TITLE: What is the geometric interpretation of Euler characteristic?
QUESTION [4 upvotes]: By this question, I know ,in the high dimensional $(\ge 3)$, the Euler characteristic is not $2-2g$, $g$ is genus. When dimension equal to $2$ , genus is the holes of surface. It is easy to understand, but when dimension $\ge3$, what is the geometric interpretation of Euler characteristic .

REPLY [1 votes]: There are many geometric definitions of Euler characteristic, here are 2 examples:
1) Suppose $M$ is a manifold. Take the diagonal $\Delta: M \to M\times M$, and consider another embedding of $M$ into $M\times M$. Then in most cases these 2 embeddings will intersect transversely and counting points of intersection with appropriate orientation gives exactly $\chi(M)$.
2) This one is much more popular equivalent definition:Let $M$ be a compact smooth manifold. Let $V$ be a vector field on $M$ with isolated zeroes. Then:
$\chi(M)$ is the analytical index of the vector field $V$.


TITLE: why do my exam boards use $t > 0$ not $t\ge 0$?
QUESTION [0 upvotes]: In my mechanics exams we often have to state when $v = 0$:
$$v = t^2 + 4t -45 = 0$$
$$v = (t+9)(t-5)$$
$$\therefore t = 5 \Rightarrow v= 0 \because t > 0$$
However my tendency is to think that it should read $t \ge 0$ but all the mark schemes seem to use $t > 0$,so i thought there might be a reason we can never say $t=0$ because of some property of time,  but some other questions in the same exams refer to $t=0$. Is there some reason for this? or is the exam board just being weird?
(posted this on maths and not physics due to it being covered in my mechanics class as part of my further maths A level, is it in the right place?)

REPLY [1 votes]: So in order to expand on the comment, the choice of using $t>0$ vs $t\ge 0$ mostly depends on context since the laws of classical mechanics are deterministic (related to the time symmetry comment I made), in other words a negative time means we can follow a particles trajectory backwards in time even before we started keeping time at the zero mark and it follows physical rules perfectly fine. However most of the time we are interested in times after we start our time keeping of the system, it's common for projectiles to be fired from tops of buildings and so when one solves, for example, when the projectile is at ground level we arrive at one positive time and a negative time. These are often mentioned to students as nonphysical answers to the question because "time can't be negative." Of course it really means you can't follow the particle backwards because it didn't exist before being fired. Later courses cover this deterministic nature of physics (quantum systems are special in this regard) and the time symmetry which does away with such strange notions of the limits on our time variable.


TITLE: Next step in proof of sets
QUESTION [1 upvotes]: Proposition to prove : (A-B)∩(B-A) = 0
So, I understand why this is 0, I'm just not sure what propositions should be used in proving so.
I have this so far
1)(A-B)∩(B-A)            :Premise
2)(A∩B')∩(B∩A')          :Alternative Representation for Set Diff.
My question is, Is this allowed:
3)(A)∩(A')∩(B)∩(B')     :?
4)0∩0                   :Intersection and Union with Complement
5)0                     :Intersection with 0
I'm not sure if I'm allowed to go to step 3, from step 2. And if I'm not able to do so, what process should I go about next in proving so?n
*I need to use laws/identities. I've already written a proof that does "element chasing" (let x be a number...)

REPLY [1 votes]: Looks legal to me. Set intersection is associative and commutative.


text:
\begin{document}
\maketitle

\begin{abstract} We present an upper bound on the mixing rate
  of the equilibrium state of a dynamical systems defined by the
  one-sided shift and a non H\"{o}lder potential of summable
  variations.  The bound follows from an estimation of the relaxation
  speed of chains with complete connections with summable decay, which
  is obtained via a explicit coupling between pairs of chains with
  different histories.
\end{abstract}

\section{Introduction}
Let $\mu_\phi$ be the equilibrium state associated to the continuous
function $\phi$.  In this paper we obtain
upper bounds for the speed of convergence of the limit
\beq
\label{mixing}
\int_X f \circ T^n\, g \,d\mu_\phi \tiende{n\to \infty} \int_X f \,
d\mu_\phi \int_X g \, d\mu_\phi
\eeq
for $\phi$ with summable
variations and $T$ the one-sided shift.  We show that this speed is
(at least) summable, polynomial or exponential according to the decay
rate of the variations of $\phi$.  The bounds apply for $f \in
L^1(\mu_\phi)$ and $g$ with variations decreasing proportionally to
those of $\phi$.

\paragraph{}

Previous approaches to the study of the mixing properties of the
one-sided shift rely on the use of
the transfer operator  $L_\phi$,
defined by the duality,
\beq
\label{duality}
\int_X f \circ T^n \,g\, d\mu_\phi = \int_X f \,L_\phi^n g\,
d\mu_\phi\;.  \eeq
If $\phi$ is H\"{o}lder, this operator, acting on the subspace of
H\"older observables, has a spectral gap and the limit (\ref{mixing})
is attained at exponential speed (Bowen, 1975).  When $\phi$ is not
H\"{o}lder, the spectral gap of the transfer operator may vanish and
the spectral study becomes rather complicated.  To estimate the mixing
rate, Kondah, Maume and Schmitt (1996) proved first that the operator
is contracting in the Birkhoff projective metric, while Pollicott
(1997), following Liverani (1995), considered the transfer operator
composed with conditional expectations.  In contrast, our approach is
based on a probabilistic interpretation of the duality~(\ref{duality})
in terms of expectations, conditioned with respect to the past, of a
chain with complete connections The convergence (\ref{mixing}) is
therefore related to the relaxation properties of this chain.  In this
paper, such relaxation is studied via a coupling method.

\paragraph{}
Coupling ideas were first introduced by Doeblin in his 1938 work on
the convergence to equilibrium of Markov chains.  He let two
independent trajectories evolve simultaneously, one
starting from the stationary measure and the other from an arbitrary
distribution.  The convergence follows from the fact that both
realizations meet at a finite time.  Instead of letting the
trajectories evolve independently, one can couple them from the
beginning, reducing the ``meeting time'' and, hence, obtaining a
better rate of convergence (leading to the so-called Dobrushin's
ergodic coefficient).  Doeblin published his results in a hardly known
paper in the Revue Math\'ematique de l'Union Interbalkanique. (For a
description of Doeblin's contributions to probability theory we refer
the reader to Lindvall 1991). His ideas were taken up and exploited
only much later in papers by Athreya, Ney, Harris, Spitzer and
Toom among others.
The sharpness of the convergence rates provided by different types of
Markovian couplings has been recently discussed by Burdzy and Kendall
(1998).

\paragraph{}
In the context of dynamical systems, the recent papers by Coelho and
Collet (1995) and Young (1997) consider the time two independent
systems take to become close.  This is reminiscent of the original
coupling by Doeblin.  In Bressaud, Fern\'andez, Galves (1997), the
coupling approach was generalized to treat chains with complete
connections.  These processes, introduced by Doeblin and Fortet (1937)
(see also Lalley, 1986) appear in a natural way in the context of
dynamical systems.  They are characterized by having transition
probabilities that depend on the whole past, albeit in a continuous
manner.  Due to this fact, the coupling can not ensure that two
different trajectories will remain equal after their first meeting
time.  But the coupling used in the present paper, and in our
preceeding one, has the property that if the trajectories meet they
have a large probability of remaining equal, and this probability
increases with the number of consecutive agreements.  In the summable
case, the coupling is such that with probability one the trajectories
disagree only a finite number of times.  In fact, our approach can
also be applied under an assumption weaker than summability
[\reff{weak.1} below], leading to trajectories that differ infinitely
often but with a probability of disagreement that goes to zero.  The
method leads, in particular, to a criterium of uniqueness for
$g$-measures proven by Berbee (1987).  The mean time between succesive
disagreements provides a bound on the speed of relaxation of the chain
and hence, through our probabilistic interpretation of
(\ref{duality}), of the mixing rate.

\paragraph{}
The paper is organized as follows. The main results and definitions
relevant to dynamical systems are stated in Section~\ref{definitions}.
The relation between chains with complete connections and the transfer
operator is spelled out in Section~\ref{transferoperator}.  In
Section~\ref{relaxation}, we state and prove the central result on
relaxation speeds of chains with complete connections.
Theorem~\ref{normalizedth} on mixing rates for normalized functions
is proven in Section~\ref{normalized}, while Theorem~\ref{general} on
rates for the general case is proven in Section~\ref{proof}.  The
upper bounds on the decay of correlations depend crucially on
estimations of the probability of return to the origin of an auxiliary
Markov chain, which are presented in Section~\ref{markovchain}.

\section{Definitions and statement of the results}
\label{definitions}

\paragraph{}
Let $A$ be a finite set henceforth called \emph{alphabet}. Let us
denote
\begin{equation}
  \label{eq:40}
  \underline{A} \;=\; \bigl\{x = (x_j)_{j\leq-1}\;,\; x\in A\bigr\}
\end{equation}
the set of sequences of elements of the alphabet indexed by the
strictly negative integers.  Each sequence $x\in\underline A$ will be
called a {\em history}.  Given two histories $x$ and
$y$, the notation $x \stackrel{m}{=}y$ indicates that $x_j = y_j$ for
all $-m \leq j \leq -1$.

As usual, we endow the set $\underline A$
with the product topology and the $\sigma$-algebra generated by the
cylinder sets.  We denote
by ${\cal C}^0(\underline{A}, \real)$ the space of real-valued
continuous functions on $\underline{A}$.

\paragraph{}
We consider the one-sided shift $T$ on $\underline{A}$,
$$ \begin{array}{llcl}
T:& \underline{A} & \longrightarrow & \underline{A} \\
& x& \longmapsto & T(x) = (x_{i-1})_{i\leq -1}.
\end{array}
$$
Given an element $a$ in $A$ and an element $x$ in $\underline{A}$,
we shall denote by $xa$ the
element $z$ in $\underline{A}$ such that $z_{-1} = a$ and
$T(z) = x$.

\subparagraph{}
Given a function $\phi$ on $\underline{A}$,
$ \phi:  \underline{A}  \to  \real$,
we define its
sequence of \emph{variations} $(\var_{m}(\phi))_{m \in \nat}$,
\beq
\label{defgamma}
\var_{m}(\phi) = \sup_{x \stackrel{m}{=}y}|\phi(x) - \phi(y)| \;.
\eeq

We shall say that it has {\em summable variations} if,
\begin{equation}
  \label{rr.10}
\sum_{m\geq 1} \var_m(\phi) < + \infty\;,
\end{equation}
and that it is {\em normalized} if it satisfies,
\begin{equation}
  \label{rr.15}
  \forall x \in \underline{A}\;, \;\;\; \sum_{a \in A} e^{\phi(xa)} =
1\;.
\end{equation}
We say that a shift-invariant measure $\mu$ on $\underline{A}$
is {\em compatible with the normalized function $\phi$} if
and only if, for $\mu_\phi$-almost-all $x$ in $\underline{A}$,
\begin{equation}
  \label{eq:50}
\esp_{\mu_\phi}\bigl( 1_{\{x_{-1}=a\}}|{\cal F}_{\le -2}\bigr)(x)
\; =\; e^{\phi(T(x)a)}\;,
\end{equation}
where the left-hand side is the usual conditional expectation of the
the indicator function of the event $\{x_{-1}=a\}$ with respect to the
$\sigma$-algebra of the past up to time $-2$.

An equivalent way of expressing this is by saying that $\mu_\phi$ is a
$g$-measure for $g = e^\phi$. If $\phi$ has summable variations, and
even under a slightly weaker conditions, then such a measure is unique
and will be denoted $\mu_\phi$.  The measure $\mu_\phi$ can also be
characterized via a variational principle, in which context it is
called \emph{equilibrium state} for $\phi$.  For details see
Ledrappier (1974), Walters (1975), Quas (1996) and Berbee (1987).

\paragraph{}
For a non-constant $\phi$, we consider the seminorm
\begin{equation}
  ||g||_{\phi} \;=\; \sup_{k \geq 0}
\frac{ \var_k(g)}{\var_k(\phi)}
\end{equation}
and the subspace of ${\cal C}^0(\underline{A})$ defined by,
\begin{equation}
  \label{rr.20}
  V_{\phi} \;=\; \biggl\{ g \in {\cal C}^0(\underline{A}, \real)\,, \;
 ||g||_\phi < + \infty \biggr\} \;.
\end{equation}

\paragraph{}
Given a real-valued sequence $(\gamma_{n})_{n \in \nat}$, let
$(S^{(\gamma)}_{n})_{n \in \nat}$ be the Markov chain taking values
in the set $\nat$ of natural numbers starting from the origin
\begin{equation}
\pro(S^{(\gamma)}_0 =0)\;=\;1
\label{rr.5.-1}
\end{equation}
whose transition probabilities are defined by
\begin{equation}
  \label{rr.5}
\begin{array}{rcl}
p_{i,i+1} &=& 1- \gamma_i \\
p_{i,0} &=& \gamma_i \;,
\end{array}
\end{equation}
for all $i\in\nat$.
For any $n\ge 1$ we define
\begin{equation}
\gamma^*_n = \pro(S^{(\gamma)}_n = 0)\;.
\label{rr.1}
\end{equation}

\paragraph{}
We now state our first result.
\begin{theorem}
\label{normalizedth}
Let $\phi: \underline{A} \to \real$ be a normalized function with summable
variations and set
\begin{equation}
  \label{rr.30}
  \gamma_n \;=\; 1 - e^{-\var_n(\phi)} \;.
\end{equation}
Then,
\begin{eqnarray}
  \label{rr.35}
    \left| \int f \circ T^n \,g\, d\mu_\phi - \int f\,
  d\mu_\phi \int g \, d\mu_\phi \right|
&\leq& \,  ||f||_1\,   ||g||_{\phi} \, \sum_{k=0}^n \var_k(\phi)\,
\gamma^*_{n-k} \label{eq:x}\\[5pt]
&\leq& C \,  ||f||_1\,   ||g||_{\phi} \, \gamma^*_{n}\;,
\label{eq:xx}
\end{eqnarray}
for all $f \in L^1(\mu_\phi)$ and $g \in V_{\phi}$, for a computable
constant $C$.
\end{theorem}
This theorem is proven in Section \ref{normalized}, using the results
obtained in Section \ref{relaxation} on the relaxation speed of chains
with complete connections.

\paragraph{}
For each non-normalized function $\phi$ with summable variations there
exists a unique positive function $\rho$ such that the function
\begin{equation}
  \label{rr.25}
  \psi = \phi + \log{\rho} - \log{\rho \circ T}
\end{equation}
is normalized (Walters, 1975). We call $\psi$ the {\em normalization}
of $\phi$.  The construction of compatible measures given in
\reff{eq:50} looses its meaning for non-normalized $\phi$.  It is
necessary to resort to an alternative characterization in terms of a
variational principle (see eg.\ Bowen 1975) leading to equilibrium
states.  In Walters (1975) it is proven that:
\begin{itemize}
\item[(a)] $\phi$ with
summable variations admits a unique equilibrium state, that we denote
also $\mu_\phi$;
\item[(b)] the corresponding normalized $\psi$, given by
\reff{rr.25}, admits a unique compatible measure $\mu_\psi$ (even when
the variations of $\psi$ may not be summable), and
\item[(c)]  $\mu_\phi=\mu_\psi$.
\end{itemize}

\paragraph{}
Our second theorem generalizes Theorem~\ref{normalizedth} to
non-normalized functions.
\begin{theorem}
\label{general}
Let $\phi: \underline{A} \to \real$ be a function with summable
variations and let $\psi$ be its normalization.  Let $(n_m)_{m \in \nat}$
be an increasing subadditive sequence such that the subsequence of the
rests, $\bigl(\sum_{k \geq n_{m}} \var_k(\phi)\bigr)_{m \geq 0}$, is
summable, and
\begin{equation}
  \label{rr.40}
  \overline{\gamma}_m = 1 - e^{- 3 \sum_{k \geq n_{m}}  \var_k(\phi)}\;;
\end{equation}
then,
\begin{eqnarray}
  \label{rr.45}
   \left| \int f \circ T^n \,g\, d\mu_\phi - \int f\,
  d\mu_\phi \int g\,  d\mu_\phi \right| &\le&
||f||_1\,   ||g||_{\phi} \, \sum_{k=0}^n \var_{n_k} (\phi)\,
\overline\gamma^*_{n-k} \\[5pt]
&\leq & C \,  ||f||_1\,   ||g||_{\phi} \, \overline{\gamma}^*_{n} \;,
\end{eqnarray}
for all  $f \in L^1(\mu_\phi)$ and $g \in V_{\phi}$, for a computable
constant $C$.  Here
$\overline\gamma^*$ is defined as in \reff{rr.1} but using the
sequence $(\overline\gamma^*_{n})_{n \in \nat}$.
\end{theorem}

\paragraph{}
The estimation of the large-$n$ behavior of the
sequence $(\gamma^*_{n})_{n \in \nat}$ given the behavior of the
original $(\gamma_{n})_{n \in \nat}$ only requires elementary
computations. For the convenience of the reader we summarize some
results in Appendix~\ref{markovchain}.

\section{Transfer operators and chains.}
\label{transferoperator}

\paragraph{}
Let $P$ be a family of transition probabilities on $A
\times \underline{A}$,
\begin{equation}
\begin{array}{llcl}
\trans:& A \times \underline{A} & \longrightarrow & [0;1] \\
& (a,z)& \longmapsto & \trans(a\,|\,z)\;.
\end{array}
\label{eq:-1}
\end{equation}
Given a history $x$, a \emph{chain with past $x$ and transitions
  $\trans$}, is the process $(Z^{x}_{n})_{n  \in{\bf Z}}$ whose conditional
probabilities satisfy
\begin{equation}
  \label{eq:10}
  \pro(Z^{x}_{n}= a \,|\,
Z^{x}_{n+j} = z_j, j\leq -1) \;=\; \trans(a \,|\,z)
\hbox{ for } n\geq 0 \;,
\end{equation}
for all $a \in A$ and all histories $z$ with $z_{j-n}
= x_j, j\leq -1$, and such that
\begin{equation}
  \label{eq:mais}
Z^{x}_{n}=x_n \;,\;  \hbox{ for } n\leq -1\;.
\end{equation}
This chain can be interpreted as a conditioned version of the process
defined by the transition probabilities \reff{eq:-1}, given a past $x$
(for more details, see Quas 1996).

\paragraph{}
Let $\phi: \underline{A} \to \real$ be a continuous normalized
function.  The transfer operator associated to $\phi$ is the
operator $L_\phi$ acting on ${\cal C}^0(\underline{A}, \real)$ defined
by,
\begin{equation}
  \label{eq:100}
   L_\phi f (x) = \sum_{y\, :\, T(y) =
  x} e^{\phi(y)} f(y)\;.
\end{equation}
This operator is related to the conditional probability \reff{eq:50}
  in the form
  \begin{equation}
    \label{eq:105}
    \esp_{\mu_\phi}\bigl( f \,|\,{\cal F}_{\le -2}\bigr) \;=\;
\bigl(L_\phi f\bigr) \circ T\;.
  \end{equation}
This relation shows the equivalence of \reff{duality} and
\reff{eq:100} as definitions of the operator.
In addition, if $\phi$ is normalized we can construct, for each
history $x \in \underline{A}$, the chain
$Z^{x}_{\phi}=(Z^{x}_{n})_{n \in{\bf Z}}$ with past $x$ and transition
probabilities
\begin{equation}
  \label{eq:20}
  \trans(a\, |\, x ) = e^{\phi(xa)}\;.
\end{equation}

\paragraph{}
Iterates of the transfer operator, $L^n_\phi g (x)$, on functions
$g \in {\cal C}^0(\underline{A})$ can be interpreted
as expectations $\esp[g((Z^{x}_{n+j})_{j \leq -1})]$ of the chain.
Indeed,
\bqy
L^n_\phi g (x) &=& \sum_{a_1, \ldots, a_n \in A}e^{\sum_{k = 1}^n
  \phi(x a_1 \cdots a_k )} g(x a_1 \cdots a_n)
\nonumber \\
&=& \sum_{a_1, \ldots, a_n \in A} \left( \prod_{k=1}^n
  \trans(a_k\,|\,a_{k-1}
  \cdots a_1 x) \right) g(x a_1 \cdots a_n) \nonumber \\
&=& \esp[ g((Z^{x}_{n+j})_{j \leq -1})]\;. \nonumber
\eqy

\paragraph{}
From this expression and the classical duality~(\ref{duality}) between
the composition by the shift and the transfer operator $L_\phi$ in
$L^2(\mu_\phi)$, we obtain the following expression for the decay of
correlations,
\begin{eqnarray}
\lefteqn{ \int f \circ T^n \,g\, d\mu_\phi - \int f\, d\mu_\phi
  \int g\,   d\mu_\phi }\nonumber\\
 &= &  \int f(x)\, L_{\phi}^n g(x)\,
  d\mu_\phi(x) - \int f(x) \left( \int L_{\phi}^n g(y) \,d\mu_\phi(y)
  \right) d\mu_\phi(x)  \nonumber \\
& =&  \int f(x) \int \Bigl(
    \esp[g((Z^{x}_{n+j})_{j \leq -1})] - \esp[g((Z^{y}_{n+j})_{j \leq
      -1})]\Bigr)\, d\mu_\phi(y)\, d\mu_\phi(x) \;.
\label{interpretation}
\end{eqnarray}
This inequality shows how the speed of decay of correlations can be
bounded by the speed with which the chain loosses its memory.  We deal
with the later problem in the next section.

\section{Relaxation speed for chains with complete connections}
\label{relaxation}
\subsection{Definitions and main result}
\paragraph{}

We consider chains whose transition probabilities satisfy
\beq
\label{p.3}
\inf_{  x , y : x \stackrel{m}{=}y }
{\trans(a \,|\, x) \over \trans(a \,|\, y)}  \geq \; 1 -   \gamma_m \;,
\eeq

for some real-valued sequence $(\gamma_m)_{m \in \nat}$,
decreasing to $0$ as $m$ tends to $+ \infty$.   Without loss of
generality, this decrease can be assumed to be monotonic.  To avoid
trivialities we assume $\gamma_0<1$.  In the literature, a
\emph{stationary} process satisfying \reff{p.3} is called a
\emph{chain with complete connections}.

\paragraph{}

\paragraph{}
For a set of transition probabilities satisfying \reff{p.3}, we
consider, for each $x\in\underline A$, the chain
$(Z^{x}_n)_{n\in\rel}$ with past $x$ and transitions $\trans$ [see
\reff{eq:10}--\reff{eq:mais}].  The following proposition
  plays a central role in the proof of our results.
\begin{prop}
\label{existcoupling}
For all histories
$x,y \in \underline{A}$, there is a coupling
$\bigl((\widetilde{U}^{x,y}_n,\widetilde{V}^{x,y}_n)\bigr)_{n\in\rel}$
of $(Z^{x}_n)_{n\in\rel}$ and $(Z^{y}_n)_{n\in\rel}$ such that the
    integer-valued  process $(T^{x,y}_n)_{n\in {\bf Z}}$  defined by
\beq
\label{defcompteur}
T^{x,y}_n \;=\; \inf \{ m \geq 0 \, : \, \widetilde{U}^{x,y}_{n-m} \neq
\widetilde{V}^{x,y}_{n-m} \},
\eeq
satisfies
\beq
\label{compteur}
\pro(T^{x,y}_n = 0) \;\leq \;  \gamma^*_n
\eeq
for $n\ge 0$, where $\gamma^*_n$ was defined in \reff{rr.1}.
\end{prop}
The proof of this proposition is given in Section \ref{sepro}.

\paragraph{}
An immediate consequence of this proposition is the following bound on
the relaxation rate of the processes $Z^{x}$.
\begin{cor}
\label{relaxccc}
For all histories $x$ and $y$, for all $a \in A$,
\begin{equation}
\label{le.10}
 \Bigl| \pro(Z^{x}_{n}= a) - \pro(Z^{y}_{n} = a) \Bigr| \;\leq\;
\gamma^*_n\;,
\end{equation}
and, for $k\geq 1$,
\begin{eqnarray}
\lefteqn{\hspace{-3cm}\left| \pro\Bigl((Z^{x}_{n}, \ldots, Z^{x}_{n+k})= (a_0,
  \ldots, a_k) \Bigr) - \pro\Bigl((Z^{y}_{n}, \ldots, Z^{y}_{n+k})= (a_0,
  \ldots, a_k) \Bigr) \right|} \nonumber\\
&& \qquad\qquad \le \quad\sum_{j=0}^{k}   \left(\prod_{m=1}^{j-1}(1- \gamma_{m})
  \right) \gamma^*_{n-j}\;.
\label{le.7}
\end{eqnarray}
\end{cor}
This lemma is proved in Section~\ref{proofcor}.

\begin{rem}
\label{unicitelarge}
Whenever
\begin{equation}
  \label{eq:pp}
 \gamma^*_n\to 0 \;,
\end{equation}
inequality \reff{le.10} implies the
existence and uniqueness of the invariant
measure compatible with a system of conditional probabilities
satisfying \reff{p.3}.  In fact, property \reff{eq:pp} holds under
the condition
\begin{equation}
  \label{weak.1}
\sum_{m\geq 1} \prod_{k=0}^{m}(1 - \gamma_k) \;=\;  + \infty \;. ,
\end{equation}
which is weaker than summability.
In this case, the Markov chain $(S^{(\gamma)}_n)_{n \in \nat}$ is no
longer transient but it is null recurrent and the property
$\pro(S^{(\gamma)}_n =0) \to 0$ remains true.

\end{rem}

\begin{rem}
If $X = (X_n)_{n\in {\bf Z}}$ is a stationary process with
transition $\trans$ satisfying \reff{p.3}, then
Corollary~\ref{relaxccc} implies
\begin{equation}
 \Bigl| \pro(Z^{x}_{n}= a) - \pro(X_{n} = a) \Bigr| \;\leq\;
 \gamma^*_n\;,
\label{le.12}
\end{equation}
uniformly in the history $x$.
\end{rem}

\subsection{Maximal coupling}
Given two probability distributions $\mu = (\mu(a))_{a \in A}$ and
$\nu = (\nu(a))_{a \in A}$ we denote by $\mu \tilde{\times} \nu = (\mu
\tilde{\times} \nu (a,b))_{(a,b) \in A \times A}$ the so-called {\em
  maximal coupling} of the distributions $\mu$ and $\nu$ defined
as follows:
\begin{equation}
\left\{
\begin{array}{ll}
\mu \tilde{\times} \nu (a,a) = \mu(a) \wedge \nu(a) & \hbox{ if } a
= b \\[15pt]
\displaystyle \mu \tilde{\times} \nu (a,b) = \frac{(\mu(a) -\nu(a))^+
  (\nu(b) -\mu(b))^+}{\sum_{e \in A}(\mu(e) -\nu(e))^+}
 & \hbox{ if } a
\neq b \;.
\end{array}
\right.
\end{equation}
For more details on maximal couplings see Appendix A.1 in
Barbour, Holst and Janson (1992).
\paragraph{}
The coupling is maximal in the sense that the distribution $\mu
\tilde{\times} \nu $ on $A \times A$ maximizes the weight
$$\Delta(\zeta) = \sum_{a \in A} \zeta(a,a)$$
of the diagonal among the distributions $\zeta$ on $A
\times A$ satisfying simultaneously
$$\sum_{a \in A} \zeta (a,b) = \nu(b) \;\;\; \hbox{ and }\;\;\; \sum_{b \in A}
\zeta(a,b) = \mu(a)\;.$$
For this coupling, the weight $\Delta(\mu \tilde{\times} \nu)$ of the diagonal satisfies,
\beq
\label{coupling}
\Delta(\mu \tilde{\times} \nu) =   \sum_{a \in A} \mu(a) \wedge
\nu(a) = 1 - \sum_{a \in A} (\mu(a) - \nu(a))^+ = 1 - \frac{1}{2} \sum_{a
  \in A} |\mu(a) - \nu(a)|.
\eeq
Moreover,
\beq
\label{coupling2}
\Delta(\mu \tilde{\times} \nu) = 1 - \sum_{a \in A} \mu(a)\left(1 -
\frac{\nu(a)}{\mu(a)}\right)^+ \geq 1 - \sum_{a \in A} \mu(a)\left(1 -
\inf_{a' \in A}\frac{\nu(a')}{\mu(a')}\right) = \inf_{a \in
A}\frac{\nu(a)}{\mu(a)}.  \eeq

\subsection{Coupling of chains with different pasts}
\paragraph{}
Given a double history $(x, y)$, we consider the transition
probabilities defined by the maximal coupling
\begin{equation}
\transt((a,b) \, | \, x,y)  = \left[\trans(\cdot \,|\,
x) \tilde{\times} \trans(\cdot  \,|\, y)\right]  (a,b)\;.
\label{ant.1}
\end{equation}

By \reff{p.3} we have,
$$ \inf_{a \in A, u \stackrel{m}{=}
 v } \frac{\trans(a\,|\,u)}{\trans(a\,|\,v)} \;\geq\;
 1 - \gamma_m. $$

By (\ref{coupling2}) this implies that
\begin{equation}
\label{diagonale}
\Delta\left(\transt(\,\cdot\,,\,\cdot \, | \, x,y) \right)\; \geq \; 1
- \gamma_m\;,
\end{equation}
whenever $ x \stackrel{m}{=} y$.

\paragraph{}
Now, we fix a double history $(x, y)$ and we define
$\bigl((\widetilde{U}^{x, y}_n, \widetilde{V}^{x,
  y}_n)\bigr)_{n\in\rel}$ to be the chain taking values in $A^2$, with
past $(x, y)$ and transition probabilities given by \reff{ant.1}. If $
x \stackrel{m}{=} y$, (\ref{diagonale}) yields
\begin{equation}
\label{alpha}
\pro( \widetilde{U}^{x,y}_0 \neq \widetilde{V}^{x,y}_0) \;\leq\;
\gamma_{m} .
\end{equation}
\paragraph{}
We denote
\begin{equation}
\Delta_{m,n} \;:=\;

\Bigl\{ \widetilde{U}_j =  \widetilde{V}_j \, ,\, m \leq j \leq n
\Bigr\}\;.
\label{le.15}
\end{equation}
Notice that $\Delta_{-m,-1}$ is the reunion over all the sequences $x,
 y$ with $ x \stackrel{m}{=} y$ of the events $\{ (\widetilde{U}_j
 ,\widetilde{V}_j) = (x_j,y_j) \,;\, j\leq -1 \}$.  Using the
 stationarity of the conditional probabilities, we obtain
\begin{equation}
\label{beta}
\pro( \widetilde{U}_n \neq \widetilde{V}_n \, |\, \Delta_{n-m,n-1})
\;\leq\; \gamma_{m} \;,
\end{equation}
for all $n\ge 0$.

\subsection{Proof of Proposition~\protect{\ref{existcoupling}}}
\label{sepro}

From this subsection on, will be working with bounds which are uniform
in $x,y$, hence we will omit, with a few exceptions, the superscript
$x,y$ in the processes $T^{x,y}_n$ (defined below), $\widetilde
U^{x,y}_n$ and $\widetilde V^{x,y}_n$.

Let us consider the integer-valued
process $(T_n)_{n\in {\bf Z}}$  defined by:
\begin{equation}
T_n \;=\; \inf \{ m \geq 0 \, : \, \widetilde{U}_{n-m} \neq
\widetilde{V}_{n-m} \}\;.
\label{le.19}
\end{equation}
For each time $n$, the random variable $T_n$ counts  the number of
steps backwards needed to find a difference in the coupling.
First, notice that (\ref{beta}) implies that,
\begin{equation}
\pro(T_{n+1}=k+1 \,|\, T_{n} = k) \;\geq\; 1-\gamma_{k}
\label{le.20}
\end{equation}
and
\begin{equation}
\pro(T_{n+1}= 0 \,|\, T_{n} = k) \;\leq\; \gamma_{k}\;,
\label{le.25}
\end{equation}
all the other transition probabilities being zero.
This process $(T_n)_{n\in {\bf Z}}$ is not a Markov chain.

\paragraph{}
We now consider the integer-valued Markov chain $(S^{(\gamma)}_n)_{n
  \geq 0}$ starting from state $0$ and with transition probabilities
given by \reff{rr.5}, that is $p_{i,i+1} = 1- \gamma_i$ and $p_{i,0} =
\gamma_i$.  Proposition~\ref{existcoupling} follows from the following
lemma, setting $k=1$.

\begin{lem}
\label{dominlem}
For each $k \in {\bf N}$, the following inequality holds:

\beq
\label{domin}
 \pro(S^{(\gamma)}_n \geq k) \;\leq\;
\pro(T_n \geq k)
\eeq
\end{lem}
\paragraph{Proof}
We shall proceed by induction on $n$.  Since $\pro(S^{(\gamma)}_0
=0)=1$, inequalities \reff{domin} holds for $n=0$.  Assume now that
\reff{domin} holds for some integer $n$.  There is nothing to prove
for $k=0$.. For $k \geq 1$,
\bqy
\pro(T_{n+1} \geq k)& = &
\sum_{m=k}^{+\infty} \pro(T_{n+1} = m )\nonumber \\

& = &
\sum_{m=k}^{+\infty} \pro(T_{n+1} = m \,|\, T_{n} = m-1 )
\,\pro(T_{n} = m-1 ) \nonumber \\
&\geq  &
\sum_{m=k}^{+\infty} (1-\gamma_{m-1})\, \pro(T_{n} = m-1 ) \nonumber \\
&=  &
\sum_{m=k}^{+\infty} (1-\gamma_{m-1}) \,\Bigl(  \pro(T_{n} \geq  m-1
)-\pro(T_{n} \geq  m )\Bigr)  \nonumber \\
&=  & (1-\gamma_{k-1})\,  \pro(T_{n} \geq  k-1 ) +
\sum_{m=k}^{+\infty} (\gamma_{m-1} - \gamma_{m})
\,\pro(T_{n} \geq  m ) \;.
\eqy
By the same computation, we see that
\beq
\label{recurmarkovchain}
\pro(S^{(\gamma)}_{n+1} \geq k) \;= \;
 (1-\gamma_{k-1}) \,\pro(S^{(\gamma)}_{n} \geq  k-1 ) +
\sum_{m=k}^{+\infty} (\gamma_{m-1} - \gamma_{m})
\,\pro(S^{(\gamma)}_{n} \geq  m ) \;.
\eeq
Hence,  using  the recurrence assumption and the fact that
 $(\gamma_{n})_{n\geq 0}$ is decreasing we conclude that
$$\pro(T_{n+1} \geq k) \;\geq\; \pro(S^{(\gamma)}_{n+1} \geq k)\;,
$$
for all $k \geq 1$.~$\square$

\subsection{Proof of Corollary~\protect{\ref{relaxccc}}}
\label{proofcor}
\paragraph{}
To prove \reff{le.10}, first notice that by construction
the process $(\widetilde U_n)_{n\in\rel}$ has the same law as
$(Z^{x}_n)_{n\in\rel}$ and $(\widetilde V_n)_{n\in\rel}$ has the
same law as $(Z^{y}_n)_{n\in\rel}$.  Thus,
\begin{equation}
\Bigl| \pro(Z^{x}_{n}= a) - \pro(Z^{y}_{n}= a) \Bigr|
\;= \; \left| \pro(\widetilde{U}_n= a) - \pro(\widetilde{V}_n= a)
\right|

\; \leq \; \pro(\widetilde{U}_n \neq \widetilde{V}_n))
\label{premiere}
\end{equation}
Hence, by definition of the process $T_n$ and Lemma \ref{dominlem},
\begin{equation}
\Bigl| \pro(Z^{x}_{n}= a) - \pro(Z^{y}_{n}= a) \Bigr|
\;\leq\;
\pro(T_{n} = 0) \leq  \pro(S^{(\gamma)}_n = 0)\;.
\label{le.30}
\end{equation}

\paragraph{}
The proof of \reff{le.7} starts similarly:
\bqy \
\lefteqn{\hspace{-3cm}\Bigl|
  \pro\Bigl((Z^{x}_{n}, \ldots, Z^{x}_{n+k})= (a_0, \ldots, a_k)\Bigr)  -
  \pro\Bigl((Z^{y}_{n}, \ldots, Z^{y}_{n+k})= (a_0,
  \ldots, a_k) \Bigr) \Bigr| } \nonumber \\

& \leq & \pro(S^{(\gamma)}_{n+k} \leq k+1) . \nonumber
\eqy

To conclude, we notice that,
\begin{equation}
\pro(S^{(\gamma)}_{n} \leq k) \;=\;  \sum_{j=0}^{k} \pro(S^{(\gamma)}_{n} = j)
\;=\; \sum_{j=0}^{k}   \left(\prod_{m=1}^{j-1}(1- \gamma_m) \right)
\pro(S^{(\gamma)}_{n-j} = 0)   \;. \; \square
\label{le.33}
\end{equation}

\section{Proof of Theorem~\ref{normalizedth}}
\label{normalized}

\paragraph{}
The proof of Theorem~\ref{normalizedth} is based on the inequality
\begin{equation}
\left| \int f \circ T^n g d\mu - \int f  d\mu \int g  d\mu \right|

\;\leq  \; ||f||_1
    \,\sup_{x,y}{\esp\left[\left|g((\tilde{U}^{x,y}_{n+j})_{j\leq-1}) -
    g((\tilde{V}^{x,y}_{n+j})_{j\leq-1})\right|\right]}  \;,
\label{eq:105.1}
\end{equation}
which follows from \reff{interpretation} and the fact that
$\bigl((\widetilde{U}^{x, y}, \widetilde{V}^{x, y})\bigr)_{n\in\rel}$
is a coupling between the chains with pasts $x$ and $y$, respectively.
An upper bound to the right-hand side is provided by
Proposition~\ref{existcoupling}.  We see that the transition
probabilities \reff{eq:20} satisfy condition \reff{p.3}, since
\begin{equation}
  \label{eq:100.1}
  \frac{\trans(a \,|\, x )}{\trans(a \,|\, y )}
\;=\; e^{\phi(ax) - \phi( ay)}
\;\geq\;   e^{- \var_{m+1}(\phi)}
\end{equation}
whenever $x,y \in\underline{A}$ are such that
$x \stackrel{m}{=}y$ for some $m \in \nat$.
We can therefore apply Proposition~\ref{existcoupling} with
\begin{equation}
  \label{eq:101}
  \gamma_m \;=\; 1 - e^{-\var_{m+1}(\phi)}\;,
\end{equation}
which tends monotonically to zero if $\sum_{m\geq 1} \var_m(\phi) < +
\infty$.

\paragraph{}
To prove \reff{eq:x} we use the process $(T^{x,y}_n)_{n\in {\bf Z}}$ to
obtain the upper bound
\begin{eqnarray}
\esp\left[\left|g((\widetilde{U}^{x,y}_{n+j})_{j\leq-1}) -
  g((\widetilde{V}^{x,y}_{n+j})_{j\leq-1})\right|\right]
& =  & \esp\left[\sum_{k=0}^{+\infty}
    1_{ \{ T^{x,y}_{n}= k \} } \left|g((\tilde{U}_{n+j})_{j\leq-1}) -
      g((\tilde{V}_{n+j})_{j\leq-1})\right|\right]  \nonumber \\
&\leq  & \sum_{k=0}^{+\infty}\,

\var_k(g)\,\pro(T^{x,y}_{n}= k)    \nonumber \\

&\leq  & ||g||_{\phi}\,\sum_{k=0}^{+\infty}\, \var_k(\phi)\,
\pro(T^{x,y}_{n}= k)\;.
\label{eq:110}
\end{eqnarray}
Now, in order to use the bound \reff{compteur} of
Proposition~\reff{existcoupling} we resort to the monotonicity of the
variations of $\phi$:
\begin{eqnarray}
\sum_{k=0}^{+\infty}\, \var_k(\phi)\, \pro(T^{x,y}_{n}= k)
&\leq  & \sum_{k=0}^{n-1}   \var_{k}(\phi)\, \pro(T^{x,y}_{n}= k)
+ \var_{n}(\phi) \sum_{k=n}^{+ \infty} \pro(T^{x,y}_{n}= k)
 \nonumber \\
&=  & \sum_{k=0}^{n-1}  \var_{k}(\phi) \pro(T^{x,y}_{n-k}= 0) +
\var_{n}(\phi) \sum_{k=n}^{+ \infty} \pro(T^{x,y}_{0}= k -n ) \nonumber\\
&\leq  & \sum_{k=0}^{n}   \var_{k}(\phi)\, \pro(S^{(\gamma)}_{n-k}= 0)\;,
\label{eq:115}
\end{eqnarray}
uniformly in $x,y$.  The bound \reff{eq:x} follows from
\reff{eq:105.1}, \reff{eq:110}, \reff{eq:115} and the fact that
\begin{equation}
  \label{eq:300}
 \sum_{j=0}^{+ \infty} \pro(T^{x,y}_{0}= j ) \;=\; 1 \;=\;
\pro(S^{(\gamma)}_{0}= 0) \;.
\end{equation}

\paragraph{}
To prove \reff{eq:xx} we use the strong Markov poroperty of the
process $(S^{(\gamma)}_n)_{n\in\nat}$ to obtain
\beq
\label{relationbasique}
\pro(S^{(\gamma)}_{n} = 0) = \sum_{k=1}^{n} \pro( \tau =  k ) \, \pro(
S^{(\gamma)}_{n-k} =0 )\;,
\eeq
where
\begin{equation}
\tau  \;=\;  \inf \{ n > 0 ; S^{(\gamma)}_n = 0 \} \;.
\label{eq:107}
\end{equation}

\paragraph{}
We now use (\ref{relationbasique}) to bound the last line in \reff{eq:115}
in the form
\bqy
\sum_{k=0}^{n}   \var_{k}(\phi)\, \pro(S^{(\gamma)}_{n-k}= 0)&\leq  &
\sum_{k=1}^{n}\bigl[\var_0(\phi) \, \pro( \tau =  k ) +
\var_k(\phi)\bigr]\,  \pro(S^{(\gamma)}_{n-k}= 0) \nonumber \\
&\leq  &  C \sum_{k=1}^{n}   \pro(\tau = k)\,\pro(S^{(\gamma)}_{n-k}=
0) \nonumber \\
&= &  C \;  \pro(S^{(\gamma)}_{n}= 0) \;,
\label{eq:122}
\eqy
with
\begin{equation}
C \;=\; \var_0(\phi) + \sup_k \,{\var_k(\phi) \over \pro( \tau =
k )}\;.
\label{eq:122.1}
\end{equation}
To conclude, we must prove that the constant $C$ is finite.  By direct
computation,
\begin{eqnarray}
\label{tau}
\pro(\tau  = 1)  &= & \gamma_{0},\nonumber \\[3pt]
\pro(\tau = n)& =& \gamma_{n-1}\,\prod_{m=0}^{n-2}(1- \gamma_{m})
\quad  \hbox{ for } n\geq 2,\\[3pt]
\pro(\tau = +\infty) &= & \prod_{m=0}^{+ \infty}(1-
\gamma_{m})\;. \nonumber
\end{eqnarray}
From this and \reff{rr.30} we obtain
\begin{equation}
\lim_{k\to\infty} {\var_k(\phi) \over \pro( \tau = k )} \;=\;
\lim_{k\to\infty} \,{\var_k(\phi) \over 1 - e^{-\var_k(\phi)}}\,
{1 \over \prod_{m=0}^{k-2}(1- \gamma_{m})} \;.

\end{equation}
Since $\var_k(\phi)\to 0$, the first fraction converges to 1.  We see
from \reff{tau} that the second fraction converges to
$1/\pro(\tau = +\infty)$.  By elementary calculus, this is finite
since $\phi$ has summable variations.~\square

\begin{rem}
The previous computations lead to stronger results for more regular
functions $g$. For example, when $g$ satisfies
\begin{equation}
  \label{eq:301}
  \var_k(g) \;\le\; ||g||_{\theta} \,\theta^k
\end{equation}
for some $\theta <1$ and some $||g||_{\theta}<\infty$ (H\"older norm
of $g$), a chain of inequalities almost identical to those ending in
\reff{eq:110} leads to
\bqy \left| \int f \circ T^n g d\mu - \int f
  d\mu \int g d\mu \right| &\leq& ||f||_1\,\sum_{k=0}^{+\infty}
||g||_{\theta}\, \theta^{k}
\,\gamma^*_{n-k} \nonumber \\
&\leq & ||f||_1 \,||g||_{\theta}\,\theta^n \sum_{k=0}^{n}
\theta^{-k}\,\gamma^*_{k} \;.
\label{eq:122.2}
\eqy

On the other hand, if $g$ is a function that depends only on the first
coordinate, we get,
\bqy
\left| \int f \circ T^n g d\mu - \int f d\mu \int g d\mu
\right| &\leq & ||f||_1 \,\sup_{x,y}{\Bigl|\esp[g(Z^{x}_n)] -
\esp[g(Z^{y}_n)]\Bigr|}\nonumber \\
&\leq & ||f||_1\, ||g||_{\infty} \pro( \tilde{U}_n \neq \tilde{V}_n)
\nonumber \\
&\leq & ||f||_1\, ||g||_{\infty} \, \gamma^*_n\;.
\label{eq:130}
\eqy
\end{rem}

\section{Proof of Theorem~\ref{general}}
\label{proof}
\paragraph{}
We now consider the general case where the function $\phi$ is not
necessarily normalized.  In this case we resort to the normalization
$\psi$ define in \reff{rr.25} and we consider
chains with transition probabilities
\begin{equation}
\trans(a \,|\, x ) \;=\; e^{\phi(xa)}\,\frac{\rho(xa)}{\rho(x)}
\;=:\; e^{\psi(xa)} \;.
\label{gra.10}
\end{equation}
However, the summability of the variations of $\phi$ does not imply
the analogous condition for $\psi$, because there
are addition ``oscillations'' due to the cocycle $\log{\rho} -
\log{\rho \circ T}$.  Instead,
\begin{equation}
\left.
\begin{array}{r}
\var_m\psi\\
\var_m(\log\rho)
\end{array}\right\} \;\leq\; \sum_{k \geq m} \var_k(\phi)\;,
\label{gra.0}
\end{equation}
for all $m\ge 0$ (see Walters 1978).
Hence, we can apply Theorem~\ref{normalizedth} only under the
condition
\begin{equation}
\sum_{k=1}^{+\infty} k\, \var_k(\phi) < + \infty\;.
\label{gra.1}
\end{equation}
If this is the case, the correlations for functions $f \in L^1(\mu)$
and $g \in V_\psi$ decay faster than $\gamma^*_{m}$, where $\gamma_{m}
= e^{\sum_{k \geq m} \var_k(\phi)} -1$.

\paragraph{}
To prove the general result without assuming \reff{gra.1} we must work
with \emph{block} transition probabilities, which are less sensitive
to the oscillations of the cocycle.  More precisely,
given a family of transition probabilities $\trans$ on $A \times
\underline{A}$, let $\trans_n$ denote the corresponding transition
probabilities on $A^n \times \underline{A}$:
\begin{equation}
\trans_{n+1}(\atom{a}{0}{n} \,|\, x)

\;=\; \trans(a_n \,|\, a_{n-1} \cdots\, a_1 x) \cdots
\trans(a_2 \,|\,a_1 x)\, \trans(a_1 \,|\, x)
\label{gra.3}
\end{equation}
where
\begin{equation}
\atom{a}{0}{n} \;:=\; (a_0, \ldots, a_n) \;\in A^{n+1}\;.
\label{gra.4}
\end{equation}
If the transition probabilities $\trans$ are defined by a normalized
function $\phi$ as in \reff{eq:20}, then we see from \reff{gra.3} that
the transition probabilities $\trans_n$ obey a similar relation
\begin{equation}
  \label{gra.7}
  \trans_n(\atom{a}{0}{n-1}\, |\, x ) = e^{\phi_n(x\atom{a}{0}{n-1})}\;,
\end{equation}
with
\begin{equation}
\phi_n(x\atom{a}{0}{n-1}) \;:=\; \sum_{k=0}^{n-1}\phi(x a_0 \cdots
a_k)\;.
\label{gra.8}
\end{equation}
In particular, for transitions \reff{gra.10} the formula \reff{gra.3}
yields
\begin{equation}
\psi_n \;=\; \phi_n + \log{\rho} - \log{\rho \circ T^n} \;.
\label{gra.12}
\end{equation}

\paragraph{}
A comparison of \reff{gra.12} with \reff{gra.0} shows that it is
largely advantageous to bound directly the oscillations of $\psi_n$.
This is what we do in this section by adapting the arguments of
Section~\ref{normalized}.

\subsection{Coupling of the transition probabilities for blocks}
\paragraph{}
For every integer $n$, we define a family of transition probability
$\transb_n$ on $(A^{n})^2 \times \underline{A}^2$ by
\begin{equation}
\transt_n(\atom{a}{0}{n-1}, \atom{b}{0}{n-1}\,|\, x, y) \;=\;
\bigl[ \trans_n(\cdot \,|\,x)\, \tilde{\times}\, \trans_n(.  \,|\,
y)\bigr]
(\atom{a}{0}{n-1}; \atom{b}{0}{n-1})\;.
\label{gra.27}
\end{equation}

Let $(n_{m})_{m \in \nat}$ be an increasing sequence.
For each double history $x,y$, we consider the coupling
$\bigl((\overline{U}^{x, y}, \overline{V}^{x, y})\bigr)_{m\in\rel}$
of the chains for $n_m$-blocks with past $x$ and $y$,  defined by,
\begin{eqnarray}
\lefteqn{
\pro(\atom{\overline{U}^{x,y}}{0}{n_m}= \atom{a}{0}{n_m}\,,\,
\atom{\overline{V}^{x,y}}{0}{n_m}= \atom{b}{0}{n_m})}
\nonumber\\
&&=\ \prod_{m=1}^{M}
\transb_{n_{m+1} - n_{m}}(\atom{a}{n_m}{n_{m+1}} \,,\, \atom{b}{n_m}{n_{m+1}}
\,|\, a_{n_m} \cdots a_0 x\,,\, b_{n_m} \cdots b_0 y)\;.
\label{gra.31}
\end{eqnarray}

\subsection{The process of last block-differences}
\paragraph{}
We set
\begin{equation}
\gamma^{(n)}_k \;=\; 1 - \inf\, \left \{
{\frac{\trans_n(\atom{a}{0}{n-1}\,|\, x) }{\trans_n(\atom{a}{0}{n-1}
\,|\, y)}}
\,:\, x \stackrel{k}{=}y\,,\; a_1, \ldots, a_{n-1} \in A\right\}\;.
\end{equation}
From~(\ref{coupling2}) we see that, for $x \stackrel{k}{=}y$, the
weight of the diagonal of each coupling $\transb_n$ satisfies
\beq
\label{diago}
\Delta\bigl(\transt_n(\cdot,\cdot \, |\, x, y)\bigr) \; \geq\;
 \inf_{a_0, \ldots, a_{n-1} \in
 A}{\frac{\trans(\atom{a}{0}{n-1}\,|\, x)
 }{\trans(\atom{a}{0}{n-1} \,|\, y)}}\; \geq \; 1 - \gamma^{(n)}_k \;.
\eeq
If we denote
$$\overline{\Delta}^{x,y}_{m, m+q} \;:= \;

\Bigl\{ \overline{U}^{x,y}_j =  \overline{V}^{x,y}_j \, , \,  n_m \leq
j \leq n_{m+q}\Bigr\} \;,$$
we deduce from \reff{diago} that
\beq
\label{onestep}
\pro(\overline{\Delta}_{m+k+1} \,|\, \overline{\Delta}_{m,m+k})
\;\geq\; 1 - \gamma^{(n_{m+k+1}-n_{m+k})}_{n_{m+k} - n_m} \;.
\eeq

\paragraph{}
We construct the process $(\overline{T}_n)_{n \in \nat}$ with
\begin{equation}
\overline{T}^{x,y}_m \;=\; \inf \,\Bigl\{ p \geq 0 \,:\,
U^{x,y}_i \neq V^{x,y}_i \hbox{ for some } i, n_{m-p} \leq i \leq
n_{m-p+1} \Bigr \} \;.

\end{equation}
By \reff{onestep}, the conditional laws of this process satisfy,
\begin{equation}
\pro(\overline{T}_{m+1}=k+1 \,|\, \overline{T}_{m} = k) \;\geq\;
  1 - \gamma^{(n_{m+k+1}-n_{m+k})}_{n_{m+k} - n_m}
\label{gra.40}
\end{equation}
and
\begin{equation}
\pro(\overline{T}_{m+1}= 0 \;|\; \overline{T}_{m} = k) \;\leq\;
\gamma^{(n_{m+k+1}-n_{m+k})}_{n_{m+k} - n_m}\;.
\label{gra.41}
\end{equation}

\subsection{The dominating Markov process}
\paragraph{}
Let us choose the length of the blocks in such a way that the sequence
$(n_{m})_{m \in \nat}$ is subadditive, i.e.
\begin{equation}
n_{m+k} - n_m \;\leq\; n_k
\label{gra.sub}
\end{equation}
for $m,k\ge 0$, and that
\begin{equation}
\sup_{n\ge 0} \gamma^{(n)}_{\ell} \;<\; 1
\label{gra.sup}
\end{equation}
for all $\ell\ge 0$.

These two properties together with \reff{gra.40}--\reff{gra.41} imply
that, for all histories $x$ and $y$,
\begin{equation}
\pro(\overline{T}^{x,y}_{m+1}=k+1 \,|\, \overline{T}^{x,y}_{m} = k) \;\geq\;
  1 - \overline{\gamma}_{k}
\label{gra.50}
\end{equation}
and
\begin{equation}
\pro(\overline{T}^{x,y}_{m+1}= 0 \,|\, \overline{T}^{x,y}_{m} = k) \;\leq\;
\overline{\gamma}_{k}.
\label{gra.51}
\end{equation}
with
\begin{equation}
\overline{\gamma}_{k} \;:=\; \sup_{n \geq 1} \gamma^{(n)}_{n_k}\;,
\label{gra.52}
\end{equation}
for $m \geq 1$.

\paragraph{}
We now define the ``dominating'' Markov chain $(S^{(\overline\gamma)}_n)_{n \in
  \nat}$ as in \reff{rr.5.-1}--\reff{rr.5}. Lemma~\ref{dominlem} yields
\begin{equation}
\pro(\overline{T}^{x,y}_m = 0) \;\leq\; \pro(\overline{S}_m = 0)
\;\leq\; \overline{\gamma}^*_m\;.
\end{equation}
Hence, if $n_m \leq n \leq n_{m+1}$,
\begin{equation}
\pro(\overline{U}^{x,y}_n \neq \overline{V}^{x,y}_n) \;\leq\;
\pro(T^{x,y}_m = 0) \;\leq\;  \overline{\gamma}^*_m\;.
\end{equation}

\subsection{Decay of correlations}

\paragraph{}
We can now mimick the proof of Theorem~\ref{normalized} in terms of
barred objects.

\paragraph{}
As $( \var_m(\phi))_{m \in \nat}$ is summable, there exists a
subadditive sequence $(n_{m})_{m \in \nat}$ such that the sequence
$\alpha_m$ of the tails

\begin{equation}
\alpha_m \;=\; \sum_{k \geq n_{m}}  \var_k(\phi)
\end{equation}
is summable:
\begin{equation}
\sum_{m \geq 0}  \alpha_m < + \infty\;.
\label{gra.51.1}
\end{equation}

\paragraph{}
The transitions for blocks of size $n$ satisfy
\begin{equation}
\frac{\trans_n(\atom{a}{0}{n-1} \,|\, x) }{\trans_n(\atom{a}{0}{n-1}
  \,|\, y)} \;\ge\; e^{-\var_k(\psi_n)}
\label{gra.55}
\end{equation}
if $x\stackrel{k}{=} y$.
But from \reff{gra.12}, \reff{gra.8} and \reff{gra.0} we have
\begin{eqnarray}
\var_k(\psi_n) &\le& \left(\sum_{m=k}^{k+n}  + \sum_{m\ge k+n} +
\sum_{m\ge k}\right) \var_m(\phi) \nonumber\\
&\le& 3\, \sum_{m\ge k}\var_m(\phi)\;.
\label{gra.56}
\end{eqnarray}
Hence we can choose in \reff{gra.52}
\begin{equation}
\overline{\gamma}_{k}  \;\leq\; 1 - e^{- 3 \alpha_k}\;,
\label{gra.57}
\end{equation}
a choice for which
\begin{equation}
\sum_{k\geq 1} \overline{\gamma}_{k} < + \infty \;.
\label{gra.58}
\end{equation}

\paragraph{}
To prove the theorem, we now proceed as in \reff{eq:105.1} and
\reff{eq:110}--\reff{eq:122.1} but replacing tildes by bars and
putting bars over the processes $(T_n)$ and $(S_n^{(\gamma)})$.  We just
point out that, due to the subadditivity of $n_m$,
$$\var_{(n_{m+k} - n_m)}(\phi) \; \leq\; \var_{n_k}(\phi)$$
uniformly in $m$.~\square

\appendix

\section{Returns to the origin of the dominating Markov chain}
\label{markovchain}
\paragraph{}
In this appendix we collect a few results concerning the
probability of return to the origin of the Markov chain
$(S^{(\gamma)}_{n})_{n \in \nat}$ defined via \reff{rr.5}.
(In the sequel we omit the superscript ``$(\gamma)$'' for simplicity.)

\begin{prop}
\label{asymptotic}
Let $(\gamma_{n})_{n \in \nat}$ be a real-valued sequence decreasing
to $0$ as $n \to + \infty$.
\bid
\item[(i)] If $ \displaystyle \sum_{m\geq 1} \prod_{k=0}^{m}(1 -
  \gamma_k) = + \infty $, then $\pro(S_n = 0) \to 0$.
\item[(ii)] If $ \displaystyle \sum_{m\geq 1} \gamma_k < + \infty $,
  then $\sum_{n \geq 0} \pro(S_n = 0) < + \infty$.
\item[(iii)] If $(\gamma_m)$ decreases exponentially, then so does
  $\pro(S_n = 0)$.
\item[(iv)] If $(\gamma_m)$ decreases polynomially, then $\pro(S_n = 0)
  = O(\gamma_n)$.
\eid
\end{prop}

\paragraph{Sketch of the proof}

\paragraph{}
Statement $(i)$ follows from the well known fact that the
Markov chain $(S_n)_{n\in\nat}$ is positive recurent if and only if,
$$\sum_{m\geq 1} \prod_{k=0}^{m}(1 - \gamma_k) <  + \infty. $$

\paragraph{}
To prove parts (ii) and (iii) we introduce the series
\begin{equation}
F(s) = \sum_{n=1}^{+ \infty} \pro(\tau=n) \,s^{n}\;,
\label{gra.61}
\end{equation}
and
\begin{equation}
G(s) \;=\; \sum_{n=0}^{+ \infty} \pro(S_n =0)\, s^{n}
\label{gra.60}
\end{equation}
where the random variable $\tau$ is the time of first return to zero,
defined in \reff{eq:107}.  The probabilities $\pro(\tau=n)$ were
computed in \reff{tau} above.
The relation (\ref{relationbasique})
implies that these series are related in the form

\beq
\label{avecF}
G(s) = \frac{1}{1 -F(s)}\;,
\eeq
for all $s \geq 0$ such that $F(s)<1$.

\paragraph{}
It is clear that the radius of convergence of $F$ is at least 1.  In fact,
  \begin{equation}
    \label{eq:151}
    F(1) \;=\; \pro(\tau < +\infty)\;.
  \end{equation}
Moreover, if $ \sum_{m\geq 1} \gamma_k < + \infty $, the radius of
  convergence of $F$ is
  \begin{equation}
    \label{eq:152}
    \lim_{n\to\infty} \, [\gamma_n]^{-1/n}\;.
  \end{equation}
This is a consequence of the fact that
$\pro(\tau=n)/\gamma_{n-1}\to \pro(\tau=+\infty) > 0$, as concluded
from \reff{tau}.

\paragraph{}
Statement $(ii)$ of the proposition is a consequence of the fact that
the radius of convergence of the series $G$ is at
  least 1 if $ \sum_{m\geq 1} \gamma_k < + \infty $.  This follows
  from the relation \reff{avecF} and the fact that the right-hand side
  of \reff{eq:151} is strictly less than one when the chain
  $(S_n^{(\gamma)})$ is transient.

\paragraph{}
To prove statement (iii) let us assume that $\gamma_m \leq C \gamma^m$
for some constants $C < +\infty $ and $0<\gamma<1$.  By \reff{eq:152},
the radius of convergence of $F$ is $\gamma^{-1} > 1$ while, by
\reff{eq:151}, $F(1)<1$.  By continuity it follows that there exists
$s_0>1$ such that $F(s_0)=1$ and, hence, by \reff{avecF},
$G(s)<+\infty$ for all $s<s_0$.  By definition of $G$, this implies
that $\pro(S_n = 0)$ decreases faster than $\zeta^n$ for any
$\zeta\in(1,s_0^{-1})$.

\paragraph{}
Statement (iv) is a consequence of the following lemma.

\begin{lem}
If
\beq
\label{condpoly}
\alpha \;:=\; \sup_i\, \overline{\lim}_{k\to\infty}
\left[ \frac{\pro(\tau = i)}{\pro(\tau =ki)}\right]^{1/k}
\;<\; {1\over \pro(\tau<+\infty)}\;,
\eeq
then
$$  \pro(S_n = 0) = O\left( \pro( \tau = n) \right).$$
\end{lem}

\paragraph{Proof}
We start with the following observation.  If
$i_1 + \cdots + i_k = n$, then $\max_{1 \leq m \leq k} i_m . n/k$ and
thus, for $g$ is an increasing
$$
g(n) \;\leq\; g\left( k\, i_{\rm max}\right) \;,
$$
where $i_{\rm max} = \max_{1\le m\le k}\,i_m$.
If we apply this to $g(n) = 1/\pro(\tau=n)$, which is increasing by
\reff{tau}, we obtain
\begin{equation}
1 \;\le\; \frac{\pro( \tau = n)}{\pro(\tau = k\,i_{\rm max})}\;.
\label{fra.1}
\end{equation}

\paragraph{}
We now invoke the following a explicit relation between the
coefficients of $F$ and $G$.
\beq
\label{generalexpression}
\pro(S_n =0) = \sum_{k = 1}^{n} \sum_{ \scriptsize
\begin{array}{c}
i_1, \ldots, i_k \geq 1 \\
i_1 + \cdots + i_k  = n
\end{array}
} \prod_{m=1}^{k} \pro(\tau = i_m) \;,
\eeq
for $n \geq 1$.

Multiplying and dividing each factor in the rightmost
product by $\pro(\tau<+\infty)$, this formula can be rewritten as
\beq
\label{autreexpression}
\pro(S_n =0) = \sum_{k = 1}^{n} \pro(\tau < + \infty)^k  \sum_{ \scriptsize
\begin{array}{c}
i_1, \ldots, i_k \geq 1 \\
i_1 + \cdots + i_k  = n
\end{array}
} \prod_{m=1}^{k} \pro(\tau = i_m \,|\, \tau <
      +\infty).
\eeq
Combining this with \reff{fra.1} we obtain
\begin{equation}
\label{fra.2}
\pro(S_n =0) \;\le\; \pro(\tau=n)\,\sum_{k = 1}^{n} \pro(\tau < +
\infty)^k  \sum_{ \scriptsize
\begin{array}{c}
i_1, \ldots, i_k \geq 1 \\
i_1 + \cdots + i_k  = n
\end{array}
} \prod_{m=1}^{k} {\pro(\tau = i_m \,|\, \tau < +\infty)\over
\pro(\tau = k\, i_{\rm max})}\;,
\end{equation}
If we single out the factor $\pro(\tau=i_{\rm max}\,|\, \tau <
+\infty)=\pro(\tau=i_{\rm max})/\pro(\tau<+\infty)$ from the rightmost product
of \reff{fra.2} and use the hypothesis \reff{condpoly} we get
\begin{equation}
\label{fra.3}
\pro(S_n =0) \;\le\; C\,\pro(\tau=n)\,\sum_{k = 1}^{n} \alpha^k\,\pro(\tau < +
\infty)^{k-1}  \sum_{ \scriptsize
\begin{array}{c}
i_1, \ldots, i_k \geq 1 \\
i_1 + \cdots + i_k  = n
\end{array}
} \prod_{\scriptsize 1\le m \le k\atop \scriptstyle i_m\neq i_{\rm max}}
\pro(\tau = i_m \,|\, \tau < +\infty)\;,
\end{equation}
for some constant $C>0$.
To bound the last sum on the right-hand side
we introduce a sequence of independent random variables
$(\tau^{(i)})_{i \in \nat}$ with common distribution
\begin{equation}
\pro(\tau^{(i)} = j) \;=\; \pro(\tau = j \,|\, \tau < +\infty)\;.
\label{fra.0}
\end{equation}
Then
\begin{equation}
\sum_{j= 1}^{n-k+1} \pro\Bigl( \sum_{i=1}^{k-1} \tau^{(i)} =
n-j\Bigr) \;\le\; 1\;.
\label{fra.5}
\end{equation}
Hence, \reff{fra.3} implies
\begin{equation}
\label{fra.4}
\pro(S_n = 0) \;\leq\; C\, \alpha\, \sum_{k=1}^{\infty}
\left[\alpha^{-1}\pro(\tau<+ \infty)\right]^{k-1}\,
\pro(\tau = n) \;\leq\; \hbox{const}\, \pro(\tau = n)\;. \; \square
\end{equation}

\paragraph{}
We notice that, according to~(\ref{tau}), $\gamma_n \sim
\pro(\tau = n)/\pro(\tau=+ \infty)$. Hence, a sufficient condition for
\reff{condpoly} is a similar condition for the sequence
$(\gamma_n)$.  Such a condition holds, for instance, if the later
sequence decays polynomially. Statement $(iii)$ of the
proposition follows.~\square

\subsection*{Acknowledgements}  It is a pleasure to thank Pablo Ferrari
and Luis Renato Fontes for useful discussions.  We also thank Pablo
Ferrari for informing us about his work in progress with Alejandro
Maass and Servet Mar\'{\i}nez on a related regenerative representation
of chains with complete connections.
\smallskip

config: arxiv


TITLE: The curvature is equal to the derivative of the angle between the curve and the x-axis?
QUESTION [4 upvotes]: I'm trying to prove that if $\vec{x}:I\rightarrow\mathbb{R}^2$ is a curve parametrized by arc length and $\theta(t)$ is the angle between the tangent line to $\vec{x}$ at point $t$ and the $x$ axis, then $\kappa=\theta'$, where $\kappa$ denotes the curvature.
I know that for a curve in $\mathbb{R}^2$, the curvature is given by
$\kappa=\frac{1}{|\vec{x}'|^3}det(\vec{x}'\vec{x}'')$.
I also know that the tangent line to $\vec{x}$ at point $\alpha\in\mathbb{R}$ is given by $y(t)=\vec{x}(\alpha)+t\vec{x}'$.
I can picture the problem, but can't write the solutions. Any ideas?

REPLY [0 votes]: A simple alternative is to use the tangent vector to the curve
$$\vec{t} = (\cos(\theta), \sin(\theta))$$
and plug it in into
$$ \kappa = \left|\frac{d\vec{t}}{ds}\right| $$
to get
$$ \kappa = \frac{d\theta}{ds} $$
via chain rule.


TITLE: Proving that the limit of a function doesn't exist using negation of epsilon delta
QUESTION [1 upvotes]: $\lim \limits_{x \to \frac{1}{2}}\frac{1}{4x-2}$
I want to use the negation, $\exists \epsilon>0$ such that  $ \forall \delta>0$ , $\lvert\frac{1}{4x-2}-L \rvert \ge \epsilon$, $\forall x$ with $0<\lvert x-\frac{1}{2} \rvert <\delta$
So can I say that because $\lvert \frac{1}{4x-2}\rvert =\lvert \frac{1}{4(x-\frac{1}{2})} \rvert = \frac{1}{4}\lvert \frac{1}{x-\frac{1}{2}} \rvert \ge \epsilon $
Then $\frac{1}{4} \ge \epsilon \lvert x-\frac{1}{2} \rvert$
Can I then let $\epsilon =\frac{1}{4 \delta}$

REPLY [1 votes]: No, behold! The choice of $\varepsilon$ should not depend on that of $\delta$.
In fact we can prove something stronger than necessary:
If $x \neq 1/2$, then
$$
\bigg| \frac{1}{4x-2} \bigg| = \frac{1}{4}\frac{1}{|x- \frac{1}{2}|};
$$
If $\varepsilon > 0$, then
$\frac{1}{4}\frac{1}{|x - \frac{1}{2}|} > \varepsilon$ if $|x-\frac{1}{2}| < \varepsilon/4$; hence $0 < |x-\frac{1}{2}| < \varepsilon/4$ only if
$$
\bigg| \frac{1}{4x-2} \bigg| > \varepsilon,
$$
which says that
$$
\bigg| \frac{1}{4x-2} \bigg| \to \infty
$$
as $x \to 1/2$.

REPLY [0 votes]: Let $\epsilon = 1$, then for any $\delta > 0$, you can find an $x$ such that: $|x-\dfrac{1}{2}| < \delta$, and $\dfrac{1}{\left|4x-2\right|} \geq 1\iff \left|x-\dfrac{1}{2}\right| \leq \dfrac{1}{4}$. You can take $x$ such that $|x-\dfrac{1}{2}| < \min{{\dfrac{1}{4}, \delta}}$


text:
\section{Euler characteristics via logarithmic differential forms}
\label{sec:backEuler}

This section connects Euler characteristic to integrals of characteristic
classes of the sheaf of logarithmic differential forms. The following
proposition is certainly known, but not easy to locate in the
literature. We use the occasion to give a self-contained proof,
see also \cite{fiori}, and recall some standard exact sequences.
\par
\begin{prop} \label{prop:chiviaTlog}
Let $\overline{B}$ be a compact smooth $k$-dimensional

manifold, let~$D$ be a normal crossing divisor and
$B = \overline{B} \smallsetminus D$. Then the Euler characteristic
of~$B$ can be computed as integral
\be \label{eq:chiviaint}
\chi(B) \= (-1)^k \int_{\overline{B}} \,c_k(\Omega^1_B(\log D))
\ee
over the top Chern class of the logarithmic cotangent bundle.
\end{prop}
\par
In all our applications, $\overline{B}$ will be a compact orbifold
or proper smooth Deligne-Mumford stack.
We work throughout with orbifold Euler characteristics, and since
then both sides of~\eqref{eq:chiviaint} are multiplicative in the
degree of a covering, we can apply Proposition~\ref{prop:chiviaTlog}
verbatim.

\subsection{The compact case and the Riemann-Roch theorem}

We start with the proof of the special case of the main
theorem.
\par
\begin{prop} \label{prop:chiviaTB}
If $B = \overline{B}$ is smooth, compact and $k$-dimensional, then
\be
\chi(B) \=  \int_{\overline{B}} \, c_k(T_B) \,.
\ee
\end{prop}
\par
We start by recalling some intersection theory. Let $\cE$ be a
holomorphic vector bundle on $B$. Denote by $c_i\coloneqq c_i(\cE)\in \CH^i(B)$
the $i$th Chern class of $E$. Recall that $c_0 = 1$ and $c_i = 0$
for $i > \rk E \eqqcolon r$. The \emph{total Chern class} of~$\cE$
is the formal sum $\c(\cE) = 1 + c_1 + \dotsc + c_r.$ in $\CH(B)$.
Splitting formally $\c(\cE) = \prod_{i=1}^r (1+\alpha_i)$ into the Chern roots,
the \emph{Chern character} is defined as the formal power series
\[\ch(E) = \sum_{i=1}^r \exp(\alpha_i) \= \sum_{s\geq 0}\frac{1}{s!}
 \sum_{i=1}^r \alpha_i^s
\= {\rm rk}(E) + c_1 + \frac{1}{2}(c_1^2 - 2c_2) + \cdots .\]
Furthermore, the Todd class is defined as
\[\td(E) = \prod_{i=1}^r \frac{\alpha_i}{1-\exp(-\alpha_i)}
\= 1 + \frac{1}{2} c_1 + \frac{1}{12}(c_1^2 + c_2) + \frac{1}{24} c_1c_2 + \cdots
.\]
\par
The Grothendieck-Riemann-Roch theorem in the case of a map $f:X \to Y$ and
for the special case of that the  higher direct images $R^if_* \cE$
vanish, states that
\be \label{GRR}
\ch(f_* \cE)\cdot \td(T_Y) \= f_*(\ch(\cE)\cdot\td(T_X))\,.
\ee
\par
\begin{proof}[Proof of Proposition~\ref{prop:chiviaTB}]
For a topological proof, see e.g.\ \cite[Proposition~11.24]{BottTu}.
Using the notations already set up, we give a quick proof if
moreover $B$ is K\"ahler. From the Borel-Serre identity
(\cite[Example~3.2.5]{Fulton}) on a $k$-dimensional manifold
\bes
c_k(T_B) \= \ch\Bigl(\sum_{j=1}^k (-1)^j \Omega_B^j\Bigr) \cdot \td(T_B)
\ees
and the application
\bes
 \int_{\overline{B}} \ch((-1)^j \Omega_B^j) \cdot \td(T_B) \=
\sum_{\ell \geq 0} (-1)^{\ell+j} h^\ell(B,\Omega^j_B)
\ees
of Grothendieck-Riemann-Roch theorem for the map from~$B$ to a point
we get
\bes
\int_{\overline{B}} \,c_k(T_B) \= \sum_{\ell,j \geq 0} (-1)^{\ell+j} h^\ell(B,\Omega^j_B)
\= \chi(B)
\ees
by the Hodge decomposition.
\end{proof}

\subsection{The non-compact case and log differential forms}

We suppose throughout that $D = \cup_{j=1}^s D_j$ is a reduced normal
crossing divisor, i.e., with distinct irreducible components~$D_i$
intersecting each other transversally. In this situation
$\Omega^1_{\overline{B}}(\log D)$ is defined to be the vector bundle
of rank~$n$ with the following local generators. In a neighborhood~$U$ of
a point where (say) the first $r \leq s$ divisors meet
and where $x_1,\ldots,x_k$ is a local coordinate system with
$D_j = \{x_j = 0\}$, then logarithmic cotangent bundle is defined by
\be
\Omega^1_{\overline{B}}(\log D)(U) \= \Bigl\langle \frac{dx_1}{x_1}, \ldots,
\frac{dx_r}{x_r}, x_{r+1}, \ldots, x_{k} \Bigr \rangle\,
\ee
as an $\cO_{\overline B}(U)$-module. There is a fundamental exact sequence for
log differential forms, namely
\ba \label{eq:LogDiffSeq}
0&\to \Omega^1_{\overline B} \to \Omega^1_{\overline B}(\log D) \to \oplus_{j=1}^s
(\fraki_{j})_* \cO_{D_j} \to 0\,,
\ea
where $\fraki_j: D_j \to \overline B$ is the inclusion map.
More details can be found e.g.\ in \cite[Proposition~2.3]{esvibook}.
\par
\begin{proof}[Proof of Proposition~\ref{prop:chiviaTlog}]
We first reduce to the case that $D$ has simple normal crossings, i.e.,
to the case the the $D_j$ are all smooth. This can always be achieved
by an \'etale covering. Since both sides of \eqref{eq:chiviaint} are
multiplied by the degree under such a covering, we can assume simple
normal crossings. Our goal is to prove
\bes
\int_{\overline B} c_k\Bigl(\Omega^1_{\overline{B}} \Bigl(\log \sum_{i \geq 2} D_i\Bigr)\Bigr)
\= \int_{\overline B} c_k\Bigl(\Omega^1_{\overline{B}}(\log D)\Bigr) - \int_{D_1}
c_{k-1}\Bigl(\Omega^1_{D_1} \Bigl(\log (\sum_{i \geq 2} D_i \cap D_1 \Bigr)\Bigr)\,,
\ees
The claim follows then from the additivity $\chi(B) + \chi(D)
= \chi(\overline{B})$ of the Euler characteristic,
Proposition~\ref{prop:chiviaTB} and an application to the preceding
identity to $B_j = \overline{B} \smallsetminus \cup_{i=j}^s D_j$.
\par
We consider the inclusion of the boundary divisor~$D_1$
and deduce from the ideal sheaf sequence that $c((\fraki_1)_*\cO_{D_1})
=(1-[D_1])^{-1}$ and that $c(\cN_{D_1}) = 1 + \fraki_1^*[D_1]$.
Moreover the normal bundle sequence $0\to \cT_{D_1} \to
\fraki_1^*\cT_{\overline{B}} \to \cN_{D_1} \to 0$ implies
\be \label{eq:LogdiffPB}
c(\Omega^1_{D_1}) \= \fraki_1^* \Bigl( c(\Omega^1_{\overline{B}}) \cdot
\frac{1}{1-[D_1]} \Bigr)\,.
\ee
On the other hand, the sequence~\eqref{eq:LogDiffSeq}  gives
\be \label{eq:OmlogMult}
c(\Omega^1_{\overline{B}}(\log D)) \=  c(\Omega^1_{\overline{B}})
\cdot \frac{1}{1-[D_1]} \cdot \prod_{j=2}^s  \frac{1}{1-[D_j]}
\ee
and also
\be
 \quad c(\Omega^1_{D_1}(\log \Bigl(\sum_{i \geq 2}
D_i \cap D_1 \Bigr))) \=  c(\Omega^1_{D_1})
\cdot \prod_{j=2}^s  \frac{1}{1-[ D_1\cap D_j]}.
\ee

Hence comparing with~\eqref{eq:LogdiffPB} we get
\be \label{eq:OmRelPullback}
c\Bigl(\Omega^1_{D_1} \bigr(\log \Bigl(\sum_{i \geq 2}
D_i \cap D_1 \Bigr)\big) \Bigr)\=\fraki_1^*c\bigl(\Omega^1_{\overline{B}}(\log D)\bigr)\,.
\ee
Finally from \eqref{eq:OmlogMult} and from the appropriate version of the sequence~\eqref{eq:LogDiffSeq} we also get
\be
c(\Omega^1_{\overline{B}}(\log D)) \=\frac{1}{1-[D_1]} c\Bigl(\Omega^1_{\overline{B}}
\Bigl(\log \sum_{i \geq 2} D_i\Bigr)\Bigr).
\ee
The claim now follows by multiplying this last expression with $1-[D_1]$, integrating and taking the $k$-th coefficient,
using that $\int_{\overline{B}} [D]  \cdot c_{k-1}
(\Omega^1_{\overline{B}} (\log D)) = \int_D \fraki_1^*c_{k-1}
(\Omega^1_{\overline{B}}(\log D))$.
\end{proof}


TITLE: $\mathbb{Q}$ closure, interior and boundary in ($\mathbb{R}$,d)
QUESTION [0 upvotes]: What is the closure, interior and boundary of $\mathbb{Q}$ in ($\mathbb{R}$,d) a discrete metric space and how to interpret or prove this?

REPLY [2 votes]: If $(\mathbb{R},d)$ is a discrete metric space, then every subset of $\mathbb{R}$ is both open and closed with respect to the given metric.
Thus every subset $X \subseteq \mathbb{R}$ satisfies the following. Interior of $X$ is equal to $X$. Closure of $X$ is equal to $X$ and the boundary of $X$ is equal to the empty set.


text:
\begin{document}

\theoremstyle{amit}
\newtheorem{defn}{Definition}[section]
\newtheorem{prop}[defn]{Proposition}
\newtheorem{lem}[defn]{Lemma}
\newtheorem{thm}[defn]{Theorem}
\newtheorem{cor}[defn]{Corollary}
\newtheorem{rmk}[defn]{Remark}
\newtheorem{ex}[defn]{Example}

\newcommand\blfootnote[1]{
  \begingroup
  \renewcommand\thefootnote{}\footnote{#1}
  \addtocounter{footnote}{-1}
  \endgroup
}

\title{Bottleneck Stability for Generalized Persistence Diagrams}
\author[1]{Alex McCleary}
\author[1]{Amit Patel}
\affil[1]{Department of Mathematics, Colorado State University}
\date{}

\maketitle

\abstract{In this paper, we extend bottleneck stability to the setting of one dimensional
constructible persistences module valued in any skeletally small abelian category.

}

\section{Introduction}

Persistent homology is a way of quantifying the topology of a function.
Given a function $f : X \to \Rspace$, persistence scans the homology of the sublevel
sets $f^{-1}(-\infty, r]$ as $r$ varies from $-\infty$ to $\infty$.
As it scans, homology appears and homology disappears.
This history of births and deaths is recorded as a \emph{persistence diagram} \cite{CSEdH}
or a \emph{barcode} \cite{ZC2005}.
What makes persistence special is that the persistence diagram of $f$ is stable
to arbitrary perturbations to $f$.
This is the celebrated \emph{bottleneck stability} of Cohen-Steiner, Edelsbrunner, and Harer \cite{CSEdH}.
Bottleneck stability makes persistent homology a useful tool in data analysis and in pure mathematics.
All of this is in the setting of vector spaces where each homology group is computed using coefficients in a field.

Fix a field $\field$ and let $\Vect$ be the category of $\field$-vector spaces.
As persistence scans the sublevel sets of $f$, it records its homology
as a functor $\Ffunc : (\Rspace, \leq) \to \Vect$ where
$\Ffunc(r) := \Hfunc_\ast \big( f^{-1}(-\infty, r]; \field \big)$ and
$\Ffunc( r \leq s) : \Ffunc(r) \to \Ffunc(s)$ is the map induced by the inclusion of the
sublevel set at $r$ into the sublevel set at $s$.
The functor $\Ffunc$ is called the \emph{persistence module} of~$f$.
Assuming some tameness conditions on $f$, the persistence diagram of $\Ffunc$ is equivalent to its barcode,
but the two definitions are very different.
The \emph{barcode} of $\Ffunc$ is its list of indecomposables.
This list is unique up to a permutation and furthermore, each indecomposable is an
\emph{interval persistence module} \cite{ZC2005, Carlsson2010, Boevey}.
The barcode model is how most people now think about persistence.
However in \cite{CSEdH} where bottleneck stability was first proved, the persistence diagram
is defined as a purely combinatorial object.
The \emph{rank function} of $\Ffunc$ assigns to each pair of values $r \leq s$
the rank of the map $\Ffunc(r \leq s)$.
The M\"obius inversion of the rank function is the \emph{persistence diagram} of $\Ffunc$.
Remarkably, these two very different approaches to persistence give equivalent answers.

The persistence diagram of \cite{CSEdH}
easily generalizes \cite{patel} to the setting of constructible persistence modules valued
in any skeletally small abelian category $\Ccat$.
The rank function of such a persistence module records the image
of each $\Ffunc(r \leq s)$ as an element of the Grothendieck group
of $\mathcal{C}$.
Here we are using the Grothendieck group of an abelian category: this is the abelian group
with one generator for each isomorphism class of objects and one relation for each short exact sequence.
The persistence diagram of $\Ffunc$ is then the M\"obius inversion of this rank function.
A weak form of stability was shown in \cite{patel}.
In this paper, we prove bottleneck stability.
Our proof is an adaptation of the proofs of \cite{CSEdH} and~\cite{crazy_persistence}.

We were hoping that the M\"obius inversion model for persistence would lead to a
good theory of persistence for multiparameter persistence modules
$\Ffunc : (\Rspace^k, \leq) \to~\Ccat$ \cite{Carlsson2009,Lesnick2015}.
The M\"obius inversion applies to arbitrary finite posets.
Assuming some finiteness conditions on $\Ffunc$, we may define its persistence diagram
as the M\"obius inversion of its rank function.
The proof of bottleneck stability presented in this paper requires positivity of the persistence diagram;
see Proposition \ref{prop:positivity}.
Unfortunately, there are simple examples of multiparameter persistence modules
whose persistence diagrams are not positive.
Therefore the proof presented here does not generalize to the multiparameter setting.
It seems that the M\"obius inversion model for persistence works well only in the setting of
one-parameter constructible persistence modules.

\section{Persistence Modules}

Fix a skeletally small abelian category $\mathcal{C}$.
By skeletally small, we mean that the collection of isomorphism classes of objects in $\mathcal{C}$
is a set.
For example, $\mathcal{C}$ may be the category of finite dimensional $\mathsf{k}$-vector spaces,
the category of finite abelian groups, or the category of finite length $R$-modules.
Let $\bar{\Rspace} := \Rspace \cup \{ \infty \}$ be the totally ordered set of real numbers with the point
$\infty$ satisfying $p < \infty$ for all $p \in \Rspace$.
For any $p \in \bar \Rspace$, we let $\infty + p = \infty$.

\begin{defn}
A \define{persistence module} is a functor $\Ffunc : \Rspace \to \mathcal{C}$.
Let
$$S=\{s_1 < s_2 < \cdots < s_k < \infty \} \subseteq \bar \Rspace$$ be a finite subset.
A persistence module $\Ffunc$ is \define{$S$-constructible} if it satisfies the following two conditions:
\begin{itemize}
\item For $p \leq q < s_1$, $\Ffunc(p \leq q) : 0 \to 0$ is the zero map.
\item For $s_i \leq p \leq q< s_{i+1}$, $\Ffunc(p \leq q)$ is an isomorphism.
\item For $s_k \leq p \leq q \leq \infty$, $\Ffunc(p \leq q)$ is an isomorphism.
\end{itemize}
\end{defn}

For example, let $f : M \to \Rspace$ be a Morse function on a compact manifold $M$.
The function $f$ filters $M$ by sublevel sets $M_{\leq r}^f := \{ p \in M \; |\; f(p) \leq r \}$.
For every $r \leq s$, $M_{\leq r}^f \subseteq M_{\leq s}^f$.
Now apply homology with coefficients in a finite abelian group.
The result is a persistence module of finite abelian groups that is constructible
with respect to the set of critical values of $f$ union $\{ \infty\}$.
If one applies homology with coefficients in a field $\mathsf{k}$, then the result
is a constructible persistence module of finite dimensional $\mathsf{k}$-vector spaces.
In topological data analysis, one usually starts with a constructible filtration of a finite simplicial complex.

There is a natural distance between persistence modules called the \emph{interleaving
distance} \cite{proximity}.
For any $\ee \geq 0$, let $\Rspace \times_\ee \{ 0, 1\}$ to be the poset
$\big(  \Rspace \times \{0\} \big) \cup \big(  \Rspace \times\{1\} \big)$
where $(p,t) \leq (q,s)$ if
\begin{itemize}
\item $t=s$ and $p \leq q$, or
\item $t \neq s$ and $p + \ee \leq q$.
\end{itemize}
Let $\iota_0,\iota_1: \Rspace \hookrightarrow  \Rspace \times_\ee \{ 0, 1\}$ be the poset maps
 $\iota_0: p \mapsto (p,0)$ and $\iota_1: p \mapsto(p,1)$.

\begin{defn}
\label{defn:interleaving}
An \define{$\ee$-interleaving} between two constructible persistence modules
$\Ffunc$ and $\Gfunc$ is a functor~$\Phi$ that makes the following diagram commute
up to a natural isomorphism:
\begin{equation}
\label{dgm:interleaving}
\begin{gathered}
\xymatrix{
&  \Rspace \times_\ee \{ 0, 1\} \ar@{-->}[dd]^{\Phi} \\
 \Rspace \ar@{^{(}->}[ru]^{\iota_0} \ar[rd]_{\Ffunc} &&  \Rspace \ar@{^{(}->}[lu]_{\iota_1} \ar[ld]^{\Gfunc} \\
& \mathcal{C}. &
}
\end{gathered}
\end{equation}
Two constructible persistence modules $\Ffunc$ and $\Gfunc$ are \define{$\ee$-interleaved} if there is an
$\ee$-interleaving between them.
The \define{interleaving distance} $d_I(\Ffunc,\Gfunc)$ between $\Ffunc$ and $\Gfunc$ is the infimum
over all $\ee\geq 0$
such that $\Ffunc$ and $\Gfunc$ are $\ee$-interleaved.
This infimum is attained since both $\Ffunc$ and $\Gfunc$ are constructible.
If $\Ffunc$ and $\Gfunc$ are not interleaved, then we let $d_I(\Ffunc,\Gfunc)=\infty$.
\end{defn}

\begin{prop}[Interpolation]
\label{prop:interpolation}
Let $\Ffunc$ and $\Gfunc$ be two $\ee$-interleaved constructible persistence modules.
Then there exists a one-parameter family of constructible persistence modules
$\{ \Kfunc_t \}_{t \in [0,1]}$
such that $\Kfunc_0 \cong \Ffunc$, $\Kfunc_1 \cong \Gfunc$, and
$d_I ( \Kfunc_t, \Kfunc_s ) \leq \ee |t-s|$.
\end{prop}
\begin{proof}
Let $\Ffunc$ and $\Gfunc$ be $\ee$-interleaved by $\Phi$ as in Definition \ref{defn:interleaving}.
Define $ \Rspace \times_\ee [0,1]$ as the poset with the underlying set $ \Rspace\times[0,1]$
and $(p,t) \leq (q,s)$ whenever $p+\ee |t-s| \leq q$.
Note that $\Rspace \times_\ee \{ 0, 1\}$ naturally embeds into $ \Rspace \times_\ee [0,1]$ via
$\iota:(p,t) \mapsto (p,t)$.
See Figure~\ref{fig:interpolation}.
Finding $\{\Kfunc_t \}_{t\in [0,1]}$ is equivalent to finding a functor $\Psi$ that makes the following diagram
commute up to a natural isomorphism:
\begin{center}
\begin{tikzcd}
 \Rspace \times_\ee \{ 0, 1\} \ar[rr,"\Phi"]\ar[d,hook,"\iota"']	&& \Ccat\\
 \Rspace \times_\ee[0,1] . \ar[urr,"\Psi"',dashed]
\end{tikzcd}
\end{center}
This functor $\Psi$ is the right Kan extension of $\Phi$ along $\iota$ for which
we now give an explicit construction.
For convenience, let $\Pcat :=  \Rspace \times_\ee \{ 0, 1\}$ and
$\Qcat :=  \Rspace\times_\ee[0,1]$.
For $(p,t) \in \Qcat$, let $\Pcat \uparrow
(p,t)$ be the subposet of $\Pcat$ consisting of all elements
$(p', t') \in \Pcat$ such that $(p, t) \leq (p',t')$.
The poset $\Pcat \uparrow (p,t)$, for any $p \in \Rspace$ and $t \notin \{ 0, 1 \}$, has two minimal elements: $(p + \ee t , 0)$ and $\big(p + \ee (1-t), 1 \big)$.
	\begin{figure}
	\centering
	\includegraphics[scale = 0.90]{interpolation}
	\caption{An illustration of the poset relation on $ \Rspace \times_\ee [0,1]$.}
	\label{fig:interpolation}
	\end{figure}
For $t \in \{ 0, 1 \}$, the poset $\Pcat \uparrow (p,t)$ has one minimal element,
namely $(p,t)$.
Let $\Psi \big( (p,t) \big) := \lim \Phi |_{\Pcat \uparrow (p,t)}$.
For $(p,t) \leq (q,s)$, the poset $\Pcat \uparrow (q,s)$ is a subposet of
$\Pcat \uparrow (p,t)$.
This subposet relation allows us to define the morphism $\Psi \big( (p,t) \leq (q,s) \big)$
as the universal morphism between the two limits.
Note that $\Psi \big( (p,0) \big)$ is isomorphic
to $\Ffunc(p)$ and $\Psi \big( (p,1) \big)$ is isomorphic to $\Gfunc(p)$.

We now argue that each persistence module $\Kfunc_t := \Psi(\cdot, t)$ is constructible.
As we increase $p$ while keeping $t$ fixed, the limit $\Kfunc_t(p)$ changes only when one of the
two minimal objects of $\Pcat \uparrow (p,t)$ changes isomorphism type.
Since $\Ffunc$ and $\Gfunc$ are constructible,
there are only finitely many such changes to the isomorphism type of $\Kfunc_t(p)$.
\end{proof}

\section{Persistence Diagrams}

Fix an abelian group $\Ggroup$ with a translation invariant partial ordering $\preceq$.
That is for all $a, b, c \in \Ggroup$, if $a \leq b$, then $a + c \leq b +c$.
Roughly speaking, a persistence diagram is the assignment to each interval
of the real line an element of $\Ggroup$.
In our setting, only finitely many intervals will have a nonzero value.

\begin{defn}
Let $\Dgm$ be the \define{poset of intervals} consisting of the following data:
	\begin{itemize}
	\item The objects of $\Dgm$ are intervals $[p,q) \subseteq \bar\Rspace$
	where $p \leq q$.
	\item The ordering is inclusion $[p_2,q_2) \subseteq [p_1,q_1)$.
	\end{itemize}
Given a finite set $S=\{s_1< s_2< \cdots <s_k < \infty\}\subseteq\bar{\Rspace}$, we use
$\Dgm(S)$ to denote the subposet of $\Dgm$ consisting of all intervals $[p,q)$
with $p,q \in S$.
The \define{diagonal} $\Delta \subseteq \Dgm$ is the subset of intervals of the form
$[p,p)$.
See Figure \ref{fig:dgm}.
\end{defn}

\begin{figure}
\begin{center}
\includegraphics[scale = 0.80]{Dgm.pdf}
\end{center}
\caption{An interval $I = [p,q)$ is visualized as the point $(p,q)$ in the plane.
The poset $\Dgm$ is therefore the set of points in the plane
on and above the diagonal.
In this example, $S  = \{ s_1 < s_2 < s_3 < s_4 < \infty\}$ and
$\Dgm(S)$ is its set of grid points.
Given an $S$-constructible persistence module $\Ffunc$ and an interval $[s_2, s_3)$,
$\tilde{\Ffunc} \big( [s_2, s_3) \big) = d \Ffunc \big( [s_2, s_3) \big) - d \Ffunc \big( [s_2, s_4) \big) + d \Ffunc \big( [s_1, s_4) \big) - d \Ffunc \big( [s_1, s_3) \big)$.}
\label{fig:dgm}
\end{figure}

\begin{defn}
A \define{persistence diagram} is a map
$Y : \Dgm \to \Ggroup$ with finite support.
That is, there are only finitely many intervals $I \in \Dgm$
such that $Y(I) \neq 0$.
\end{defn}

We now introduce the bottleneck distance between persistence diagrams.

\begin{defn}
A \define{matching} between two persistence diagrams $Y_1, Y_2 : \Dgm \to \Ggroup$ is a
map $\gamma: \Dgm \times \Dgm \to \Ggroup$
satisfying
	\begin{align*}
	Y_1(I) & = \sum_{J \in \Dgm} \gamma(I,J) \textrm{ for all }I \in \Dgm \backslash \Delta \\
	Y_2(J) & = \sum_{I \in \Dgm} \gamma(I,J) \textrm{ for all } J \in \Dgm \backslash \Delta.
	\end{align*}
The \define{norm} of a matching $\gamma$ is
$$ ||\gamma|| := \max_{\big \{I=[p_1,q_1),J=[p_2,q_2) \,\big |\, \gamma(I,J)\neq 0 \big \}}
\big \{|p_1-p_2|,|q_1-q_2| \big\}. $$
If either $q_1$ or $q_2$ is $\infty$, then $ | q_1 - q_2 | = \infty$.
The \define{bottleneck distance} between two persistence diagrams $Y_1, Y_2 : \Dgm \to \Ggroup$ is
$$d_B(Y_1,Y_2) := \inf_\gamma ||\gamma||$$
over all matchings $\gamma$ between $Y_1$ and $Y_2$.
This infimum is attained since persistence diagrams have finite support.
\end{defn}

\section{Diagram of a Module}

We now describe the construction of a persistence diagram
from a constructible persistence module.
Fix a skeletally small abelian category $\Ccat$.

\begin{defn}
The \define{Grothendieck group} $\Ggroup(\mathcal{C})$ of $\mathcal{C}$ is the abelian group
with one generator for each
isomorphism class $[a]$ of objects $a \in \mathsf{ob}\; \mathcal{C}$ and one relation
$[b] = [a] + [c]$
for each short exact sequence
$0 \to a \to b \to c \to 0.$
The Grothendieck group has a natural translation invariant partial ordering where
$[a] \preceq [b]$ whenever $a\hookrightarrow b$.
For each $a\hookrightarrow b$, we have
$a \oplus c \hookrightarrow b \oplus c$ for any object $c$ in $\mathcal{C}$.
This makes $\preceq$ a translation invariant partial ordering.
\end{defn}

\begin{ex}
Here are three examples of $\mathcal{C}$ with their Grothendieck groups.
	\begin{itemize}
	\item Let $\Vect$ be the category of finite dimensional $\field$-vector spaces
	for some fixed field~$\field$.
	The isomorphism class of a finite dimensional $\field$-vector space is completely determined
	by its dimension.
	This means that the free abelian group generated by the set of isomorphism classes in
	$\Vect$ is $\bigoplus_{n} \Zspace$ where $n \geq 0$ is a natural number.
	Since every short exact sequence in $\Vect$ splits, the only relations are of the form $[A] + [B] = [C]$ whenever $A \oplus B \cong C$.
	Therefore $\Ggroup (\Vect) \cong \Zspace$ where the translation invariant partial
	ordering $\preceq$ is the usual total ordering on the integers.

	\item Let $\Finab$ be the category of finite abelian groups.
	A finite abelian group is isomorphic to
		$$\dfrac{\Zspace}{p_1^{n_1} \Zspace} \oplus \cdots \oplus
		 			\dfrac{\Zspace}{p_k^{n_k} \Zspace}$$
	where each $p_i$ is prime.
	The free abelian group generated by the set of isomorphism classes in $\Finab$
	is $\bigoplus_{(p,n)} \Zspace$
	over all pairs $(p, n)$ where $p$ is prime and $n \geq 0$ a natural number.
	Every primary cyclic group $\dfrac{\Zspace}{p^n \Zspace}$ fits into a short exact sequence
		\begin{equation*}
		\begin{tikzcd}
		0 \arrow[r] & \dfrac{\Zspace}{p^{n-1} \Zspace} \arrow[r, "\times p"] &
		\dfrac{\Zspace}{p^{n} \Zspace} \arrow[r, "/"] &
		\dfrac{\Zspace}{p\Zspace} \arrow[r] & 0
		\end{tikzcd}
		\end{equation*}
	giving rise to a relation
	$\left[ \dfrac{\Zspace}{p^{n} \Zspace} \right] =
	\left[ \dfrac{\Zspace}{p^{n-1} \Zspace} \right] + \left[ \dfrac{\Zspace}{p \Zspace} \right].$
	By induction, $\left[ \dfrac{\Zspace}{p^{n} \Zspace} \right] = n \left[ \dfrac{\Zspace}{p \Zspace} \right]$.
	Therefore $\Ggroup( \Finab ) \cong \bigoplus_p \Zspace$ where $p$ is prime.
	For two elements $[a], [b] \in \Ggroup( \Finab )$, $[a] \preceq [b]$
	if the multiplicity of each prime factor of $[a]$ is at most the multiplicity of each
	prime factor of $[b]$.

	\item
	Let $\Ab$ be the category of finitely generated abelian groups.
	A finitely generated abelian group is isomorphic to
		$$ \Zspace^m \oplus \dfrac{\Zspace}{p_1^{n_1} \Zspace} \oplus \cdots \oplus
		 			\dfrac{\Zspace}{p_k^{n_k} \Zspace}$$
	where each $p_i$ is prime.
	The free abelian group generated by the set of isomorphism classes in $\Ab$
	is $\Zspace \oplus \bigoplus_{(p,n)} \Zspace$
	over all pairs $(p, n)$ where $p$ is prime and $n \geq 0$ a natural number.
	In addition to the short exact sequences in $\Finab$, we have
		\begin{equation*}
		\begin{tikzcd}
		0 \arrow[r] & \Zspace \arrow[r, "\times p"] &
		\Zspace \arrow[r, "/"] &
		\dfrac{\Zspace}{p\Zspace} \arrow[r] & 0
		\end{tikzcd}
		\end{equation*}
	giving rise to the relation $\left[ \dfrac{\Zspace}{p \Zspace} \right] = [0]$.
	Therefore $\Ggroup( \Ab) \cong \Zspace$ where $\preceq$ is the usual total ordering on the integers.
	Unfortunately all torsion is lost.
	\end{itemize}
\end{ex}

Given a constructible persistence module, we now record the images of all its maps
as elements of the Grothendieck group.

\begin{defn}
Let $S = \{s_1 < \cdots < s_k < \infty\}$ be a finite set and $\Ffunc$ an
$S$-constructible persistence module valued in $\mathcal{C}$.
Choose a $\delta > 0$ such that $s_{i-1} < s_i - \delta$, for all $1< i \leq k$.
The \define{rank function} of $\Ffunc$ is the map
$d \Ffunc : \Dgm \to \Ggroup( \mathcal{C} )$ defined as follows:
$$d \Ffunc (I) =
\begin{cases}
\big[ \image \, \Ffunc(p<s_i-\delta) \big]	& \text{for } I = [p,s_i) \text{ where } 1 \leq i \leq k \\
\big[ \image \, \Ffunc(p \leq q) \big]	& \text{for all other } I=[p,q).
\end{cases}
$$
Note that for any $[p,q) \in \Dgm$, $d \Ffunc \big ( [p,q) \big)$ equals
$d \Ffunc(I)$ where $I$ is the largest interval in $\Dgm(S)$ containing $[p,q)$.
This means that $d \Ffunc$ is uniquely determined by its value on $\Dgm(S)$.
\end{defn}

\begin{prop}
Let $\Ffunc$ be a constructible persistence module valued in a skeletally small abelian category
$\mathcal{C}$.
Then its rank function $d \Ffunc : \Dgm \to \Ggroup(\mathcal{C})$ is a poset
reversing map.
That is for any pair of intervals
$[p_2, q_2) \subseteq [p_1,q_1)$,
$d \Ffunc \big( [p_1, q_1) \big) \preceq d \Ffunc \big( [p_2,q_2) \big)$.
\end{prop}
\begin{proof}
Suppose $\Ffunc$ is $S = \{s_1 < \cdots < s_k < \infty\}$-constructible.
Consider the following commutative diagram:
	\begin{equation*}
	\xymatrix{
	\Ffunc(p_1) \ar[d]_{h := \Ffunc (p_1 \leq q_1)} \ar[rrr]^{e := \Ffunc(p_1 \leq p_2)}
	&&& \Ffunc(p_2) \ar[d]^{f := \Ffunc(p_2 \leq q_2)} \\
	\Ffunc(q_1) &&& \Ffunc(q_2). \ar[lll]^{g := \Ffunc(q_2 \leq q_1)}
	}
	\end{equation*}
We may assume $q_1, q_2 \notin S$.
If this is not the case, replace $q_1$ and/or $q_2$ in the above diagram with
$q_1 - \delta$ and $q_2 - \delta$ for some sufficiently small $\delta > 0$.
We have $d \Ffunc \big( [p_1, q_1) \big) = [ \image\, h ]$ and
$d \Ffunc \big( [p_2, q_2) \big) =  [ \image\, f ]$.
Let $I := \image\, ( f \circ e )$ and $K := I \cap \ker g$.
Then $K \mono I \mono \image\, f$ and $\image\, h \cong \sfrac{I}{K}$.
Therefore $d \Ffunc \big ( [p_1, q_1) \big) \preceq d \Ffunc \big( [p_2, q_2 ) \big)$.
\end{proof}

Given the rank function $d \Ffunc : \Dgm \to \Ggroup(\mathcal{C})$
of an  $S$-constructible persistence module~$\Ffunc$, there is a unique
map $\tilde \Ffunc : \Dgm \to \Ggroup$ such that
	\begin{equation}
	\label{eq:mobius}
	d \Ffunc (I) =  \sum_{J \in \Dgm: J \supseteq I } \tilde \Ffunc (J)
	\end{equation}
for each $I \in \Dgm$.
This equation is the \emph{M\"obius inversion formula}.
For each $I = [s_i, s_j)$ in $\Dgm(S)$,

	\begin{equation}
	\label{eq:inversion}
	\tilde \Ffunc (I) = d \Ffunc  \big( [s_i, s_j) \big) - d \Ffunc \big( [s_{i}, s_{j+1}) \big)
	+ d \Ffunc \big( [s_{i-1}, s_{j+1}) \big) - d \Ffunc \big( [s_{i-1}, s_j) \big).
	\end{equation}
For each $I = [s_i, \infty)$ in $\Dgm(S)$,
	\begin{equation}
	\label{eq:inversion_infty}
	\tilde \Ffunc (I) = d \Ffunc  \big( [s_i, \infty) \big) - d \Ffunc \big( [s_{i-1}, \infty) \big).
	\end{equation}
For all other $I \in \Dgm$, $\tilde \Ffunc(I) = 0$.
Here we have to be careful with our indices.
It is possible $s_{j+1}$ or $s_{i-1}$ is not in $S$.
If $s_{j+1}$ is not in $S$, let $s_{j+1} = \infty$.
If $s_{i-1}$ is not in $S$, let $s_{i-1}$ be any value strictly less than $s_1$.
We call $\tilde \Ffunc$ the \emph{M\"obius inversion} of $d \Ffunc$.

\begin{defn}
\label{defn:diagram}
The \define{persistence diagram of a constructible persistence module} $\Ffunc$
is the M\"obius inversion $\tilde \Ffunc$ of its rank function $d \Ffunc$.
\end{defn}

The Grothendieck group of $\Ccat$ has one relation for each short exact sequence in $\Ccat$.
These relations ensure that the persistence diagram of a persistence module is positive which
plays a key role in the proof of Lemma \ref{lem:box}.

\begin{prop}[Positivity \cite{patel}]
\label{prop:positivity}
Let $\Ffunc$ be a constructible persistence module
valued in a skeletally small abelian category $\Ccat$.
Then for any $I\in \Dgm$, we have~$[0] \preceq \tilde \Ffunc(I)$.
\end{prop}

\section{Stability}

We now begin the task of proving bottleneck stability.
Throughout this section, persistence modules are valued in a fixed skeletally small abelian
category $\Ccat$.

\begin{defn}
For an interval $I = [p,q)$ in $\Dgm$ and a value $\ee \geq 0$, let
$$ \square_{\ee} I :=  \big\{ [r,s) \in \Dgm \, \big |\, p-\ee < r \leq p+\ee \text{ and } q-\ee \leq s < q+ \ee \big \} $$
be the subposet of $\Dgm$ consisting of intervals $\ee$-close to $I$.
If $I$ is too close to the diagonal, that is if $q - \ee \leq p+\ee$, then we let
$\square_\ee I$ be empty.
We call $\square_\ee I$ the $\ee$-\emph{box} around $I$.
See Figure \ref{fig:integration}.
Note that if $q = \infty$, then $\square_\ee I = \big \{ [r, \infty) \; \big | \;  p-\ee < r \leq p+ \ee \big \}$.
\end{defn}

	\begin{figure}
	\begin{center}
	\includegraphics[scale = 0.80]{Dgm_box.pdf}
	\end{center}
	\caption{The shaded area is a box $\square_\ee I$.
	Note that $\square_\ee I$ is closed on the bottom and right, and it is open on the top and left.
	}
	\label{fig:integration}
	\end{figure}

\begin{lem}
\label{lem:corners}
Let $\Ffunc$ be an $S$-constructible persistence module, $I = [p,q)$, and $\ee > 0$.
If $\square_\ee I$ is nonempty, then
	\begin{equation*}
	\label{eq:sum}
	\sum_{J \in \square_\ee I} \tilde{F}(J) = d \Ffunc \big( [p+\ee, q-\ee) \big) -
	d \Ffunc \big( [p+\ee, q+\ee) \big)
	+ d \Ffunc \big( [p-\ee, q+\ee) \big) - d \Ffunc \big( [p-\ee, q-\ee) \big)
	\end{equation*}
whenever $q \neq \infty$ and
	\begin{equation*}
	\label{eq:sum}
	\sum_{J \in \square_\ee I} \tilde{F}(J) = d \Ffunc \big( [p+\ee, \infty) \big)
	- d \Ffunc \big( [p-\ee, \infty) \big)
	\end{equation*}
whenever $q = \infty$.
\end{lem}

\begin{proof}
Both equalities follow easily from the M\"obius inversion formula; see Equation~\ref{eq:mobius}.
If $q \neq \infty$, then
	\begin{align*}
	\sum_{J \in \square_\ee I} \tilde \Ffunc(J) &= \sum_{\substack{J \in \Dgm: \\ J \supseteq [p+\ee,q-\ee)}}
	\tilde \Ffunc(J)
	- \sum_{\substack{J \in \Dgm: \\ J \supseteq [p+\ee, q+\ee)}} \tilde \Ffunc(J) +
	\sum_{\substack{J \in \Dgm: \\ J \supseteq [p-\ee, q+\ee)}} \tilde \Ffunc(J) -
	\sum_{\substack{J \in \Dgm: \\ J \supseteq [p-\ee, q-\ee)} } \tilde \Ffunc(J) \\
	& = d \Ffunc \big( [p+\ee,q-\ee) \big) - d \Ffunc \big( [p+\ee,q+\ee) \big) -
	d \Ffunc \big( [p-\ee,q+\ee) \big) + d \Ffunc \big( [p-\ee,q-\ee) \big).
	\end{align*}
If $q = \infty$, then
	\begin{align*}
	\sum_{J \in \square_\ee I} \tilde \Ffunc(J) &= \sum_{\substack{J \in \Dgm: \\ J
	\supseteq [p+\ee,\infty)}} \tilde \Ffunc(J)
	- \sum_{\substack{J \in \Dgm: \\ J \supseteq [p-\ee, \infty)}} \tilde \Ffunc(J) \\
	& = d \Ffunc \big( [p+\ee,\infty) \big) - d \Ffunc \big( [p-\ee,\infty) \big).
	\end{align*}
\end{proof}

\begin{lem}[Box Lemma]
\label{lem:box}
Let $\Ffunc$ and $\Gfunc$ be two $\ee$-interleaved constructible persistence modules, $I \in \Dgm$,
and $\mu > 0$.
Then
$$\sum_{J \in \square_{\mu} I} \tilde \Ffunc(J) \preceq \sum_{J \in \square_{\mu + \ee} I} \tilde \Gfunc(J)$$
whenever $\square_{\mu + \ee} I$ is nonempty.
\end{lem}

\begin{proof}
Suppose $\Ffunc$ and $\Gfunc$ are
$\ee$-interleaved by $\Phi$ in Diagram \ref{dgm:interleaving}.
Define $\varphi_{r}: \Ffunc(r) \to \Gfunc(r+\ee)$ as
$\Phi \big( (r,0) \leq (r+\ee,1) \big)$ and define
$\psi_{r}: \Gfunc (r) \to \Ffunc (r+\ee)$ as $\Phi \big( (r,1) \leq (r+\ee,0) \big)$.

Suppose $I = [p,q)$ where $q \neq \infty$. By Lemma \ref{lem:corners},
\begin{align*}
\sum_{J \in \square_{\mu} I } \tilde{F}(J) =\; &
	d \Ffunc \big( [p+\mu, q-\mu) \big) - d \Ffunc \big( [p+\mu, q+\mu) \big) \\
	&+ d \Ffunc \big( [p-\mu, q+\mu) \big) - d \Ffunc \big( [p-\mu, q-\mu) \big) \\
\sum_{J \in \square_{\mu+\ee} I} \tilde{G}(J) =\; &
	d \Gfunc \big( [p+\mu+\ee, q-\mu-\ee) \big) - d \Gfunc \big( [p+\mu+\ee, q+\mu+\ee) \big) \\
	& + d \Gfunc \big( [p-\mu-\ee, q+\mu+\ee) \big) - d \Gfunc \big( [p-\mu-\ee, q-\mu-\ee) \big).
\end{align*}
Choose a sufficiently small $\delta > 0$ so that we have the following equalities:
	\begin{align*}
d \Ffunc \big( [p+\mu, q-\mu) \big) &= \big[ \image\; \Ffunc ( p + \mu < q-\mu-\delta) \big] \\
d \Ffunc \big( [p+\mu, q+\mu) \big) &= \big[ \image\; \Ffunc ( p+\mu < q+\mu-\delta) \big] \\
d \Ffunc \big( [p-\mu, q+\mu) \big) &= \big[ \image\; \Ffunc ( p-\mu < q+\mu-\delta) \big] \\
d \Ffunc \big( [p-\mu, q-\mu) \big) &= \big[ \image\; \Ffunc ( p-\mu < q-\mu-\delta) \big] \\
d \Gfunc \big( [p+\mu+\ee, q-\mu-\ee) \big) &= \big[ \image\; \Gfunc ( p+\mu+\ee < q-\mu-\ee-\delta) \big] \\
d \Gfunc \big( [p+\mu+\ee, q+\mu+\ee) \big) &= \big[ \image\; \Gfunc ( p+\mu+\ee < q+\mu+\ee-\delta) \big]\\
d \Gfunc \big( [p-\mu-\ee, q+\mu+\ee) \big) &= \big[ \image\; \Gfunc ( p-\mu-\ee < q+\mu+\ee-\delta) \big] \\
d \Gfunc \big( [p-\mu-\ee, q-\mu-\ee) \big) &= \big[ \image\; \Ffunc ( p-\mu-\ee < q-\mu-\ee-\delta) \big].
	\end{align*}
Consider the following commutative diagram where the horizontal and vertical arrows are the appropriate
morphisms from $\Ffunc$ and $\Gfunc$:
\begin{center}
\begin{tikzcd}
\Gfunc(p-\mu-\ee) \ar[rrrr] \ar[dddd] \ar[dr,"\psi_{p-\mu -\ee}"]
&&&& G(p+\mu+\ee) \ar[dddd] \\
& \Ffunc (p-\mu) \ar[rr] \ar[dd]&& \Ffunc(p+\mu) \ar[dd] \ar[ur,"\varphi_{p+\mu}"]\\
\\
  &\Ffunc(q+\mu-\delta) \ar[dl,"\varphi_{q+\mu-\delta}"]&& \Ffunc(q-\mu-\delta) \ar[ll] \\
\Gfunc(q+\mu+\ee-\delta)	&&&& \Gfunc(q-\mu-\ee-\delta). \ar[llll] \ar[ul,"\psi_{q-\mu-\ee-\delta}"]
\end{tikzcd}
\end{center}
Choose two values $a < b$ such that $a+ \mu+\ee < b-\mu-\ee$ and let
$$T := \{ a-\mu-\ee < a-\mu < a+\mu < a+\mu+\ee < c < b-\mu-\ee < b-\mu < b+\mu < \infty \}
\subseteq \bar \Rspace.$$
Let $\Hfunc: \Rspace \to \mathcal{C}$ be the $T$-constructible persistence module determined by the following diagram:
\begin{equation*}
\xymatrix{
\Hfunc(a-\mu-\ee) = \Gfunc(p-\mu-\ee) \ar[r] & \Hfunc(a-\mu) = \Ffunc(p-\mu) \ar[r] &
\Hfunc(a+\mu) = \Ffunc(p+\mu) \ar[d] \\
\Hfunc(b-\mu-\ee) = \Ffunc(q-\mu-\delta) \ar[d] & \Hfunc(c) = \Gfunc (q-\mu-\ee-\delta) \ar[l] &
\Hfunc(a+\mu+\ee) = \Gfunc(p+\mu+\ee) \ar[l] \\
\Hfunc(b-\mu) = \Ffunc(q+\mu-\delta) \ar[r] & \Hfunc(b+\mu) = \Gfunc(q+\mu+\ee-\delta).
}
\end{equation*}
Here the value of $\Hfunc$ is given on each value in $T$ and morphisms between adjacent objects are the connecting
morphisms in the above commutative diagram.
For example, for all $a+\mu+\ee \leq r < c$, $\Hfunc(r) = \Gfunc(p+\mu+\ee)$ and
$\Hfunc(a+\ee+\mu \leq r) = \id$.
The morphism $\Hfunc(c \leq b-\mu-\ee)$ is $\psi_{q-\mu-\ee-\delta}$.
We have the following equalities:
\begin{align*}
\big[ \image\; \Ffunc ( p + \mu < q-\mu-\delta) \big] &= d \Hfunc \big( [a+\mu, b-\mu) \big) \\
\big[ \image\; \Ffunc ( p + \mu < q+\mu-\delta) \big] &= d \Hfunc \big( [a+\mu, b+\mu) \big)  \\
\big[ \image\; \Ffunc ( p - \mu < q+\mu-\delta) \big] &= d \Hfunc \big( [a-\mu, b+\mu) \big) \\
\big[ \image\; \Ffunc ( p - \mu < q-\mu-\delta) \big] &= d \Hfunc \big( [a-\mu, b-\mu) \big) \\
\big[ \image\; \Gfunc ( p+\mu+\ee < q-\mu-\ee-\delta) \big] &= d \Hfunc \big( [a+\mu+\ee, b-\mu-\ee) \big) \\
\big[ \image\; \Gfunc ( p+\mu+\ee < q+\mu+\ee-\delta) \big] &=
	d \Hfunc \big( [a+\mu+\ee, b+\mu+\ee) \big) \\
\big[ \image\; \Gfunc ( p-\mu-\ee < q+\mu+\ee-\delta) \big] &= d \Hfunc \big( [a-\mu-\ee, b+\mu+\ee) \big) \\
\big[ \image\; \Gfunc ( p-\mu-\ee < q-\mu-\ee-\delta) \big] &= d \Hfunc \big( [a-\mu-\ee, b-\mu-\ee) \big).
\end{align*}
By Lemma \ref{lem:corners} along with the above substitutions, we have
$$\sum_{J \in \square_{\mu}[a,b)}  \tilde \Hfunc (J) = \sum_{J \in \square_{\mu} I} \tilde{\Ffunc}(J)$$
$$\sum_{J \in \square_{\mu+\ee}[a,b)} \tilde \Hfunc (J) = \sum_{J \in \square_{\mu+\ee} I } \tilde{G}(J).$$
By the inclusion $\square_{\mu} [a,b) \subseteq \square_{\mu + \ee} [a,b) $ along with Proposition \ref{prop:positivity}, we have
$$\sum_{J \in \square_{\mu} [a,b)} \tilde \Hfunc (J) \preceq
\sum_{J \in \square_{\mu+\ee} [a,b)} \tilde \Hfunc (J).$$
This proves the statement.

Suppose $I = [p,\infty)$ and let $z \in \Rspace$ be larger than $\mu + \ee$ and $s_k$ where $\Ffunc$ and $\Gfunc$ are $\{ s_1 < \cdots < s_k < \infty \}$-constructible.
By Lemma \ref{lem:corners},
\begin{align*}
\sum_{J \in \square_{\mu} I } \tilde{F}(J) =\; &
	d \Ffunc \big( [p+\mu, \infty) \big) - d \Ffunc \big( [p-\mu, \infty) \big) \\
\sum_{J \in \square_{\mu+\ee} I} \tilde{G}(J) =\; &
	d \Gfunc \big( [p+\mu+\ee, \infty) \big) - d \Gfunc \big( [p-\mu-\ee, \infty) \big).
\end{align*}
We have the following equalities:
	\begin{align*}
d \Ffunc \big( [p+\mu, \infty) \big) &= \big[ \image\; \Ffunc ( p+\mu < \infty) \big] \\
d \Ffunc \big( [p-\mu, \infty) \big) &= \big[ \image\; \Ffunc ( p-\mu < \infty) \big] \\
d \Gfunc \big( [p+\mu+\ee, \infty) \big) &= \big[ \image\; \Gfunc ( p+\mu+\ee < \infty) \big] \\
d \Gfunc \big( [p-\mu-\ee, \infty) \big) &= \big[ \image\; \Gfunc ( p-\mu-\ee < \infty) \big].
	\end{align*}
Consider the following commutative diagram where the vertical and horizontal arrows are the appropriate
morphisms from $\Ffunc$ and $\Gfunc$:
\begin{center}
\begin{tikzcd}
\Gfunc(p-\mu-\ee) \ar[rrrr] \ar[dddd]  \ar[dr,"\psi_{p-\mu-\ee}"] &&&& \Gfunc(p+\mu+\ee) \ar[dddd] \\
& \Ffunc (p-\mu) \ar[rr] \ar[dd] && \Ffunc(p+\mu) \ar[dd] \ar[ur,"\varphi_{p+\mu}"] & \\
&&&& \\
& \Ffunc(\infty) \ar[dl,"\varphi_{\infty}"]&& \Ffunc(\infty) \ar[ll]  & \\
\Gfunc(\infty) &&&& \Gfunc(\infty). \ar[llll]  \ar[ul,"\psi_{\infty}"]
\end{tikzcd}
\end{center}
Note that $\psi_\infty$ and $\varphi_\infty$ are isomorphisms.
Let
$$T := \{ -\mu-\ee < -\mu < \mu < \mu+\ee < \infty \}
\subseteq \bar \Rspace.$$
Let $\Hfunc: \Rspace \to \mathcal{C}$ be the $T$-constructible persistence module determined by the following diagram:
\begin{equation*}
\xymatrix{
\Hfunc(-\mu-\ee) = \Gfunc(p-\mu-\ee) \ar[r] & \Hfunc(-\mu) = \Ffunc(p-\mu) \ar[r] &
\Hfunc(\mu) = \Ffunc(p+\mu) \ar[d] \\
& \Hfunc(z) = \Ffunc (z)  &  \Hfunc(\mu+\ee) = \Gfunc(p+\mu+\ee). \ar[l]
}
\end{equation*}
Here the value of $\Hfunc$ is given on each value in $T$ and morphisms between adjacent objects are the connecting
morphisms in the above commutative diagram.
We have the following equalities:
\begin{align*}
\big[ \image\; \Ffunc ( p + \mu <\infty) \big] &= d \Hfunc \big( [\mu, \infty) \big) \\
\big[ \image\; \Ffunc ( p - \mu < \infty) \big] &= d \Hfunc \big( [-\mu, \infty) \big) \\
\big[ \image\; \Gfunc ( p+\mu+\ee < \infty) \big] &= d \Hfunc \big( [\mu+\ee, \infty) \big) \\
\big[ \image\; \Gfunc ( p-\mu-\ee < \infty) \big] &= d \Hfunc \big( [-\mu-\ee, \infty) \big).
\end{align*}
By Lemma \ref{lem:corners} along with the above substitutions, we have
$$\sum_{J \in \square_{\mu}[0,\infty)}  \tilde \Hfunc (J) = \sum_{J \in \square_{\mu} I} \tilde{\Ffunc}(J)$$
$$\sum_{J \in \square_{\mu+\ee}[0,\infty)} \tilde \Hfunc (J) = \sum_{J \in \square_{\mu+\ee} I } \tilde{G}(J).$$
By the inclusion $\square_{\mu} [0,\infty) \subseteq \square_{\mu+\ee} [0,\infty) $ along with
Proposition \ref{prop:positivity}, we have
$$\sum_{J \in \square_{\mu} [0,\infty)} \tilde \Hfunc (J) \preceq
\sum_{J \in \square_{\mu+\ee} [0,\infty)} \tilde \Hfunc (J).$$
This proves the statement.

\end{proof}

\begin{defn}
The \define{injectivity radius}
of a finite set $S = \{ s_1 < s_2 < \cdots < s_k < \infty\}$ is
$$\rho := \min_{1 < i \leq k} \frac{s_{i}-s_{i-1}}{2}.$$
Note that if a persistence module $\Ffunc$ is $S$-constructible and $I \in \Dgm(S)$, then
	$$\tilde \Ffunc (I) = \sum_{J \in \square_\rho I} \tilde \Ffunc (J).$$
\end{defn}

\begin{lem}[Easy Bijection]
\label{lem:bijection}
Let $\Ffunc$ be an $S$-constructible persistence module and $\rho > 0$ the injectivity
radius of $S$.
If $\Gfunc$ is a second constructible persistence module such that $d_I(\Ffunc,\Gfunc) < \rho/2$, then
$d_B(\tilde \Ffunc, \tilde \Gfunc) \leq d_I(\Ffunc,\Gfunc)$.
\end{lem}

\begin{proof}
Let $\ee = d_I (\Ffunc, \Gfunc)$.
Choose a sufficiently small $\mu > 0$ such that $\mu + 2 \ee < \rho$.
We construct a matching $\gamma_\mu : \Dgm \times \Dgm \to \Ggroup(\Ccat)$ such that
	\begin{align}
	\tilde \Ffunc(I) & = \sum_{J \in \Dgm} \gamma_\mu(I,J) \textrm{ for all }I \in \Dgm \backslash \Delta
	\label{eq:forward} \\
	\tilde \Gfunc(J) & = \sum_{I \in \Dgm} \gamma_\mu(I,J) \textrm{ for all } J \in \Dgm \backslash \Delta
	\label{eq:backward} .
	\end{align}

Fix an  $I \in \Dgm(S)$.
By Lemma \ref{lem:box},
$$\tilde \Ffunc(I) = \sum_{J \in \square_{\mu} I} \tilde \Ffunc (I) \leq \sum_{J \in \square_{\mu + \ee} I} \tilde \Gfunc(J) \leq \sum_{J \in \square_{\mu + 2\ee} I}\tilde \Ffunc(J) = \tilde \Ffunc(I).$$
Let $\gamma_\mu(I,J) := \tilde \Gfunc(J)$ for all $J \in \square_{\mu + \ee}(I)$.
Repeat for all $I \in \Dgm(S)$.
Equation \ref{eq:forward} is satisfied.
We now check that $\gamma_\mu$ satisfies Equation \ref{eq:backward}.
Fix an interval $J = [p,q)$ and suppose $\tilde \Gfunc (J) \neq 0$.
If $q-p > \ee$, then by Lemma \ref{lem:box}
	$$\tilde \Gfunc(J) = \sum_{I \in \square_{\mu} J} \tilde \Gfunc (I) \preceq \sum_{I \in \square_{\mu+\ee} J}
	\tilde \Ffunc(J).$$
This means $\gamma_\mu(I, J) \neq 0$ for some $I \in \square_{\mu+\ee} J$.
If $q-p \leq \ee$, then we match $J$ to the diagonal.
That is, we let $\gamma_\mu \big ( [\sfrac{q-p}{2}, \sfrac{q-p}{2} ), J \big) := \tilde \Gfunc(J)$.

By construction, $||\gamma_\mu|| \leq \mu + \ee $ for all $\mu > 0$ sufficiently small. Therefore $d_B ( \tilde \Ffunc, \tilde \Gfunc ) \leq \ee = d_I(\tilde \Ffunc, \tilde \Gfunc)$.
\end{proof}

We are now ready to prove our main result.

\begin{thm}[Bottleneck Stability]
Let $\mathcal{C}$ be a skeletally small abelian category and $\Ffunc, \Gfunc:  \Rspace \to \mathcal{C}$
two constructible persistence modules.
Then $d_B \big( \tilde \Ffunc, \tilde \Gfunc \big) \leq d_I (\Ffunc, \Gfunc)$
where $\tilde \Ffunc$ and $\tilde \Gfunc$ are the persistence diagrams of $\Ffunc$ and $\Gfunc$,
respectively.
\end{thm}

\begin{proof}
Let $\ee = d_I(\Ffunc, \Gfunc)$.
By Proposition \ref{prop:interpolation}, there is a one parameter family of constructible persistence modules
$\{\Kfunc_t\}_{t \in [0,1]}$ such that
$d_I(\Kfunc_t, \Kfunc_s) \leq \ee |t-s|$, $\Kfunc_0 \cong \Ffunc$, and $\Kfunc_1 \cong \Gfunc$.
Each $\Kfunc_t$ is constructible with respect to some set $S_t$, and each set $S_t$ has
an injectivity radius $\rho_t > 0$.
For each time $t \in [0,1]$, consider the open interval
$$U(t) = ( t-\sfrac{\rho_t}{4\ee},t+\sfrac{\rho_t}{4\ee} ) \cap [0,1]$$
By compactness of $[0,1]$, there is a finite set
$Q = \{ 0 = t_0 < t_1 < \cdots < t_n = 1 \}$
such that $\cup_{i=0}^n U(t_i) = [0,1]$.
We assume that $Q$ is minimal, that is, there does not exists a pair $t_i, t_j \in Q$ such that
$U(t_i) \subseteq U(t_j)$.
If this is not the case, simply throw away $U(t_i)$ and we still have a covering of $[0,1]$.
As a consequence, for any consecutive pair $t_i < t_{i+1}$, we have $U(t_i) \cap U(t_{i+1}) \neq \emptyset$.
This means
$$t_{i+1} - t_i \leq \frac{1}{4\ee} (\rho_{t_{i+1}}+\rho_{t_i}) \leq \frac{1}{2\ee}
\max\{\rho_{t_{i+1}},\rho_{t_i}\}$$
and therefore $d_I( \Kfunc_{t_{i}}, \Kfunc_{t_{i+1}}) \leq \frac{1}{2} \max \{\rho_{t_{i}},\rho_{t_{i+1}} \}$.
By Lemma \ref{lem:bijection},
$$ d_B \big( \tilde \Kfunc_{t_{i}},\tilde \Kfunc_{t_{i+1}} \big) \leq
d_I \big( \Kfunc_{t_{i}}, \Kfunc_{t_{i+1}}),$$
for all $1\leq i \leq n-1$.
Therefore
$$d_B( \tilde \Ffunc, \tilde \Gfunc \big) \leq
\sum_{i=0}^{n-1} d_B \big( \tilde \Kfunc_{t_{i}},\tilde \Kfunc_{t_{i+1}} \big) \leq
\sum_{i=0}^{n-1} d_I \big( \Kfunc_{t_{i}}, \Kfunc_{t_{i+1}}) \leq \ee.$$
\end{proof}

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