lem}
  Suppose that $M$ is a model set containing the origin and satisfying
  H1 and H2. The acceptance domain of every (non-empty) patch of $M$
  containing the origin can be written as a finite union of closed
  convex sets that have non-empty interior.
\end{lem}
\begin{proof} Since $W$ is a finite union of connected polyhedra this
  is also the case for $W_P$.  We can decompose $W_P$ into a finite
  union of connected polyhedra and each of those into a finite union
  of convex polyhedra. Since $W_P$ is the closure of its interior the
  convex polyhedra can be taken to have non-empty interior.
\end{proof}
We formulate the result of the above lemma as a hypothesis on the
window $W$ that is weaker than H2.
\begin{enumerate}
\item[H$2'$.] The acceptance domain of every patch containing the origin
can be written as a finite union of closed convex sets that have non-empty interior.
\end{enumerate}

\begin{lem}\label{lem-W} Consider a cut \& project scheme
  $(\Gamma,H,\R^d)$ and a compact subset $W\subset H$ that is the
  closure of its interior.
  For any neighborhood $U$ of $0\in H$ there exists a finite set
  $J\subset W\cap\pi^\perp(\Gamma)$ such that
  $\emptyset\neq\bigcap_{u\in J} W-u \subset U$.
\end{lem}

\begin{proof}
  Without loss of generality we can assume that $U$ is contained in
  the compact set $W-W$, so that $K = (W-W) \backslash U$ is compact.
  Since $W \cap \pi^\perp(\Gamma)$ is countable, we can find a
  sequence of nested finite sets $J_1 \subset J_2 \subset \cdots$ with
  $\bigcup J_i = W \cap \pi^\perp(\Gamma)$.  If every intersection
  $\bigcap_{u\in J_i} W-u$ contains a point $x_i \in K$ then, by
  compactness, there is a limit point $x_\infty \in K$ such that
  $x_\infty \in \bigcap_{u\in \pi^\perp(\Gamma)} W-u$.  But
  Schlottmann proved \cite{Schlottmann}[Lemma~4.1] that $\bigcap_{u\in
    W \cap \pi^\perp(\Gamma)} W-u = \{0\}$, which is a contradiction.
\end{proof}

\begin{cor}\label{cor-W} Let $M$ be a model set containing the origin.
  Any open subset of the window of $M$ contains the acceptance domain
  for a patch of $M$ that contains the origin.
\end{cor}

\begin{proof} Let $M$ be a model set with window $W$ and let $U\subset
  W$ be an open subset.
Pick $x \in M$ such that $\sigma_M(x) \in U$, and let
$U' = U - \sigma_M(x)$, which is an open neighborhood of $0$.
We apply Lemma~\ref{lem-W} to obtain a finite set $J\subset
W\cap\pi^\perp(\Gamma)$ such that
  $\emptyset\neq\bigcap_{u\in J} W-u \subset U'$. For each $u$ pick
$q\in M$ such that $\sigma_M(q) -\sigma_M(x) = \sigma(q-x) =
u$. Denoting by $Q$ the set of such  points $q$ we have

$\emptyset \neq \bigcap_{q \in Q} W - \sigma(q)\subset U$.
Let $B\subset \R^d$ be any compact neighborhood of the origin containing $Q$.
The acceptance domain of $P = B\cap M$ is then a subset of $U$. Furthermore, $P$ contains the origin.
\end{proof}

\subsection{Singular model sets}
We wish to describe the singular model sets, that is, the elements in
the hull $\Omega(W)$ that are not
themselves  projection sets with   non-singular parameter.
\begin{prop}\label{prop-nonsing}
Let $M$ be a model set with window $W$ and $\xi = \pim(M)$. Then
$$ \m_\xi(Int(W))\subset M \subset \m_\xi(W).$$
\end{prop}
\begin{proof}
  A model set $M$ with window $W$ is a limit of a sequence $(\m_\eta
  -x_n)_n$ where $\eta$ is non-singular and $x_n\in \R^d$. But
  $\m_\eta - x_n = \m_{\xi_n}$, where $\xi_n = \eta +
  [0,x_n]_\Gamma$. By continuity of $\pim$ we thus have $\xi = \lim
  \xi_n$ and so the sequence $(\xi_n)_n$ lifts to a sequence
  $(\tilde\xi_n)_n\subset H\times\R^d$ which converges to a lift
  $\tilde\xi$ of $\xi$. We may quickly restrict to the case that
  $\tilde\xi_n^\|=0$, because otherwise we can replace $x_n$ by
  $x_n-\tilde\xi_n^\|$ and $\eta$ by $\eta-[0,\tilde\xi^\|]_\Gamma$ to
  reduce to that situation. Then $M$ and all $\m_{\xi_n}$ are subsets
  of $\Gamma^\|$.  In particular, $\Gamma^\|$ is the common domain of
  all functions $\sigma_{\xi_n}$, and the sequence of functions
  $(\sigma_{\xi_n})_n$ converges uniformly to $\sigma_{\xi}$, since
  $(\tilde\xi_n)_n$ converges to $\tilde\xi$. Hence if
  $\sigma_{\xi}(x)\in Int(W)$ then an open neighborhood of
  $\sigma_{\xi}(x)$ in $Int(W)$ contains all $\sigma_{\xi_n}(x)$ for
  $n$ sufficiently large which shows that $\sigma_{\xi_n}(x)\in W$ for
  all $n$ sufficiently large. The latter means that $x$ belongs to
  $\m_{\xi_n}$ for all $n$ sufficiently large and thus also to
  $M$. This shows that $\m_\xi(Int(W))\subset M$.

  To obtain the other inclusion we use the same kind of argument but
  for the complement $W^c$ instead of $Int(W)$.  Indeed, if
  $\sigma_{\xi}(x)\in W^c$ then an open neighborhood of
  $\sigma_{\xi}(x)$ in $W^c$ contains all $\sigma_{\xi_n}(x)$ for $n$
  sufficiently large which shows that $x$ cannot belong to $M$.
\end{proof}
We can now generalize one direction of Proposition \ref{prop-acc} to
singular model sets.
\begin{cor}\label{cor-acc}
  Let $P$ be a $B$-patch for a model set with window $W$ and
  $M'\in\Omega(W)$ a possibly singular model set. We assume that $P$
  and $M'$ contain the origin.  Let $x\in M'$. If $\sigma_{M'} (x) \in
  Int(W_P)$ then $P = (M'-x)\cap B$.  In particular $\{\sigma_{M'}
  (x): P = (M'-x)\cap B\}$ is dense in $W_P$.
\end{cor}
\begin{proof} Set $\xi = \pim(M')$.
We see from the description of $M'$ given in Theorem \ref{thm-sms} that
the projection set $\m_\xi(Int(W_P))$ is contained in $M'$. Hence
condition (\ref{accept}) also applies to a singular $M'$,
as long as $\sigma_{M'}(0)$ does not lie on the boundary of $W_p$.
Hence if $\sigma_{M'}(x) \in Int(W_P)$ then $P = (M'-x)\cap B$.
Denseness of
$\{\sigma_{M'} (x): P = (M'-x)\cap B\}$ follows directly from that fact that
$\{\sigma_{M'} (x): x\in M'\}$ is dense in $W$.
\end{proof}

We now describe the potential difference between two singular model
sets with the same parameter.  For that we assume that the window $W$
is polyhedral and decompose it as a polyhedral complex. Let $\F(W) $
be the set of open faces of $W$.  Thus $W$ is the disjoint union of
its interior $Int(W)$ with the $f\in\F(W)$ (where vertices are
considered open 0-cells). Let $V(f)$ be the vector space parallel to
$f$, that is, the space spanned by $f-f$.

Let $\tilde H_\xi(f)$ be the closure of $V(f)\cap
(\Gamma+\xi)^\perp$. $\tilde H_\xi(f)$ might be empty, but if it is
not empty it is of the form $\tilde H_\xi(f) = H_\xi(f) +
\Delta_\xi(f)$ where $H_\xi(f)$ is a real vector space and
$\Delta_\xi(f)$ a discrete subset of $(\Gamma+\xi)^\perp$. In that
case $\sigma^{-1}(H_\xi(f)\cap \Gamma^\perp) $ is a sublattice and we
let $E_\xi(f)$ be its real span. We define $E_\xi(f)$ to be empty if
$\tilde H_\xi(f)$ is empty.

Furthermore, given that $f$ is bounded, there is a
finite subset $\Phi_{\xi}(f)\subset \Delta_\xi(f)$
such that
\begin{equation}\label{eq-f}
f\cap (\Gamma+\xi)^\perp \subset \bigcup_{\eta\in\Phi_{\xi}(f)} \eta + H_\xi(f).
\end{equation}

\begin{thm} \label{thm-sms}
Let $M$ be an arbitrary model set with polyhedral window $W$ and let $\xi = \pim(M)$.
Then
$$ M \backslash \m_\xi(Int(W)) \subset \bigcup_{f\in \F(W)}  \bigcup_{\eta\in \Phi_{\xi}(f)}
E_\xi(f)+\sigma_\xi^{-1}(\eta).$$
\end{thm}

\begin{proof} By Proposition \ref{prop-nonsing} we have
$$ M \backslash \m_\xi(Int(W)) \subset \bigcup_{f\in \F(W)} \m_\xi(f) .$$
If $\m_\xi(f)$ is not empty then (\ref{eq-f}) implies that
$$ \m_\xi(f) = \bigcup_{\eta\in\Phi_{\xi}(f)} \eta +\m_\xi(f_\eta) $$
where $f_\eta = (f-\eta) \cap H_\xi(f)$ is an
open subset of $H_\xi(f)$. Moreover,
$$\m_\xi(f_\eta) = \{\pi^\|(x):x\in\Gamma, \pi^\perp(x)\in f_\eta\}\subset \sigma^{-1}(H_\xi(f)\cap \Gamma^\perp)\subset E_\xi(f).$$
\end{proof}

\begin{lem}\label{lem-codim}

$E_f$ has codimension at least $1$.
\end{lem}

\begin{proof}
Let $\Gamma_\xi(f) = {\pi_{\Gamma}^\perp}^{-1}(H_\xi(f)\cap\Gamma^\perp)$ and $V$ a bounded open subset of $H_\xi(f)$. Then we can rewrite
$\m_\xi(V) = \{\pi^\|(x):x\in\Gamma_\xi(f), \pi^\perp(x)\in V\}$ and so we see that
$\m_\xi(V)$ is also the projection set with window $V$ and parameter $0$
for the cut \& project scheme
$(\Gamma_\xi(f),H_\xi(f),E_\xi(f))$. Since $V$ is open
$\m_\xi(V)$ is relatively dense in $E_\xi(f)$ \cite{BaakeGrimm13} and thus has strictly positive lower density in $E_\xi(f)$.

Since $\pi^\perp(\Gamma)$ is dense in $H$ we can find a subset
$V\subset f_\eta$, open in the topology of $H_\xi(f)$, and an infinite
subset $\Psi\subset \pi^\perp(\Gamma)$ such that $V+\psi\subset W$ for
all $\psi\in\Psi$ and such that the sets $V+\psi$ have pairwise empty
intersection. Hence $\m_\xi(W)$ contains the disjoint union of all
$\m_\xi(V+\psi)$, $\psi\in\Psi$. Now the lower density of
$\m_\xi(V+\psi)$ in $E_f$ is independent of $\psi$. Therefore, if
$E_f$ has dimension $d$ and hence is all of $\R^d$ than the lower
density of $\m_\xi(W)$ must be infinite, which is a contradiction.
\end{proof}

\begin{cor}
  Consider two singular elements $M_1,M_2$ of the hull of a model set.
  If $\pim(M_1)=\pim(M_2)=\xi$ then the difference set $M_1\Delta M_2$
  is contained in a finite affine hyperplane arrangement which we
  denote $\aaa(M_1,M_2)$.
\end{cor}
\begin{proof}
  By Theorem \ref{thm-sms} the symmetric difference $M_1\Delta M_2$ is
  contained in $\bigcup_{f\in \F(W)} \bigcup_{\eta\in \Phi_{\xi}(f)}
  E_\xi(f)+\sigma_\xi^{-1}(\eta)$.  By Lemma~\ref{lem-codim} the sets
  $E_\xi(f)+\sigma_\xi^{-1}(\eta)$ are proper affine
  hyperplanes. Indeed, if $\sigma_\xi^{-1}(\eta)$ is not finite that
  it contains a period which must be contained in $E_\xi(W)$. There
  are only finitely many affine hyperplanes, because $\F(W)$ and $
  \Phi_{\xi}(f)$ are finite.
\end{proof}

The affine hyperplanes making up $\aaa(M_1,M_2)$  depend on the pair $M_1,M_2$. However, their number is uniformly bounded:

\begin{lem}\label{bound-N} Given the hull $\Omega(W)$ of a model set.
There is a finite number $N$ such that for all  $M_1,M_2\in\Omega(W)$
with $\pim(M_1)=\pim(M_2)$

the number of hyperplanes in the arrangement $\aaa(M_1,M_2)$ is bounded by $N$.

\end{lem}

\begin{proof}
  There is a finite number of faces and each face $f$ gives rise to a
  set $\Phi_{\xi}(f)$ which can vary with $\xi$, but the number of its
  elements is bounded from above since $f$ is bounded and $\Delta_f$
  discrete.
\end{proof}

\begin{prop}\label{rel-dense}
Given the hull $\Omega(W)$ of a model set.
Given $r>0$ there exists a $\rho>0$ such that for all $M_1,M_2\in\Omega(W)$ which
satisfy $\pim(M_1)=\pim(M_2)$ and
all $x\in \R^d$ the ball $B_{\rho}(x)$ contains at least one
point at which $M_1$ and $M_2$ agree out to distance $r$.
\end{prop}

\begin{proof}
  Let $\xi=\pim(M_1)=\pim(M_2)$. Pick a patch $(P,B)$ where $B$
  contains a ball of radius $r$.  Since the interior of the acceptance
  domain $W_P$ is in the interior of $W$, $\m_\xi(W_P^0)$ is a subset
  of both $M_1$ and $M_2$.

  This is a relatively dense subset of
  $M$,
  so there exists a $\rho$ such that for all $x\in \R^d$ the ball
  $B_{\rho}(x)$ contains at least one point of $\m_\xi(Int(W_P))$, and
  around this point $M_1$ and $M_2$ agree out to distance $r$.
\end{proof}

\subsection{Pattern equivariant functions}
Let $\Lambda\subset\R^d$ be a FLC-Delone set and $Y$ some set.  A
function $f:\R^d \to Y$ is called {\em \spe\/} if there exists an
$R>0$ (called the radius) such that, whenever $x_1,x_2 \in \R^d$ are
such that $\Lambda - x_1$ and $\Lambda -x_2$ agree exactly on the ball
$B_R(0)$, then $f(x_1)=f(x_2)$. In other words, each function value
$f(x)$ is determined exactly by the pattern of $\Lambda$ in a ball of
radius $R$ around $x$. For most of our purposes we need functions only
defined on $\Lambda$, or the $CW$-complex it defines. So we call a
function $\phi: \Lambda \to Y$ \spe\ if there exists an $R>0$ such
that, whenever $x_1,x_2 \in \Lambda$ are such that $\Lambda - x_1$ and
$\Lambda -x_2$ agree exactly on the ball $B_R(0)$, then
$\phi(x_1)=\phi(x_2)$. It can be shown that if $Y$ is a finite
dimensional real vector space then any \spe\ function on $\Lambda$ is
the restriction of a smooth \spe\ function on $\R^d$ \cite{Kel}.

Any locally constant function $\tilde f:\Xi_\Lambda\to Y$ defines a
\spe{} function on $f:\Lambda\to Y$ via $f(x) = \tilde f(\Lambda-x)$;
this defines a bijective correspondence.

Now let $Y$ be a topological space.  Any continuous function $\tilde
f:\Omega_\Lambda\to Y$ is uniquely determined by the function
$f:\R^d\to Y$, $f(x) = \tilde f(\Lambda-x)$.  We call a function
$f:\R^d\to Y$ arising in such a way from a continuous function $\tilde
f:\Omega_\Lambda\to Y$ {\em \wpe\/}. Likewise $\phi: \Lambda\to Y$ is
{\em \wpe\/} if it arises in the above way from a corresponding
function $\tilde \phi:\Xi_\Lambda\to Y$. Equivalently we may say that
$\phi: \Lambda\to Y$ is \wpe\ for $\Lambda$ iff the function
$\{\Lambda-x : x\in \Lambda\}\ni (\Lambda-x)\mapsto \phi(x)$ is
uniformly continuous in the topology of $\Xi_\Lambda$.  It is not
difficult to see that if $Y=\R^k$ is a finite dimensional vector space
then $\phi$ is \wpe\ if and only if it is the uniform limit of \spe\
functions.

The following is a very important example.  A direct proof not using
the maximal equicontinuous factor map can obtained from Corollary
\ref{cor-W}, see also \cite{BaakeLenz}.

\begin{lem}\label{lem-starwpe}
The star map $\sigma_M:M\to H$ of a model set is \wpe.
\end{lem}

\begin{proof}

Recall that the image of $\sigma_M$ lies in $W$. We can hence rewrite (\ref{eq-pistar}) as
$\sigma_M = \imath_H^{-1}\left|_W\right.\circ \pim(M-\cdot)$.
Since $W\subset H$ is compact
$\imath_{H}^{-1}\left|_W\right. : [W,0]_\Gamma \to H$ is uniformly continuous.
Since also $\pim$ is uniformly continuous $\sigma_M$ is \wpe.
\end{proof}
A point pattern $\Lambda$ can always be realized as the vertex set of
a polygonal tiling of $\R^d$. Such a tiling gives a cell decomposition of
$\R^d$. Functions on $\Lambda$ with values in an abelian group $A$
can then be viewed as 0-cochains on this CW complex.
Likewise, we can consider 1-cochains, 2-cochains, etc. These
cochains are considered \spe{} if their values on a cell are determined exactly
by the pattern of $\Lambda$ in some fixed finite radius around that cell, and
\wpe{} if they can be uniformly approximated by \spe{} cochains.

The coboundary of a \spe{} cochain is \spe{}.  The cohomology of the
complex of \spe{} cochains with values in $A$ is called the $A$-valued cohomology of $\Lambda$ and denoted
$H^*(\Lambda,A)$. It is naturally isomorphic to the \v Cech
cohomology $\check H^*(\Omega_\Lambda,A)$  \cite{integer}.

There is an equivalent description of the cohomology of $\Lambda$
provided that $A=\R$ (or $\R^k$).  We can consider de Rham forms on
$\R^d$ which are \spe\ for $\Lambda$. These form a sub complex of the
usual de Rham complex for $\R^d$ and $H^*(\Lambda,\R)$ can be seen as
the cohomology of this sub complex \cite{Ke1}.

\section{Shape conjugacies and \ns\ generators}
\label{sec-tove}
A shape conjugation is a particular shape deformation in the sense of
\cite{CS2,Kel}, and shape deformations arise as follows:

Consider a function $F:\Lambda\to\R^d$, defined on an FLC Delone set
$\Lambda$.  It defines a new set $$\Lambda^F :=
\{x+F(x):x\in\Lambda\}.$$ We assume that the coboundary $\delta F$ of
$F$ is \spe.  This means that the elements of $\Lambda^F-\Lambda^F$
can be locally derived from $\Lambda-\Lambda$ and implies in
particular that $\Lambda^F$ has FLC. We think of $\Lambda^F$ as a
deformation of $\Lambda$ (as if we had turned on $F$ slowly) and call
the function $F$ the {\em generator\/} of the deformation.

One possible way for $\delta F$ to be \spe\ is for $F$ already to be \spe. This is the case precisely if
 $\Lambda^F$ can be locally derived from $\Lambda$.

 A deformation is a {\em shape semi-conjugacy\/} if the map
 $\Lambda-x\mapsto \Lambda^F-x$ extends from the orbit of $\Lambda$ in
 $\Omega_\Lambda$ to a topological semi-conjugacy
 $\s_F:\Omega_\Lambda\to\Omega_{\Lambda^F}$.

Our ultimate aim is to understand the extent to which the dynamical system of a
Delone set determines the Delone set. More specifically, given an
FLC-Delone set $\Lambda$, we ask which FLC-Delone sets are
topologically equivalent to it? To investigate this question we recall the following theorem.
\begin{thm}[\cite{KS13}, Theorem 5.1]\label{thm-shape}
  Let $\Lambda$ and $\Lambda'$ be FLC Delone sets that are pointed
  topologically conjugate.  For each $\epsilon >0$ there exists a FLC
  Delone set $\Lambda_\epsilon$ that is MLD with $\Lambda$ and a
  function $F_\epsilon:\Lambda_\epsilon \to \R^d$ whose coboundary is
  \spe\ such that $\Lambda'=\Lambda^{F_\epsilon}$ and
  $\s_{F_\epsilon}:\Omega_{\Lambda_\epsilon} \to \Omega_{\Lambda'}$ is
  a topological conjugacy; in other words $F_\epsilon$ is a generator
  of a shape conjugation mapping $\Lambda_\epsilon$ to $\Lambda'$.
  Moreover $\sup_x |F_\epsilon(x)|\leq\epsilon$.

\end{thm}

\begin{remarks}
  The term ``deformation of a model set" has been used in the
  literature (see \cite{BaakeLenz,BernuauDuneau}) also for other
  kinds of deformations for which, in particular, the deformed set is
  no longer necessarily FLC. These could be achieved by functions $F$
  whose co-boundaries are not \spe.  Without the FLC requirement there
  are many more possible deformations and our rigidity results do not
  apply.
\end{remarks}
\subsection{Auto-conjugacy}
Theorem~\ref{thm-shape} is about pointed topological conjugacy. To
understand what it means for the two patterns to be merely
topologically conjugated we need to understand when, given two
elements $\Lambda_1,\Lambda_2$ of the same hull, the map
$\Lambda_1\to\Lambda_2$ extends to a topological conjugacy. This is
then an auto-conjugacy.

A first observation to make is that an auto-conjugacy $\varphi$ must
preserve the equivalence relation given by $\pim$, i.e.\ if
$\pim(\Lambda_1)=\pim(\Lambda_2)$ then
$\pim(\varphi(\Lambda_1))=\pim(\varphi(\Lambda_2))$. Furthermore, the
map induced by $\varphi$ on $\Omega_{max}$ must be the rotation by
$\eta:=\pim(\Lambda_2)-\pim(\Lambda_1)$, since this is a homeomorphism
(and hence the only one) on $\Omega_{max}$ mapping $\pim(\Lambda_1)-x$
to $\pim(\Lambda_2)-x$ for all $x\in\R^d$.  It follows then that for
model sets the rotation by $\eta$ must leave the set $NS$ of
nonsingular points invariant.  But for a generic choice of the window the only translations
leaving $NS$ invariant are the elements of $(\{0\}\times \R^d
+\Gamma)/\Gamma$. It follows that on the orbit of a non-singular model set $\varphi$ is given by
a global translation. By continuity $\varphi$ must then be everywhere this global translation.
Thus for generic model sets auto-conjugacies are global translations and hence in particular MLD transformations.
The more general case in which
$NS$ admits symmetries is presently under investigation.

\subsection{Asymptotically negligible co-chains}
It turns out that the map $\Lambda-x\mapsto \Lambda^F-x$ extends to a
topological semi-conjugacy $\s_F:\Omega_\Lambda\to\Omega_{\Lambda^F}$
if and only if $F$ is \wpe\ \cite{CS2,Kel}.  So the generator of a
shape semi-conjugacy is a \wpe\ function $F:\Lambda\to\R^d$ whose
coboundary is \spe. If $F$ were \sP, then the shape deformation would
be a local derivation.  If $F$ is small enough, then this procedure
can be inverted \cite{Kel}, so that $\s_F$ is a shape {\em conjugacy\/}.

Shape semi-conjugacies (and small shape conjugacies), up to MLD shape
conjugacies, are thus parametrized by the sub-group of the first
cohomology group consisting of \spe\ co-cycles which are coboundaries
of \wpe\ functions. Such co-cycles are called {\em asymptoticaly
  negligible\/} and we denote the subgroup by
$H^1_{an}(\Lambda,\R^d)$.

\begin{prop}
The coboundary of the star map $\sigma_M$ of a model set is an asymptotically negligible
$1$-cocycle.
\end{prop}
\begin{remarks}
The above proposition can be understood as a generalization of a
result from Boulmezaoud's thesis \cite{B-thesis}.\footnote{Boulmezaoud
  showed that in the case that the internal group is $\R^n$ the star
  map $\R^d\to\R^n$, $x\mapsto x^*$ extends to a \wpe\ smooth function
  whose differential is \spe.}

\end{remarks}
\begin{proof} By Lemma~\ref{lem-starwpe} $\sigma_M:M\to H$ is \wpe. That its coboundary is \spe\ is a direct consequence of its additivity. In fact,
if $e$ is an edge between points $x_1,x_2$ in $M$ then $\delta \sigma_M(e) =
\sigma_M(\partial e) = \sigma_M(x_2)-\sigma_M(x_1) = (x_2-x_1)^*$,
and so depends only on the displacement between $x_1$ and
$x_2$.  $\delta \sigma_M(e)$ is thus \spe{} for any radius $R$ greater than the maximal distance between two neighboring points.
\end{proof}

\begin{cor} \label{cor-repr}
Consider a model set $M$ with
  internal group $H$. Let
  $L:H\to \R^d$ be a continuous group homomorphism. Then
  $F_L:M\to\R^d$, $F_L(x) = L(\sigma_M(x))$ is a generator of a shape
  conjugation. The model set $M^{F_L}$ resulting from the shape change
  is the reprojection along $H' = \{(h,- L(h))\in H\times\R^d: h\in
  H\}$.
\end{cor}
\begin{proof}
Continuity of $L$ implies that $F_L$ is \wpe. Additivity of $L$
implies that $F_L$ has \spe\ coboundary. Finally,
we have $M^{F_L} =\{x+{{L}}(\sigma_M(x)):x\in M\} =
\{(\pi^\|+{{L}}\circ\pi^\perp)(\sigma_M(x),x) : x\in M \}$.
Now the elements in the kernel of $\pi^\|+{{L}}\circ\pi^\perp$ have the form $(h, -L(h))$, $h\in H$ and so $\pi^\|+{{L}}\circ\pi^\perp$ is the
projection onto $\R^d$ along the subspace
$H' = \{(h,- L(h))\in H\times\R^d: h\in  H\}$.
\end{proof}

\subsection{Nonslip generators}

We will introduce a property for generators of shape conjugations
which characterizes those we have discussed above in the context of model sets. We state the definition for
arbitrary \wpe{} functions, but the primary application is for vector-valued functions.
Recall that $F$ being \wpe\ means that there exists a continuous
function $\tilde F:\Xi_\Lambda\to \R^d$ such that $F(x) = \tilde
F(\Lambda-x)$. We denote $\Rmax^\Xi = \Rmax\cap \Xi\times\Xi$.

\begin{dfn}\label{def-repr}
  Let $\Lambda$ be a Meyer set.  A \wpe{} function $F:\Lambda\to Y$ is
  {\em \ns} if there exists an $\epsilon>0$ such that,
  for all $(\Lambda_1,\Lambda_2)\in \Rmax^\Xi$ we have
  $\tilde F({\Lambda_1}) = \tilde F({\Lambda_2})$ whenever

$\Lambda_{1}$ and $\Lambda_2$ agree out to a radius of $\epsilon^{-1}$.

We call $R=\epsilon^{-1}$ the \ns\ radius of $F$.
\end{dfn}
A \spe\ function is manifestly \ns. The star map is not \spe, but we
shall see below that it is \ns.

\begin{lem}
The star map of $M$ is \ns.
\end{lem}

\begin{proof}
Let $M_1,M_2\in\Omega_M$.
If $\pim(M_1)=\pim(M_2)=\xi$ and $x \in M_1\cap M_2$, then
equation (\ref{eq-pistar}) implies that
$\sigma_{M_1}(x)=\sigma_{M_2}(x)$.
Since $\tilde\sigma_{M}(M_i-x) = \sigma_{M_i}(x)$
for $i=1,2$ the star map is non-slip
for every positive radius.
\end{proof}
\begin{cor}
Every generator of a shape conjugation of the form $F_L$ is \ns.
\end{cor}

\subsection{Nonslip = reprojection}

We have just seen that continuous linear functions $L: H \to \R^d$ define \ns\
generators of shape conjugations, and hence that reprojections are generated
by \ns{} generators. We
next show the converse, that under hypotheses H1 and H$2'$, all \ns{} generators are
essentially of this form.  The following theorem is the main result of
this section, and proves half of Theorem~\ref{NewMainTheorem}.

\begin{thm}\label{mainlemma}
 Let $M$ be a model set satisfying hypotheses H1 and H$2'$.
Let $F:M\to\R$ be a \wpe\ function that  is \ns, and whose coboundary $\delta F$ is \spe.

Then $F$ can be written as $F(x) = L(\sigma_M(x)) + \psi(x)$, where $L:H\to\R$ is a
  continuous linear map and  $\psi: M\to\R$ is \spe.
\end{thm}

\begin{proof}
Note that since all elements of $C$ have finite order, $\{0\}\times C$
must lie in the kernel of $L$
and $L$ is determined on $\R^n$ alone.

To view the \ns{} property from a different angle we consider an
inverse limit construction for the  canonical transversal
$\Xi$.
Let $M_1\sim_\epsilon M_2$ if
$\pim(M_1)=\pim(M_2)$ and the patterns  $M_1$, $M_2$ agree out to radius
$\epsilon^{-1}$.  This is the intersection of two closed equivalence
relations and therefore itself a closed equivalence relation; the
quotient space $\Xi/\sim_\epsilon$ is a compact Hausdorff
space. Denote by $\pi_\epsilon:\Xi\to\Xi/\sim\epsilon$ the canonical
projection. Then
$$ \Xi = \lim_{0 \leftarrow \epsilon} \Xi/\sim_\epsilon$$
and a \wP\ $F$ is \ns\ iff $\tilde F$ is the pullback by $\pi_\epsilon$ of
a continuous function on some approximant $\Xi/\sim_\epsilon$.

Let $F:M\to\R$ be a \ns, \wpe\ function, whose coboundary $\delta F$ is \spe.
Since the star map $\sigma_M$ is also \ns,
there exists an $\epsilon>0$ such that we have the commutative diagram
$$
\begin{array}{rcccc}
\Xi & \stackrel{\pi_\epsilon}{\to}& \Xi/\sim_\epsilon &\stackrel{\widetilde{\sigma_M}^\epsilon}\to & H \\
\tilde F \downarrow & &  \tilde F^\epsilon \downarrow & & \\
\R & = &\R & = &\R
\end{array}
$$
where $\tilde F^\epsilon$ and $\widetilde{\sigma_M}^\epsilon$ are the pullbacks of
the continuous maps $\tilde F$ and $\widetilde{\sigma_M}$ induced by $F$ and $\sigma_M$. We need to show
that there exists a $0<\eta\leq\epsilon$, a \spe{} function $\psi$ and a continuous group
homomorphism $L$ such that the right side of the diagram can be
completed to a commutative diagram
$$
\begin{array}{rcl}
\Xi/\sim_\eta &\stackrel{\widetilde{\sigma_M}^\eta}\to & H \\
\tilde F^\eta-\tilde\psi^\eta \downarrow & & \downarrow L\\
\R & = &\R
\end{array}
$$
($\eta^{-1}$ is hence at least as large as the \spe\ radius of $\psi$ and the \ns\ radius of $F$).
Given that a global translation of a point set is an MLD transformation which may be absorbed in the definition of $\psi$ we may assume that $M$ contains the origin.

By FLC there are a finite number of possible $\epsilon^{-1}$-patches
at $0$ which contain the origin. By H$2'$ the acceptance domain of each
$\epsilon^{-1}$-patch can be written as a finite union of closed
convex sets that have non-empty interior.  We call these convex sets
{\em sectors\/} and let $I$ index the sectors of all
$\epsilon^{-1}$-patches.  Thus $\alpha \in I$ denotes both a patch
(located for reference at the origin) and a closed convex subset
$W^\alpha \subset W$ of non-empty interior.  For each such $\alpha$,
let $\Xi^\alpha$ be the corresponding subset of $\Xi$, i.e.\ the set
of point patterns that (a) contain the origin, (b) have the correct
$\epsilon^{-1}$ patch around the origin, and (c) are mapped to sector
$W^\alpha$ by $\widetilde{\sigma_M}$.

The crucial observation is that on $\Xi^\alpha$ the equivalence
relation $\sim_\epsilon$ coincides with the relation defined by
$\pim$, and hence for $M_1,M_2\in \Xi^\alpha$ we have
$M_1\sim_\epsilon M_2$ iff $\widetilde{\sigma_M}(M_1) =
\widetilde{\sigma_M}(M_2)$. In other words, the restriction of
$\widetilde{\sigma_M}^\epsilon$ to $\Xi^\alpha/\sim_\epsilon$ is a
homeomorphism between $\Xi^\alpha/\sim_\epsilon$ and $W^\alpha$.  It
follows that for each $\alpha$ there is a unique function
$f^\alpha:W^\alpha\to\R$ such that
\begin{equation}\label{eq-com}
\begin{array}{rcccl}
\Xi^\alpha & \stackrel{\pi_\epsilon}{\to}&\Xi^\alpha/\sim_\epsilon &\stackrel{\widetilde{\sigma_M}^\epsilon}\to & W^\alpha \\
\tilde F\downarrow & & \tilde F^\epsilon \downarrow & & \downarrow f^\alpha\\
\R& = & \R & = &\R
\end{array}
\end{equation}
commutes.

Let $y\in D = \sigma^{-1}((W-W)\cap\Gamma^\perp)$
be a possible displacement between two points in the same pattern.
Define $$\Delta_y\tilde F:\Xi \cap (\Xi +y) \to \R$$
by
$$\Delta_y\tilde F(M') = \tilde F(M'-y) -\tilde F(M').$$
We consider also the
restriction $\Delta_y^{\alpha,\alpha'}\tilde F$ of $\Delta_y\tilde F$
to $\Xi^\alpha\cap (\Xi^{\alpha'}+y)$. Taking into account the fact that
$\widetilde{\sigma_M}(M'-y)- \widetilde{\sigma_M}(M')
=\sigma_{M'}(y) - \sigma_{M'}(0) = y^*$, the preceding
paragraph shows that
$$ \Delta_y^{\alpha,\alpha'}\tilde F(M') =
\Delta_{y^*}^{\alpha,\alpha'} f(\widetilde{\sigma_M}(M'))
$$
where  $\Delta_{v}^{\alpha,\alpha'} f(u)=f^{\alpha'}(u+v) -   f^\alpha(u)$.

Since $F$ has \spe{} coboundary, and since \spe{} functions are
locally constant on $\Xi$, $\Delta_y \tilde F$ is locally constant on
$\Xi \cap (\Xi + y)$. Hence by (\ref{eq-com}) the function
$\Delta_{y^*}^{\alpha,\alpha'} f$ is locally constant on $W^\alpha\cap
(W^{\alpha'}-y^*)$. Since $W^\alpha$ and $W^{\alpha'}$ are convex,
$W^\alpha\cap (W^{\alpha'}-y^*)$ is connected or empty. Hence the
function $\Delta_{y^*}^{\alpha,\alpha'} f$ is actually constant on
$W^\alpha\cap (W^{\alpha'}-y^*)$.

There are finitely many sectors and each sector has non-empty
interior.  So there is an open neighborhood $U\subset H$ of $0$ such
that for all $\alpha$ and all $v\in U$ we have $W^\alpha\cap
(W^{\alpha}-v)\neq\emptyset$.  We claim that for $v\in U$ the value of
$\Delta_{v}^{\alpha,\alpha} f(u)=f^{\alpha}(u+v) - f^\alpha(u)$ is
independent of $\alpha$ as well as independent of $u\in W^\alpha\cap
(W^{\alpha}-v)$ (we know already that it is independent of $u$ if $v=y^*$ with $y\in D$).

The sectors have non-empty interior, so by Corollary \ref{cor-W} for each
sector $\alpha$ we can find a patch $P^\alpha$ of $M$ whose acceptance
domain is contained in the interior of $W^\alpha$.  By repetitivity,
there is a radius such that every ball of that radius contains at
least one copy of each patch $P^\alpha$. Let $P$ be a patch of $M$ of
that radius, so that $P$ contains translates of all the patches
$P^\alpha$.  That is, there are $x^\alpha\in \R^d$ such that
$P^\alpha+x^\alpha$ is a subpatch of $P$. It follows that
$\widetilde{\sigma_M}(M-x^\alpha)$ lies in the interior of $W^\alpha$.
By Corollary \ref{cor-acc} the set $\{\sigma_M(x) : P=(M-x)\cap B\}$ is
dense in the acceptance domain $W_P$.  If $P=(M-x_1)\cap B=(M-x_2)\cap
B$ we call $x_2-x_1$ a {\em return vector\/} of $P$.  The possible
values of $y^*$ for return vectors $y$ of $P$ are thus dense in
$W_P-W_P$.

Pick a $y$ such $y^*$ is a return vector of $P$.
We then have
$$\Delta_y \tilde F(M - x^\alpha) - \Delta_y \tilde F(M - x^{\alpha'})
= \tilde F(M - x^\alpha)-\tilde F(M-x^{\alpha'}) - (\tilde F(M -y-
x^{\alpha})-\tilde F(M-y-x^{\alpha'})) .$$ Now $\tilde F(M -
x^\alpha)-\tilde F(M-x^{\alpha'})$ is obtained by adding the $\delta
F(e_i)$ over the edges $e_i$ along a path in $P$ which joins
$x^\alpha$ to $x^{\alpha'}$, while $F(M-y-x^\alpha) -
F(M-y-x^{\alpha'})$ is obtained by summing the values of $\delta
F(e_i)$ over the corresponding path in $P+y$. Since $y$ is a return
vector to $P$ and $\delta F$ is \spe, the result is the same. Hence

$$\Delta_{y^*}^{\alpha, \alpha}f(\widetilde{\sigma_M}(M-x^\alpha))
- \Delta_{y^*}^{\alpha',
  \alpha'}f(\widetilde{\sigma_M}(M-x^{\alpha'})) = \Delta_y \tilde F(M
- x^\alpha) - \Delta_y \tilde F(M - x^{\alpha'}) = 0.$$ To summarize,
we have established that for all $y^*\in (W_P-W_P)\cap \Gamma^\perp$
the value of $\Delta_{y^*}^{\alpha, \alpha}f(u)$ is the same for all
$\alpha$ and all $u\in W^\alpha\cap (W^\alpha-y^*)$. Moreover, for
fixed $u$ in the interior of $W^\alpha\cap (W^\alpha-y^*)$, the
function $v\mapsto \Delta_{v}^{\alpha, \alpha}f(u)$ is continuous in a
neighborhood of $y^*$.  It follows that $\Delta_{v}^{\alpha, \alpha}
f(u)$ is independent of $\alpha$ and $u\in W^\alpha\cap (W^\alpha-v)$
for all $v \in W_P-W_P$.

Let $\tilde U = (W_P^o - W_P^o)\cap U$.  $\tilde U$ is an open
neighborhood of the identity in $H$.
We define $L:\tilde U\to \R$ such that $L(v)$ is the constant value that the
function $\Delta_v^{\alpha,\alpha} f$ takes on
$W^\alpha\cap(W^\alpha-v)$.
We saw that $L$ is continuous. We claim that $L$ is additive where
sums are defined. Indeed, if $u_1,u_2,u_1+u_2\in U$ then there is $u\in W^\alpha$ such that
also $u+u_1$ and $u+u_1+u_2$ lie in $W^\alpha$.
It follows that, for all  $u\in \tilde U$
$$ L(u_1+u_2) =
f^{\alpha}(u+u_1+u_2) - f^\alpha(u+u_1) + f^\alpha(u+u_1) - f^\alpha(u) = L(u_2)+L(u_1).
$$
A continuous additive function on a neighborhood of the origin is
necessarily linear. That is, $L$
equals its derivative.
We may then extend $L$ to a group homomorphism on
the group generated by $\tilde U$ and thus obtain a linear function $L:H\to\R$ that is
trivial on the torsion factor.

Now let
$\psi(x) = F(x) - L(\sigma_M(x))$ for $x \in M$.
This then defines a function $\tilde\psi:\Xi \to \R$,
$\tilde \psi = \tilde F - L\circ\widetilde{\sigma_M}$ which is again
continuous (on $\Xi$).
If $\widetilde{\sigma_M}(M-x) = \widetilde{\sigma_M}(M-y)\in W^\alpha$ then
$$\psi(y)-\psi(x) = \Delta_{y-x}^\alpha \tilde F(M-x) - L((y-x)^*) = 0$$
by the construction above and so $\tilde \psi$ is a continuous
function which is constant on $\Xi^\alpha$. Moreover, if $W^\alpha\cap
W^\beta\neq\emptyset$ then continuity implies that $\tilde \psi$ takes
the same value on
$\Xi^\alpha$ and $\Xi^\beta$. It follows that $\tilde\psi$ is constant on the
the pre-images
under $\widetilde{\sigma_M}$ of the connected components of the
acceptance domains of the $\epsilon^{-1}$-patches at $0$. However, different
components of the same central $\epsilon^{-1}$ patch are separated by
a nonzero distance in $W$, and so
can be distinuished by the $R$-patches at $0$ for some (possibly large) fixed $R>0$.
Different central patches are distinguished by their
patterns out to distance $\epsilon^{-1}$. Thus $\psi(x)$ is in fact
\spe\ with radius $\eta^{-1}$ where $\eta = \min(R^{-1},\epsilon)$.
\end{proof}

\begin{cor} If $F$ is a \ns\ generator of a shape conjugation
for a model set $M$ satisfying H1 and H\/$2'$ then, up to MLD transformations,
$M^F$ is a reprojection of $M$. In particular $M^F$ is a model set.
\end{cor}
\begin{proof}
  Theorem \ref{mainlemma} applied to vector valued functions and
  Corollary \ref{cor-repr} imply that $F$ is the generator of a
  reprojection plus a \spe{} function. Hence, up to an MLD
  transformation, $M^{F}$ is a reprojection of $M$, and is a model
  set.
\end{proof}

\section{Asymptotically negligible = \ns}\label{brillig}

We now turn to the question of when an asymptotically negligible cocycle is \ns. Here we need the stronger assumption H2.

\begin{thm}\label{PolyhedralTheorem}
Let $M$ be a model set satisfying assumptions H1 and H2, and let $F:M\to \R$ be a weakly pattern equivariant function whose coboundary is strongly pattern equivariant. Then $F$ is nonslip.
\end{thm}
\begin{proof}
Let $F:M\to \R$ be \wpe\ with \spe\ $\delta F$. We need to show that
there exists $R>0$ such that, for any choice of
pair $M_1,M_2$ in the hull of $M$ and point $x \in M_1$ such that  $\pim(M_1)=\pim(M_2)$
and $B_{R}\cap (M_1-x) = B_{R}\cap
(M_2-x)$ then $F_2(x)-F_1(x)=0$.

We denote by $R_0$ the radius of pattern equivariance of $\delta F$.
Fix $R>R_0$ and consider {\em doubly pointed double
  $R$-patches\/}. These are double-$R$-patches $(P,Q;B_R(z))$ of
$(M_1,M_2)$ which are centered in a point $z\in M_1\cup M_2$, i.e.\ $P
= M_1\cap B_R(z)$ and $Q = M_2\cap B_R(z)$, together with two points
$x,y\in P\cap Q$ which are at least distance $R_0$ away from $z$.

We denote such an object by
$P^{(2)}(x,y) = (x,y;P,Q;B_R(z))$.
By FLC there are
finitely many up to translation. Since $\delta F$ is \spe\ with radius $R_0$ the
expression
$$F(P^{(2)}(x,y))=\delta F_2(x,y)-\delta F_1(x,y) = F_2(y)-F_1(y) -(
F_2(x)-F_1(x))$$
depends only on the translational congruence class of $P^{(2)}(x,y)$.
Hence the set $D_R$ of possible values $F$ can take on doubly pointed double $R$-patches is finite.

We now need the following lemma.

\begin{lem}\label{prop-R_1}
There exists $\tilde N$ and $R_1>0$ such that for all pairs $(M_1,M_2)$ with $\pim(M_1)=\pim(M_2)$
and all

$x\in M_1\cap M_2$ we have $F_2(x)-F_1(x) \in \tilde D_{R_1}: = {D_{R_1}+\cdots+D_{R_1}}$ ($\tilde N$ copies).
\end{lem}
\begin{proof}[Proof of the lemma]

Let $\V$ be the collection of subspaces $E_f$ of $\R^d$ associated to the faces of $W$.
Let $\nu\in\R^d$ be a vector of length one
and pick $\omega>0$ such that the cone
$$C := \bigcup_{\lambda\geq 0} B_{\lambda \omega}(\lambda \nu)$$
intersects a vectorspace of $\V$ only trivially. Therefore $C$ intersects each hyperplane from $\aaa(M_1,M_2)$ at most once and so, by Lemma~\ref{bound-N}, intersects in total at most $N$ hyperplanes.

Now let $M_1,M_2\in \Omega_M$ satisfy $\pim(M_1)=\pim(M_2)$ and
consider a point $x\in M_1\cap M_2$.  Application of
Proposition \ref{rel-dense} with $r>0$ guaranties that there is $\rho>0 $ so
that we may choose a point $x_n\in B_\rho(x+n\rho\nu)$ at which $M_1$
and $M_2$ agree, and this for all $n\in\N$. We choose $x_0=x$ and thus
obtain a sequence $(x_n)_n\subset M_1\cap M_2$. Let $R_1 = 2\rho +
R_0$ and consider the sequence of doubly pointed double $R_1$-patches
$P^{(2)}_n(x_n,x_{n+1}) =
(x_n,x_{n+1};P_n,Q_n;B_{2\rho}(x+(n+1/2)\rho\nu))$.  If $n\geq
4/\omega$ then $P^{(2)}_n(x_n,x_{n+1})$ is contained in $x+C$. Since
$x+C$ intersects at most $N$ hyperplanes from $\aaa(M_1,M_2)$ at most
$\tilde N = N+4/\omega$ double patches $(P_n,Q_n)$ are distinct and so
for at most $\tilde N$ values of $n$ we have $F(P^{(2)}_n)\neq
0$. Once all hyperplanes are crossed, $M_1-x_n$ and $M_2-x_n$ agree
out to distance approximately $n\rho\omega$ and hence $\lim_{n\to
  \infty} (F_2(x_n)-F_1(x_n)) = 0$ by the weak pattern equivariance of
$F$. Thus $F_2(x)-F_1(x) = \sum_n F(P^{(2)}_n) \in \tilde D_{R_1}$
with $R_1 = 2\rho + R_0$.
\end{proof}

We continue the proof of Theorem \ref{PolyhedralTheorem}. If $\tilde
D_{R_1} = \{0\}$, then $F$ is \ns{} with radius $R_1$.  Otherwise, let
$c = \min\{|d|:d\in \tilde D_{R_1}\backslash \{0\}\}$.  Since $F$ is
\wpe\ $\tilde F$ is uniformly continuous, so there exists $R$ such
that $B_{R}\cap (M_1-x) = B_{R}\cap (M_2-x)$ implies $|\tilde
F(M_1-x)-\tilde F(M_2-x)|< c$. But if $B_{R}\cap (M_1-x) = B_{R}\cap
(M_2-x)$ then $x\in M_1\cap M_2$, so the inequality implies $|\tilde
F(M_1-x)-\tilde F(M_2-x)|= 0$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{NewMainTheorem}]

  If $\mo'$ is pointed topologically conjugate to $\mo$, then $\mo'$
  is MLD to a pattern $\mo''$ that is shape conjugate to $\mo$. Let
  $F$ be the generator of that shape conjugacy. By Theorem
  \ref{PolyhedralTheorem} applied to vector-valued functions, $F$ is
  \ns. By Theorem \ref{mainlemma} applied to vector-valued functions,
  $F$ is then the sum of a linear map $L: H \to \R^d$ and a \spe{}
  function $\psi_\epsilon$. Since $L$ induces a reprojection and
  $\psi_\epsilon$ induces an MLD transformation, $\mo''$ is MLD to a
  reprojection of $\mo$.  Since $\mo'$ is MLD to $\mo''$, $\mo'$ is
  also MLD to a reprojection of $\mo$.
\end{proof}

\section{Cohomological interpretation}\label{sec-cohomology}

We have already described how the coboundary $\delta F$ of a generator
of a shape deformation defines an element in the first cohomology
$H^1(\Lambda,\R^d)$ of $\Lambda$ with coefficients in $\R^d$. Here the
cohomology is the cohomology of \spe\ cochains on the $CW$-complex
defined by $\Lambda$. Alternatively we can describe
$H^1(\Lambda,\R^d)$ as the cohomology of the complex of \spe\
$\R^d$-valued de Rham $1$-forms on $\R^d$. Its elements then arise as
differentials $dF$ of smooth functions $F:\R^d\to \R^d$ and the
relation between the two description is given by integration: Given a
$1$-chain defined by $\Lambda$, that is, essentially an oriented edge
$e$ in a tiling whose vertices are given by $\Lambda$, $\delta F(e) =
\int_e dF = F(y)-F(x)$, where $x,y\in\Lambda$ are the source and the
range vertex of the edge.

There are a number of canonical
subgroups of $H^1(\Lambda, \R^d)$, and our results can be viewed as
saying when these subgroups are, and aren't, equal.

One subgroup, denoted $H^1_{lin}(\Lambda,\R^d)$ is given by generators
$F$ that are the the restriction to $\Lambda$ of linear maps
$L:\R^d\to\R^d$. For each such $F$, the deformed set is simply the
result of applying the linear transformation $\mbox{\rm id}+L$ to the
points of $\Lambda$. If $L$ is non-zero then the deformation cannot be
a local derivation, so $H^1_{lin}(\Lambda,\R^d)$ is isomorphic to
$Hom(\R^d,\R^d) \cong \R^{d^2}$ (as a vector space).

For model sets, reprojections give another subgroup,
$H^1_{repr}(\Lambda,\R^d)$. Elements of $H^1_{repr}(\Lambda, \R^d)$
correspond to generators of the form $F_L(x) = L(\sigma_M(x))$, where
$L:H\to \R^d$ is a continuous group homomorphism.

The {\em asymptotically negligible\/} classes \cite{CS2}
$H^1_{an}(\Lambda, \R^d)$ are represented by the coboundaries of
shape semi-conjugacies. That is, by \spe{} 1-cochains that are
the coboundaries of \wpe{} functions.
Recall that a
generator of a shape semi-conjugacy is \wpe\ and hence
bounded.  (In fact, a generator $F$ is \wpe\ iff it is bounded
  \cite{KS13}). Since linear maps are unbounded, it follows that
$H^1_{lin}(\Lambda,\R^d)\cap H^1_{an}(\Lambda,\R^d)
=\{0\}$.

We also considered \ns{} generators. Recall that any
\spe\ generator is \ns{} and a \ns{} generator is asymptotically
negligible. Hence the classes of \ns\ generators define
a subgroup $H^1_{ns}(\Lambda,\R^d)$ of $H^1_{an}(\Lambda,\R^d)$.
Furthermore, in the context of  a model set $\mo$, $F_L$ is \ns\ and so
$H^1_{repr}(\mo,\R^d)$ is a subgroup of $H^1_{ns}(\mo,\R^d)$.
Thus we have a sequence of inclusions
$$ H^1_{repr}(\mo, \R^d) \subset H^1_{ns}(\mo, \R^d)
\subset H^1_{an}(\mo, \R^d).$$
A natural question is whether these groups coincide.

\subsection{Reinterpretation of Theorems ~\ref{mainlemma} and \ref{PolyhedralTheorem}}
We can also consider functions and cochains
with values in $\R$ rather than with values in $\R^d$, with
$$ H^1_{repr}(\Lambda, \R^d) =  H^1_{repr}(\Lambda, \R) \otimes \R^d, \quad
H^1_{ns}(\Lambda, \R^d) =  H^1_{ns}(\Lambda, \R) \otimes \R^d, \quad
H^1_{an}(\Lambda, \R^d) =  H^1_{an}(\Lambda, \R) \otimes \R^d.
$$

The following are immediate corollaries of Theorem \ref{mainlemma}
and Theorem \ref{PolyhedralTheorem}.
\begin{cor}
If $\mo$ is a model set satisfying
H1 and H\/$2'$, then $H^1_{repr}(\mo, \R) = H^1_{ns}(\mo, \R)$.
\end{cor}
\begin{proof} By Theorem \ref{mainlemma}, each \ns{} generator is
the sum of a linear function on $H$ and a \spe{} function, so
the 1-cohomology class of its coboundary is in $H^1_{repr}(\mo, \R)$.
\end{proof}
\begin{cor}
If $\mo$ is a model set satisfying
H1 and H2, then $H^1_{ns}(\mo, \R) = H^1_{an}(\mo, \R)$.
\end{cor}
\begin{proof}Theorem \ref{PolyhedralTheorem} gives this result on the level
of cochains. Every generator of a shape semi-conjugacy is \ns, period.
\end{proof}
The last two corollaries together give Theorem~\ref{MainTheorem2}.

\subsection{Image of the first cohomology of the maximal
  equicontinuous torus}
The maximal equicontinuous factor map $\pi_{max}:\Omega\to
\Omega_{max}$ induces an injective map in cohomology
$\pi_{max}^*:H^1(\Omega_{max},\Z)\to H^1(\Omega,\Z)$.
We have thus a fourth subgroup which is worth comparing with the other,
namely the image under $\pim^*$ of  $H^1(\Omega_{max},\R^d)$ which we denote by
$H^1_{max}(\Lambda,\R^d)$.
To do that we need a better understanding of
the image of $\pi_{max}^*$.

We recall from \cite{BKS} that $\Omega_{max}$ can be alternatively described with the
help of the topological eigenvalues of the action. Let $\hat \R^d =
Hom(\R^d,U(1))$ be the Pontryagin dual of $\R^d$, where we require
each homomorphism to be continuous on $\R^d$. An element
$\chi\in\hat{\R^d}$ is a topological eigenvalue if there exists a
non-vanishing continuous function $f:\Omega\to\C$ such that
$f(\Lambda-t) = \chi(t)f(\Lambda)$. Topological eigenvalues form a
countable subgroup $\eee$ of $\hat{\R^d}$ and $\Omega_{max}$ can be
identified with the dual $\hat\eee = Hom(\eee, U(1))$, where $\eee$ is
given the discrete topology.

For a general topological space $X$,
$H^1(X,\Z)$ is isomorphic to $[X,S^1]$, the homotopy classes of
continuous maps $X\to S^1$. Furthermore, if $\varphi:X\to Y$ is a
continuous map then $\varphi^*:H^1(Y,\Z)\to H^1(X,\Z)$ can be
identified with the mapping $[Y,S^1]\ni[f] \mapsto [f\circ\varphi]\in
[X,S^1]$. We apply this to $\pi_{max}:\Omega\to\Omega_{max}$.
Since $\Omega_{max}\cong\hat{\mathcal E}$ the elements of
$[\Omega_{max},S^1]$ are the homotopy classes of characters on
$\hat{\mathcal E}$. Hence  $[\Omega_{max},S^1]\cong \mathcal E$ and
the image of $\chi\in \mathcal E$ under $\pi_{max}^*$ in $[\Omega,S^1]$
is given by the homotopy class of an eigenfunction $f_\chi$ of $\chi$ ($f_\chi$ is
normalized so as to have modulus $1$). This describes the image of
$\pi_{max}^*$ (in degree one) in $[\Omega,S^1]$ \cite{BKS}.

To obtain the image of $\pi_{max}^*$ in pattern equivariant cohomology
we consider the restriction of a representative $f$ of an
element of $[\Omega,S^1]$ to the orbit of $\Lambda$ and define
$$\check f:\R^d\to S^1, \quad \check f(x) := f(\Lambda-x).$$
\begin{lem}
Any element of $[\Omega,S^1]$ admits a representative  $f$ such
that
$\check f(x) := f(\Lambda-x)$ is \spe.
\end{lem}
\begin{proof}
$[\Omega,S^1]$ can be seen as the direct limit of $[\mathcal G_n,S^1]$ where
$\mathcal G_n$ is the $nth$ approximant in the G\"ahler complex \cite{SadunG}.
Each element thus comes from some $[\mathcal G_n,S^1]$ and the latter
elements produce \spe\ functions when considered on the orbit.
\end{proof}
Since $\R^d$ is simply connected we can
lift $\check f$ to a continuous function
$\tau:\R^d\to\R$ such that $\check f(x) = \exp{2\pi i \tau(x)}$.
We define $F$ to be the restriction of $\tau$ to $\Lambda$. Then $\delta
F$ is \spe\ and so we have a map
$[\Omega,S^1]\to H^1(\Lambda,\Z)$: $[f] \mapsto [\delta F]$.
\begin{lem}
  With the above notation $[f] \mapsto [\delta F]$ is a group
  homomorphism whose image corresponds to the image of $[\Omega,S^1]$
  in $H^1(\Omega,\R)$ under the identification $H^1(\Omega,\Z)\cong
  H^1(\Lambda,\Z)\subset H^1(\Lambda,\R)$.
\end{lem}
\begin{proof}
  Using G\"ahler's approximation this statement boils down to consider
  the map between $[\mathcal G_n,S^1]$ and $H^1(\mathcal G_n,\R)$
  where the latter can be considered as de Rham cohomology on a
  branched manifold. If $[f]\in [\mathcal G_n,S^1]$ then (assuming
  without restriction of generality that $f$ is smooth) $\frac1{2\pi
    i}f^{-1}df$ represents the element in $H^1(\mathcal G_n,\R)$ under
  the map $[\mathcal G_n,S^1]\to H^1(\mathcal G_n,\Z)\subset
  H^1(\mathcal G_n,\R)$. This is well-known (see also \cite{KP}). Now
  in order to obtain the map on cellular cohomology of $\mathcal G_n$
  one just needs to integrate over $1$-chains. The result is a
  $1$-cochain which, when interpreted as \spe\ $1$-cochain on
  $\Lambda$ coincides presicely with $\delta F$.
\end{proof}

If we combine the two arguments we see that the image of
$H^1(\Omega_{max},\Z)\cong \mathcal E$ in pattern equivariant
cohomology can be described as follows.
\begin{cor}\label{cor-H1}
Upon the above identification of $H^1(\Omega_{max},\Z)$ with $\eee$ and the identification of
$H^1(\Omega,\Z)$ with $H^1(\Lambda,\Z)$ the map $\pim^*$ becomes
the map
$$\eee\ni \chi \mapsto [\delta\tilde\beta\left|_\Lambda\right. ]\in H^1(\Lambda,\Z)$$
where $\tilde \beta\left|_\Lambda\right.$ is the restriction to $\Lambda$ of a
continuous function $\tilde \beta :\R^d\to \R$ such that $\exp(2\pi i\tilde\beta)$ is \spe\ and
homotopic to $\chi$.
\end{cor}
Each eigenvalue $\chi$ can be lifted, that is
there exists a $\beta\in{\R^d}^*$ such that $\chi(x) = \exp 2\pi i \beta(x)$. Therefore
$\tilde \beta - \beta$ must be bounded and hence \wpe. It follows
that $[\delta\tilde\beta] - [\delta\beta]\in
H^1_{an}(\Lambda,\R)$ and, since $\beta$ is linear, we see that
$$ H^1_{max}(\Lambda,\R^d) \subset
H^1_{lin}(\Lambda,\R^d)+H^1_{an}(\Lambda,\R^d).$$
Stated differently, the image of $ H^1_{max}(\Lambda,\R^d)$ in the
quotient group\footnote{This quotient group

is the mixed group of \cite{Kel}
studied in \cite{B-thesis} for projection method tilings.}
 $$H^1_{m}(\Lambda,\R^d) :=
H^1(\Lambda,\R^d)/H^1_{an}(\Lambda,\R^d)$$
can be identified with a subspace of the vector space of linear deformations.
Indeed let
$\psi: H^1(\Omega_{max},\R^d) \to
H^1_{m}(\Lambda,\R^d)$ be the composition of $\pi_{max}^*$ with the
canonical projection. It is induced by
$[\exp 2\pi i \beta] \mapsto [\delta\beta]$.
\begin{lem}
Let $\Lambda$ be a repetitive FLC Delone set.
We have
$$\psi(H^1(\Omega_{max},\R^d)) =
\big(H^1_{lin}(\Lambda,\R^d)+H^1_{an}(\Lambda,\R^d)\big)/
H^1_{an}(\Lambda,\R^d)\cong Hom(\R^d,\R^d)$$
whenever $\Lambda$ is topologically conjugate to a Meyer set.
\end{lem}
\begin{proof}
The above shows that $\psi(H^1(\Omega_{max},\R^d)) \subset
\big(H^1_{lin}(\Lambda,\R^d)+H^1_{an}(\Lambda,\R^d)\big)/
H^1_{an}(\Lambda,\R^d)$ and the r.h.s.\ is clearly isomorphic to the vector space of all linear deformations. Thus  equality holds precisely if the real span of $\{\beta\in{\R^d}^* :\exp 2\pi i \beta \in \eee\}$ has dimension $d$. By the results of \cite{KS13} this is equivalent to saying that
$\Lambda$ is topologically conjugate to a Meyer set.
\end{proof}
\subsection{Case of model sets}
In the case of model sets which satisfy H1 we can say more, because we have a more
explicit model for the maximal equicontinuous factor, namely the torus parametrization.

Indeed, if $H = \R^n\times C$ then, by cocompactness of $\Gamma$,
$\Omega_{max} =  H\times\R^d/\Gamma$ is an $n+d$-torus
and so we can identify $H^1(\Omega_{max},\Z)\cong \eee\cong \hat\Omega_{max}$ with the
so-called recicrocal lattice  $\Gamma^{rec}$ which
is given by those continuous group homomorphisms $\alpha:H\times\R^d\to \R$ which satisfy
$\alpha(\gamma) \in\Z$ for all $\gamma\in\Gamma$. Again, $\alpha$ must be trivial on the torsion component $C$ and restricts to a linear map $\R^n\times\R^d\to\R$.
\begin{prop} Let $\mo$ a model set satisfying H1.
Upon the identification of $H^1(\Omega_{max},\Z)$ with $\Gamma^{rec}$ and the identification of
$H^1(\Omega,\Z)$ with $H^1(\mo,\Z)$ the map $\pim^*$ becomes
the map
$$\Gamma^{rec}\ni \alpha \mapsto [\delta \alpha(\sigma_M(\cdot),\cdot))]\in H^1(\mo,\Z).$$
\end{prop}
\begin{proof}
The eigenvalue $\chi\in\eee$ corresponding to $\alpha\in\Gamma^{rec}$ is
$\chi(x) = \exp 2\pi i \alpha(0,x)$, $x\in\R^d$. Recall that
$\sigma_M:M\to H$ is \wpe\ and has \spe\ coboundary. We may therefore extend it to
a \wpe\ function on all of $\R^d$ whose differential is \spe\
\cite{Kel}.  We denote the extension also by $\sigma_M$.
As  $\{\sigma_M(x):x\in M\}$ lies in a
compact subset of $H$,
$\chi_t(x) = \exp 2\pi i \alpha(t\sigma_M(x),x)$
is a homotopy between $\chi$ and $\chi_1$.
Furthermore, if $x,y\in M$ then $(\sigma_M(y),y)-(\sigma_M(x),x)=
((y-x)^*,y-x)\in\Gamma$ and so $M\ni x \mapsto \exp 2\pi i
\alpha(\sigma_M(x),x))$ is constant and hence a \spe\ function on
$M$. It follows that $\chi_1$ is a \spe\ function on $\R^d$ and thus
we may apply Corollary \ref{cor-H1} to obtain the statement.
\end{proof}

\begin{prop}\label{prop-impi} Let $\mo$ be a model set satisfying H1, that is, $H=\R^n\times C$
with some finite group $C$. Then $H^1_{repr}(\mo,\R)$ has dimension $n$ and
$$H^1_{max}(\mo,\R^d) =  H^1_{repr}(\mo,\R^d)\oplus H^1_{lin}(\mo,\R^d).$$

\end{prop}
\begin{proof}
We have $H^1_{max}(\mo,\R)\cong\Gamma^{rec}\otimes_\Z\R\cong (\R^n\times\R^d)^*\cong
{\R^n}^*\oplus{\R^d}^*$.  Thus an element corresponding to $\alpha\in\Gamma^{rec}\otimes_\Z\R$ can be split into $(\alpha^\perp,\alpha^\|)\in {\R^n}^*\oplus{\R^d}^*$. Under this splitting the coboundary
$\delta \alpha(\sigma_M(\cdot),\cdot)$ becomes $(\delta \alpha^\perp\circ\sigma_M,\delta \alpha^\|)$.
With $\R^d$ coefficients this induces exactly the splitting $H^1_{max}(\mo,\R^d) =  H^1_{repr}(\mo,\R^d)\oplus H^1_{lin}(\mo,\R^d)$. This also shows that the dimension of $H^1_{repr}(\mo,\R)$ is $n$.
\end{proof}
With this proposition at hand we see that Theorem~\ref{MainTheorem2} is equivalent to
Theorem~\ref{MainTheorem3}. Indeed, it shows that for polyhedral model sets
the statement $\dim H^1_{an}(\mo,\R)=n$ is equivalent to
$H^1_{an}(\mo,\R) \subset H^1_{max}(\mo,\R)$.

\section{Nonslip sets and the Meyer property}\label{sec-Meyer}

In this section we explore a little further the concept of \ns\
generators.  Our hope is that this may turn out useful later for the
study of shape conjugations of Delone sets which are not model.  We
consider an analogous property of sets, and show that a generator of a
shape conjugation is \ns\ if an only its associated shape conjugacy
preserves that property.

\begin{dfn} A Delone set $\Lambda$ is {\em \ns} if for all $R>0$ one
 can find an $\epsilon>0$ such that for all $\Lambda_1,\Lambda_2\in\Omega_\Lambda$
 we have that if
  $\pim(\Lambda_1)=\pim(\Lambda_2)$ and $d(\Lambda_1,\Lambda_2)\leq
  \epsilon$, then both sets agree on $B_R(0)$.
\end{dfn}

\begin{lem}
  Every Meyer set is \ns.
\end{lem}
\begin{proof}
Recall that for Meyer sets $\pim(\Lambda_1)=\pim(\Lambda_2)$ implies that $0\in \Lambda_1-\Lambda_2$. But $\Lambda_1-\Lambda_2$
  is uniformly discrete by the Meyer property. Hence if $\Lambda_1$
  and $\Lambda_2$ are close enough they have to coincide on a ball of
  radius equal to the inverse of their distance.
\end{proof}

 Recall that a  generator of shape conjugacy of an FLC Delone set $\Lambda\subset\R^d$ is a
 function $F:\Lambda\to\R^d$ such that $\Lambda\mapsto \Lambda^F = \{x+F(x):x\in\Lambda\}$ extends to an $\R^d$-equivariant homeomorphixm $\s_F:\Omega_\Lambda\to \Omega_{\Lambda^F}$.
We saw that in this case $F$ extends to a continuous map $\tilde F:\Omega_\Lambda\to\R^d$ and so we may define
$ F_{\Lambda'}:\Lambda'\to\R^d$
by $F_{\Lambda'}(x) = \tilde F(\Lambda'-x)$. It is not difficult to see that $ F_{\Lambda'}$ is \wpe\
for ${\Lambda'}$
and that $\s_F(\Lambda') = {\Lambda'}^{F_{\Lambda'}} = \{x+F_{\Lambda'}(x): x\in {\Lambda'}\}$.
Indeed $\s_F(\Lambda') = \lim \s_F(\Lambda)-x_n$ for some sequence $(\Lambda-x_n)_n$ converging to $\Lambda'$, and
$\s_F(\Lambda)-x_n = \{x+\tilde F(\Lambda - x):x\in\Lambda\}-x_n =  \{y+\tilde F(\Lambda -x_n-y):y\in\Lambda-x_n\}$. Now since $\tilde F$ is bounded and continuous we conclude that
$\lim \s_F(\Lambda)-x_n = \{y-\tilde F(\Lambda'-y):y\in\Lambda'\}$.

Nonslip generators of shape conjugacies and \ns\ sets are closely
related.

For a more general (not necessarily Meyer) Delone set we
generalize the concept of a \ns\ \wpe\ function as follows:
\begin{dfn}
Let $\Lambda$ be an FLC Delone set.
A  \wpe{} function
$F:\Lambda\to Y$ is {\em \ns\/} if there exists an $\epsilon>0$ such that for all $\Lambda_1,\Lambda_2\in\Rmax^\Xi$ we have
$\tilde F(\Lambda_1) = \tilde F(\Lambda_2)$ whenever $d(\Lambda_1,\Lambda_2)\leq \epsilon$.
\end{dfn}

Note that if $\Lambda$ is \ns, then the above definition reduces
to the definition we previously gave for Meyer sets. This
follows as in the proof of the last lemma form the fact  that
$\pim(\Lambda_1)=\pim(\Lambda_2)$ implies that $\Lambda_1$ and $\Lambda_2$
 agree on balls once they are close.
\begin{prop}
Let $\Lambda$ be \ns. $\Lambda^F$ is \ns\ iff $F$ is \ns.
\end{prop}
\begin{proof}
\noindent ``$\Rightarrow$'' We suppose that $\Lambda^F$ is \ns. Hence, given $R$
there exists $\delta$ such that for all $(\Lambda_1,\Lambda_2)\in
\Rmax$ we have $d(\s_F(\Lambda_1), \s_F(\Lambda_2))<\delta$ implies
$B_{R}[\s_F(\Lambda_1)] = B_{R}[\s_F(\Lambda_2)]$. Moreover, $\tilde F$ is
uniformly continuous so there exists $\epsilon_1$ such that for all $\Lambda_1,\Lambda_2\in
\Xi$ we have $d(\Lambda_1,\Lambda_2)<\epsilon_1$ implies
$\|\tilde F(\Lambda_1) - \tilde F(\Lambda_2)\|<R^{-1}$. Furthermore, $\s_F$
is uniformly continuous so  there exists $\epsilon_2$ such that for
all $\Lambda_1,\Lambda_2\in
\Omega$ we have $d(\Lambda_1,\Lambda_2)<\epsilon_2$ implies
$d(\s_F(\Lambda_1), \s_F(\Lambda_2))<\delta$. Finally, $\Lambda$ is \ns\
so
there exists $\epsilon_3$ such that for all $(\Lambda_1,\Lambda_2)\in
\Rmax$ we have $d(\Lambda_1,\Lambda_2)<\epsilon_3$ implies
$B_{1}[\Lambda_1] = B_{1}[\Lambda_2]$. Let $\epsilon =
\min\{\epsilon_1,\epsilon_2,\epsilon_3,\}$ (which depends on
$R$). Then for all $(\Lambda_1,\Lambda_2)\in
\Rmax^\Xi$ we have $d(\Lambda_1,\Lambda_2)<\epsilon$ implies that
$0\in \Lambda_1\cap\Lambda_2$, $B_{R}[\s_F(\Lambda_1)] =
B_{R}[\s_F(\Lambda_2)]$ and $\|F_{\Lambda_1}(0) - F_{\Lambda_2}(0)\|<R^{-1}$.
Since $\s_F(\Lambda_i) = \{ x + F_{\Lambda_i}(x) : x\in\Lambda_i\}$ we
see that, if $R$ is large enough, this implies $F_{\Lambda_1}(0) = F_{\Lambda_2}(0)$.

\medskip

\noindent ``$\Leftarrow$'' We suppose that $F$ is \ns, hence there exists $\delta$ such that
for all $(\Lambda_1,\Lambda_2)\in\Rmax^\Xi$ and all $x\in\Lambda_1\cap\Lambda_2$ we have that $d(\Lambda_1-x,\Lambda_2-x)<\delta$ implies $F_{\Lambda_1}(x) = F_{\Lambda_2}(x)$.
By definition of the metric there exists $\epsilon_1$ such that if $d(\Lambda_1,\Lambda_2)<\epsilon_1$ then  $d(\Lambda_1-x,\Lambda_2-x)<\delta$ for all $x$ of size smaller or equal to the radius of relative denseness.
Let $R>0$ and $\|F\|=\sup_{\Lambda'\in\Omega} \|\tilde F(\Lambda')\|$ which is finite, by the continuity of $\tilde F$.
Since $\Lambda$ is \ns\ there exists $\epsilon_2$ such that
for all $(\Lambda_1,\Lambda_2)\in\Rmax$ we have $d(\Lambda_1,\Lambda_2)<\epsilon_2$ implies
$B_{R+\|F\|}[\Lambda_1] = B_{R+\|F\|}[\Lambda_2]$. Let $\epsilon = \min\{\epsilon_1,\epsilon_2\}$.
Then, if $(\Lambda_1,\Lambda_2)\in\Rmax$ and $d(\Lambda_1,\Lambda_2)<\epsilon$
we have $B_{R+\|F\|}[\Lambda_1] = B_{R+\|F\|}[\Lambda_2]$ and
 $F_{\Lambda_1}(x) =
F_{\Lambda_2}(x)$ for all $x\in \Lambda_1\cap \Lambda_2$ of size smaller or equal to the radius of relative denseness. It follows that $B_{R}[\s_F(\Lambda_1)] = B_{R}[\s_F(\Lambda_2)]$.
Thus $\Lambda^F$ is \ns.
\end{proof}

The following corollary is just a special case:
\begin{cor}\label{cor-ns}
Let $\Lambda$ be a Meyer set and $F$ be a generator of a shape conjugacy.
If $\Lambda^F$  is a Meyer set then $F$ must be \ns.
\end{cor}
The contrapositive says that if $F$ is not \ns, then $\Lambda^F$ is not
Meyer. That is
 Theorem \ref{MainTheorem4}.

Applying these observations to model sets we obtain:
\begin{cor}\label{cor-last}
Given a model set $\mo$ which satisfies H1.
If $H^1_{ns}(\mo,\R) \subset H^1_{max}(\mo,\R)$
then any shape conjugation of $\mo$ which is a Meyer set is a reprojection of $\mo$.
\end{cor}
\begin{proof}
Let $F$ be the generator of a shape conjugation of $\mo$ such that $\mo^F$ is a Meyer set.
By Corollary \ref{cor-ns}, $F$ must be \ns.
By Proposition \ref{prop-impi}, $H^1_{ns}(\mo,\R) \subset H^1_{max}(\mo,\R)$
is equivalent to the equality
$H^1_{ns}(\mo,\R) = H^1_{repr}(\mo,\R)$. Hence $ \mo^F$ is a reprojection.
\end{proof}

\section{A model set that is not \brillig}\label{examples}

Theorem \ref{NewMainTheorem} states that most common examples of model sets,
constructed by direct application of the cut \& project method with polyhedral windows, are \brillig.
In this example we exhibit a model set with a Euclidean internal space (so satisfying H1) that is
not. The example is constructed by a substitution; and model sets arrising from substitutions may have very complicated (fractal) windows.

Consider the 1-dimensional substitution $\sigma$ on four letters:
\begin{eqnarray*} \sigma(a_1) & = &  a_1b_1a_2 \\
\sigma(b_1) & = &  a_1b_2 \\
\sigma(a_2) & = &  a_1b_2a_2 \\
\sigma(b_2) & =  & a_2b_1.
\end{eqnarray*}
Its substitution matrix is
$$\sm = \begin{pmatrix} 1&1&1&0 \cr 1&0&1&1 \cr 1&0&0&1 \cr 0&1&1&0 \end{pmatrix}$$
and has eigenvalues $\phi^2$, $-\phi$, $\phi^{-1}$ and $\phi^{-2}$
where $\phi = (1+\sqrt{5})/2$ is the golden mean.  We choose tile
length proportional to the right Perron Frobenius eigenvalue, namely
$b_1$ and $b_2$ tiles to have unit length, and $a_1$ and $a_2$ tiles
to have length equal to $\phi =(1+\sqrt{5})/2$.  This is then a
geometric primitive aperiodic unimodular Pisot substitution (which is
not irreducible).
By the results of \cite{BSW} the dynamical spectrum is pure point,
indeed the balanced pair $(a_1b_1,b_1a_1)$ terminates with coincidence
(see \cite{BSW} for explanations on this notion).  A lot is known
about such substitution tilings. The following can be found more or less implicit in, for instance,
\cite{BM,BBK,Sing}.
\begin{itemize}
\item Since the spectrum is pure point, the tilings are MLD to
  (possibly colored) regular model sets.
 Moreover,
  the set $M$ of left boundary points of $a_1$-tiles in a tiling is MLD
  to the tiling and hence also a regular model set.
\item Since the substitution is unimodular, the maximal
  equicontinuous factor $\Omega_{max}$ of the associated dynamical system
  is a torus of dimension $J$ where $J$ is the algebraic
degree of the Perron Frobenius eigenvalue.

This eigenvalue is here $\phi^2$ and hence $J=2$.
This, in turn implies that $M$ has a
  cut \& project scheme in which the internal space $H$ is $\R$.

\end{itemize}
One readily computes using the technique of \cite{AP} that
$H^1(\Omega_M, \R) = \R^4$, and that substitution acts on $H^1(\Omega_M,\R)$ by
the transpose $\sm^T$ of the substitution matrix.  For substitution tilings,
$H^1_{an}(\Omega_M, \R)$ is the span of all of the generalized
eigenspaces of this action

with eigenvalues strictly inside the unit circle \cite{CS2}. In our case
this means that $H^1_{an}(\Omega_M, \R)=\R^2$ is the span of the
$\phi^{-1}$ and $\phi^{-2}$ eigenvectors of $\sm^T$.

The generator of a shape conjugacy
corresponding to the $\phi^{-2}$ eigenvector induces a reprojection and hence is
\ns. It corresponds to a shape conjugacy in which all of the $a$
tiles are lengthened and the $b$ tiles are shortened (or vice-versa),
while maintaining $|a_1|=|a_2|$ and $|b_1|=|b_2|$ and preserving the quantity
$|a_1|\phi + |b_1| = \phi^2+1$.

The generator $F$ of a shape conjugacy corresponding to the $\phi^{-1}$ eigenvector is
not \ns, and results in a Delone set $\mo^F$ that is not Meyer.  To
see this, let
$A_i^n = \sigma^n(a_i)$ and $B_i^n = \sigma^n(b_i)$
be $n$-th order supertiles, and let $|A_i|$ and $|B_i|$ be the
Euclidean lengths of these supertiles after deformation. This time
$|A_1|\neq |A_2|$, in fact, $|A_1^n|-|A_2^n|$ is
proportional to $\phi^{-n}$.
We have $|A_1^n| \in \mo^F-\mo^F$, since
$A_1^nB_1^n$ appears in our tiling and $A_1^n$ and $B_1^n$ start with $a_1$.

Likewise, $|A_2^n| \in \mo^F-\mo^F$, since $A_2^nB_1^n$ appears in the tiling and also $A_2^n$ starts with $a_1$. However, since
$|A_1^n|-|A_2^n|$ is proportional to $\phi^{-n}$ the set of differences
$\mo^F-\mo^F$ cannot be uniformly discrete, so $\mo^F$ is not
Meyer.

The key feature of this example is that the substitution
matrix is reducible. The maximal equicontinuous factor is determined
by the dynamical spectrum, which for self-similar tile lengths is
determined by the Perron-Frobenius eigenvalue $\lambda_{PF}$. If
$\lambda_{PF}$ is a Pisot number, then a basis for the \ns{}
generators of shape conjugacy is given by the eigenvectors with
eigenvalues algebraically conjugate to $\lambda_{PF}$. However, a
basis for $H^1_{an}$ is given by {\em all\/} the eigenvectors with
eigenvalue strictly smaller than 1. The eigenvectors whose small
eigenvalues are not conjugate to $\lambda_{PF}$ correspond to \wpe{}
functions that are not \ns.

\medskip

{\bf Acknowledgments}: We thank Marcy Barge for showing us the doubled
Fibonacci substitution that we employed in Section \ref{examples}.
The work of the first author is partially supported by the ANR SubTile
NT09 564112.  The work of the second author is partially supported by
NSF grant DMS-1101326.

\medskip


TITLE: Checking the convergency of the following series
QUESTION [0 upvotes]: I wanted to determine when the sum $\sum n^{n^{1/3}}x^{2n}$ converges/diverges. I noticed that the series diverges when $x \leq -1$, $x \geq 1$, but I don't know that to do to check for convergency when $-1<x<1$. I tried to apply the ratio test and the root test, but I wasn't able to get anything out of it

REPLY [1 votes]: If $|x|<1$, then\begin{align}\lim_{n\to\infty}\sqrt[n]{\left|n^{n^{1/3}}x^{2n}\right|}&=\lim_{n\to\infty}n^{n^{1/3}/n}|x|^2\\&=\lim_{n\to\infty}n^{n^{-2/3}}|x|^2\\&=|x|^2\\&<1,\end{align}and therefore the series converges, by the root test. Note that$$\lim_{n\to\infty}n^{n^{-2/3}}=1$$because, for each $x>0$,$$x^{x^{-2/3}}=\exp\left(x^{-2/3}\log(x)\right)=\exp\left(\frac{\log(x)}{x^{2/3}}\right)$$and that, by L'Hopital's Rule,$$\lim_{x\to\infty}\frac{\log(x)}{x^{2/3}}=\lim_{x\to\infty}\frac{x^{-1}}{\frac23x^{-1/3}}=\lim_{x\to\infty}\frac32x^{-2/3}=0.$$


TITLE: Check convergence of following two series
QUESTION [2 upvotes]: I have following two series.
1.$\sum_{1}^{\infty}\ln\left(\frac{n}{n+1}\right)$  2.$\sum_{2}^{\infty}\ln(\left(\frac{n+1}{n-1}\right)$
First I thought both of them are convergent as they are similar to $\sum \ln(n/n)$ and this series is just $0$+$0$..+$0$ and so by limit comparison test both of them converge.
But then when I treat above two as telescoping series I think they must be divergent.
Please clarify me.

REPLY [0 votes]: 1)$\log( \dfrac{n}{n+1}) = \log n - \log (n+1)$
$S_n:= \log(1) - \log(2) +  \log(2) - \log(3)..+\log (n) -\log (n+1) =$
$ \log (1) - \log (n+1).$
2)$ \log (\dfrac{n+1}{n-1}) ={ \displaystyle  \int_{n-1}^{n+1}\dfrac{1}{x}dx }$
$\gt 2 \dfrac{1}{n+1}.$
1) Find $\lim_{ n \rightarrow \infty} S_n.$
2) Any resemblance to the harmonic series?


TITLE: Product of Dependent Discrete RVs
QUESTION [0 upvotes]: $X$ is chosen uniformly from $\lbrace 1,2,\ldots, n\rbrace$ and then $Y$ is chosen uniformly from $\lbrace 1,2,\ldots, X \rbrace$. I want to compute $P(Y=j)$ and $E(XY).$
These variables aren't independent. So if I take $W=XY,$ then $1\le W\le n^2$
I have $E(W)=\sum_{i=1}^{n^2}iP(W=i)$. I'm not sure how to compute $P(W=i).$
I can figure out this value for some $i$'s, but if $i=n+1$, what is $P(W=i)$?

REPLY [2 votes]: $E(XY)=E(E(XY\mid X))=E(XE(Y\mid X))$
$E(Y\mid X=N)=\sum_{y=1}^{N}\frac{y}{N}=\frac{N+1}{2}$
Then $$E(XE(Y\mid X))=E(X\cdot \frac{X+1}{2})=\frac{1}{2}[E(X^2)+E(X)]\\=\frac{1}{2}\cdot \Big[\frac{(n+1)(2n+1)}{6}+ \frac{n+1}{2}\Big]\\=\frac{(n+1)(n+2)}{6}$$
(Check if my calculation is ok or not!)

REPLY [1 votes]: Law of Total Expectation:
$$\mathbb E[XY] = \mathbb E[\mathbb E[XY|X]] = \mathbb E[X \mathbb E[Y|X]]$$


TITLE: Why cannot static forces on a small scale model be measured in wind tunnel if the model is not rigid?
QUESTION [0 upvotes]: I'm studying the theory of wind tunnel tests in context of structural engineering. Often we want to predict the wind loads on a real building by placing a small scale model of the building into a wind tunnel. The wind loads can then be measured on the model, and mathematically scaled for the real building.
Common way of measuring the wind load on the building foundation is by placing the model on a force balance. It is a stiff plate-like instrument upon which the model is laid, which measures the moment applied to it by the wind, through the model.
All texts about the theory of wind tunnel testing mention that the model has to be light and very rigid. Here is a quote from one:

If the model is not rigid enough and responds dynamically
to the wind force, the interaction between the dynamic response and the aerodynamic force would
contaminate the base force measurements.

I don't fully understand this sentence, and it is not elaborated more in any source I can find. What is meant by the dynamic response contaminating the base force measurements? I mean, we are trying to measure the base forces. Aren't the dynamic responses of the building part of the base force? If the building, for example, vibrates, shouldn't those forces be considered anyway, so why not include them in the measurement? Or if we really want to just measure the static force on the base, is there no way to mathematically filter the measurement to obtain the static force? If, say, we assume the model vibrates sinusoidally in the wind, the inertial forces from the vibration are measured by the balance. But cannot we then simply take the average of the sinusoid to get the constant part of the force, that is, the static force on the model?

REPLY [0 votes]: From my reading of chapter 1 of this thesis, there are different methods of wind tunnel testing. You are right that we will want to know the dynamic loading on the building, but that will depend on the construction of the building. We will likely first need the static response, for which we need the building model to be very stiff so that it doesn't vibrate in the tunnel, this is base load testing.
A very stiff model allows the base load on the building to be calculated when we only know the geometry of the building and not its construction, as might be typical early in a design. The measured base load can then inform the construction. When the construction is known, a wind tunnel model can be built that matches its properties. Additional wind tunnel testing then can give the dynamic loading.
In general, it is preferable to design your experiments to remove sources of error rather than relying on postprocessing.