# Discontinuities cause essential spectrum on surfaces

Oliver Butterley  (Oliver Butterley) Department of Mathematics, University of
Rome Tor Vergata – Via della Ricerca Scientifica 1 – 00133 Roma – Italy
<EMAIL_ADDRESS>,  Giovanni Canestrari  (Giovanni Canestrari)
Department of Mathematics, University of Rome Tor Vergata – Via della Ricerca
Scientifica 1 – 00133 Roma – Italy<EMAIL_ADDRESS>and  Roberto
Castorrini  (Roberto Castorrini) Dipartimento di Matematica, Università di
Pisa – Largo B. Pontecorvo 5 – 56127 Pisa – Italy
<EMAIL_ADDRESS>
###### Abstract.

Two-dimensional maps with discontinuities are considered. It is shown that, in
the presence of discontinuities, the essential spectrum of the transfer
operator is large whenever it acts on a Banach space with norm that is
stronger than $L^{\infty}$ or $BV$. Three classes of examples are introduced
and studied. In two dimensions there is complication due to the geometry of
the discontinuities, an issue not present in the one-dimensional case and
which is explored in this work.

##  1\. Introduction

Dynamical systems with some degree of hyperbolicity are typically known to
behave with good statistical properties, e.g., exponential decay of
correlations with respect to a physical measure, linear response, etc. A
particularly strong property for the system is to exhibit a full resonance
spectrum, in the sense that correlations can be described to arbitrary
precision (see, e.g., [14, 7] and references within). However this is not
always possible. Systems with discontinuities often occur in physically
motivated examples, e.g., in billiard flows due to glancing collisions and in
the Lorenz flow due to a shearing behaviour at a singular point. To what
extent can discontinuities show an effect in the statistical properties?

The subject of this present work is to demonstrate that discontinuities really
do affect the statistical properties of a dynamical system, besides causing
countless technical problems in proofs. In one-dimensional systems there is a
rather complete description of how discontinuities cause essential spectrum
[5]. However the restriction on dimension is unfortunate in view of
application to general systems and so here we take the step of extending to
higher dimension. For simplicity we consider the two-dimensional case although
much of the framework appears relevant for arbitrary dimensions. We work with
expanding or partially expanding systems, leaving aside for moment the equally
relevant hyperbolic case.

The question of resonance spectrum is closely related to the one of
determining the spectrum of the transfer operator acting on some Banach space.
Specifically, it is useful to know that the essential spectrum is small since
the spectrum outside of the essential spectrum consists of isolated
eigenvalues of finite multiplicity. In such a way the “true” point spectrum is
revealed and not the possible “fake” part due to the choice of the particular
Banach space. In general the spectrum of an operator depends on the choice of
Banach space but in this context there are limitations if one wants to obtain
information relevant to the statistical properties. For instance, we would
like smooth observables to be dense in the Banach space otherwise we might see
behaviour that doesn’t reflect the true system. On the other hand, the Banach
space can’t be too small because then the spectrum might be missing some
information about the observables of interest. The present question is closely
related to zeta functions, a notion of describing resonances inherent to a
system (see, e.g., [2]). The analogous result in this context would be showing
the impossibility of meromorphically extending the zeta function to the whole
complex plane.

In the case of one-dimensional expanding interval maps it was shown [5] that
there is a dichotomy, either the system is Markov or there exists an
unavoidable disc of essential spectrum with radius dependent on the asymptotic
behaviour of the weight of the chosen transfer operator. In the two-
dimensional case we haven’t been able to conclude such a simple story. It
appears that much of this is due to the fact that in higher dimensions there
are multiple different ways a system can fail to be Markov (see further
discussion in Section 3). This issue is intimately connected with the fact
that discontinuities in two dimensions (and higher) can exhibit subtle and
complex behaviour (see, e.g., [9, 10, 19]). Consequently it is much harder to
work with the discontinuity set compared to the one-dimensional case where the
singularities are isolated points and there is no geometry to consider.
Indeed, it is well established that controlling “complexity” of the
discontinuity is essential for working with such systems (see, e.g., [17] and
[12, Chapter 5] concerning controlling complexity and demonstrating that
growth dominates cutting).

In the one-dimensional case [5] there were some restrictions, due to the
method, on the Banach space for which the result applies. To be precise, the
result held only for Banach spaces which are stronger than $L^{\infty}$. This
restriction is due to the way the technique involves the behaviour in the
dual: loosely speaking, if the space is too large then the dual is too small
to complete the argument. However, these authors believe that this is an
artificial requirement since the claimed result is known to also hold for all
other Banach spaces which they are aware of [5, §2]. In two dimensions the
reference norm is less obvious and the results here are presented for two
separate assumptions, namely $L^{\infty}$ and $BV$. Neither of these spaces is
stronger than the other in dimension two or higher and so the estimates which
follow are, depending on the system being studied, sometimes better in one
case, sometimes in the other.

In this text we present the main result (Theorem 1) and three distinct example
classes of system for which this applies. The first example (Section 6.1) is a
class of piecewise affine expanding maps of the torus where the discontinuity
propagates along lines which, iterate after iterate, change in slope and so
never overlap with each other. The second class of examples (Section 6.2)
consists of expanding maps of the torus which can be written as cocycles. Here
we see that the non-Markovness of the base map leads to a discontinuity of the
two-dimensional system and hence unavoidable essential spectrum. The third
class of examples (Section 6.3) are expanding maps of the torus which are
smooth except for in a very small neighbourhood. In this way we see that a
local discontinuity of the system is sufficient to lead to large essential
spectrum. In the presentation of the main argument we take this opportunity to
isolate the abstract functional analytic result (Proposition 2.2) which is at
the core of this work and which was also present in the one-dimensional case
yet slightly obscured.

Continuing forward, there are the following pertinent questions.

######  Question 1 (dichotomy for the 2D case).

Can a lower bound on the essential spectral radius be proven for any two-
dimensional piecewise expanding map which fails to be Markov? The assumption
used in this work is arguably natural and verifiable. Nonetheless it would be
tidy to either have no additional assumption or to show the existence of
obstacles to such a relaxation of the assumptions (see Section 3).

######  Question 2 (hyperbolic case).

Can these ideas be extended to the hyperbolic case? Specifically, consider the
two-dimensional hyperbolic map associated to Sinai billiards. Can one prove a
lower bound for the essential spectral radius of the transfer operator
associated to the SRB measure for any useful Banach space? Such a study would
need to consider Banach spaces of distributions, specifically the anisotropic
spaces suitable for studying hyperbolic systems.

######  Question 3 (diverse notions of resonances).

Recall that there are several closely related notions:

  * •

Obstacles to the meromorphic extension of the zeta function,

  * •

Precision to which it is possible to describe correlations in terms of
resonances for $\mathcal{C}^{\infty}$ observables,

  * •

Lower bound on essential spectral radius of the transfer operator for any
useable Banach space.

Is it possible to formalize completely the connection between these notions?

######  Question 4 (arbitrary dimension).

Can these ideas be extended to arbitrary dimension for expanding systems? The
structure of this present argument suggests that such would be possible in a
similar way but using codimension one linear functionals.

Section 2 contains the key definitions and our results. Section 3 is devoted
to discussion of the assumptions on the discontinuities. The proof of an
abstract functional analytic result is contained in Section 4. This is a
result which relates the existence of a family of linear functionals to a
lower estimate of the essential spectral radius and so, in Section 5 we
construct the relevant linear functionals. Section 6 contains a description
and the results related to three example classes of systems.

## Acknowledgements

We are grateful to Péter Bálint, Bernardo Carvalho, Daniele Galli, Marco
Lenci, Carlangelo Liverani and Daniel Smania for several enlightening
discussions. The authors are partially supported by PRIN Grant “Regular and
stochastic behaviour in dynamical systems” (PRIN 2017S35EHN). O.B. and G.C.
are partially supported by the MIUR Excellence Department Project MatMod@TOV
awarded to the Department of Mathematics, University of Rome Tor Vergata.

##  2\. Settings and results

Let $\mathcal{M}$ be a compact, smooth and connected Riemannian manifold. We
suppose that $\mathcal{F}:\mathcal{M}\to\mathcal{M}$ is a $\mathcal{C}^{r}$
_piecewise non-singular map_ , $r\geq 1$, in the sense that $\mathcal{F}$ is a
$\mathcal{C}^{r}$ local diffeomorphism on $\mathcal{M}\setminus\Gamma$ where
$\Gamma$ is a finite union of finite length $\mathcal{C}^{1}$ curves and the
derivative map is uniformly non-singular.111We write _uniformly non-singular_
to mean that the determinant of the derivative map (in some coordinate atlas)
is uniformly bounded away from zero. For convenience we suppose that
$\mathcal{F}$ is orientation preserving.

Let $\varphi:\mathcal{M}\rightarrow\mathbb{C}$ be $\mathcal{C}^{1}$ on
$\mathcal{M}\setminus\Gamma$, uniformly bounded and uniformly bounded away
from zero. We call $\varphi$ the _weight_ of the transfer operator. The
transfer operator is defined, pointwise for any
$h:\mathcal{M}\rightarrow\mathbb{C}$, as

(2.1) $\mathcal{L}h(y)=\sum_{\mathcal{F}x=y}(h\cdot\varphi)(x).$

At the very least we would like to consider the set of observables,
$\mathcal{C}^{\infty}(\mathcal{M})$. However $\mathcal{L}$ might not leave
$\mathcal{C}^{\infty}(\mathcal{M})$ invariant, particularly in the
discontinuous case of present interest. Let
$\mathfrak{A}=\mathrm{span}\left({\\{\mathcal{L}^{n}(\mathcal{C}^{\infty}(\mathcal{M}))\\}}_{n=0}^{\infty}\right)$.

Keeping track of the discontinuity set will be crucial and so we introduce the
following notation. Let $M$ denote the closure of $\mathcal{M}\setminus\Gamma$
in the sense that $M$ is a smooth Riemannian manifold with boundary and the
boundary comprises of a finite union of $\mathcal{C}^{1}$ curves originating
from $\Gamma$. Call $F:M\to\mathcal{M}$ the unique $\mathcal{C}^{r}$ extension
of $\mathcal{F}$ to $M$. In this way we have a solid way to work with the
limits from one side and the other of $\Gamma$. Let $\iota:M\to\mathcal{M}$
denote the embedding between manifolds and observe that it is 1-to-1 except on
$\partial M$ where222Intersections of curves in $\Gamma$ (at most a finite
number) cause $\iota$ to be many-to-1. it is mostly 2-to-1. Let,

$\Gamma_{1}=F\Gamma,\quad\quad\Gamma_{j+1}=\mathcal{F}^{j}\Gamma_{1},\
j\in\mathbb{N},\quad\quad\widehat{\Gamma}=\bigcup\nolimits_{j\in\mathbb{N}}\Gamma_{j}\subset\mathcal{M}.$

We anticipate here that the transfer operator $\mathcal{L}$ (2.1) involves the
inverse of $\mathcal{F}$ and so the location of discontinuities of the images
of a smooth observable under iterates of $\mathcal{L}$ is $\widehat{\Gamma}$
(see Lemma 5.3).

If $A,B$ are two sets in $M$, we say that they are _equivalent_ , and we write
$A\approxeq B$, if $A$ and $B$ are equal modulo a set of points whose
$1$-dimensional Hausdorff measure is zero. For any curve
$\gamma\subset\mathcal{M}$ and map $\mathcal{G}:\mathcal{M}\to\mathcal{M}$,
let $\operatorname{Jac}({\mathcal{G}|}_{\smash{\gamma}})$ denote the
_Jacobian_ of $\mathcal{G}$ restricted to the curve $\gamma$.333In particular
this means that, if $\mathcal{G}$ restricted to $\gamma$ is measurable and
injective, then, for any $\phi:\gamma\to\mathbb{R}$,
$\int_{\mathcal{G}\gamma}\phi(y)\
dy=\int_{\gamma}\phi(\mathcal{G}x)\operatorname{Jac}({\mathcal{G}|}_{\gamma})(x)\
dx$.

######  Definition 2.1 (proper discontinuity).

Given $\mathcal{F}$ as above, we say that a $\mathcal{C}^{1}$ connected curve
$\gamma\subset\partial M$ is a _proper discontinuity_ if, denoting
$\gamma_{1}=F\gamma$, $\gamma_{k+1}=\mathcal{F}^{k}\gamma_{1}$,
$k\in\mathbb{N}$, the following are satisfied:

  1. (A-0)

$\mathcal{F}$ is not continuous at $x$ for all $x\in\iota\gamma$;

  2. (A-1)

There exist functions, $\alpha_{k}:\gamma_{k}\to\mathbb{C}$, $k\in\mathbb{N}$
such that $\alpha_{1}\equiv 1$ and

(2.2)
$\alpha_{k+1}(\mathcal{F}x)={\alpha_{k}(x)}\varphi(x)^{-1}\operatorname{Jac}({\mathcal{F}|}_{\gamma_{k}})(x)^{-1}\quad\text{for
a.e. $x\in\gamma_{k}$;}$

  3. (A-2)

$\gamma_{k}\cap\Gamma_{1}\approxeq\emptyset$ for all $k\geq 2$;

  4. (A-3)

$\mathcal{F}^{-1}\gamma_{k+1}\cap\widehat{\Gamma}\approxeq\gamma_{k}$ for all
$k\in\mathbb{N}$.

Notice that the right hand side of (2.2) is well defined since $\varphi(x)$ is
uniformly bounded away from zero and $\mathcal{F}$ is non-singular. We denote
by $BV(\mathcal{M})$ the usual Banach space of functions of bounded variation
on $\mathcal{M}$. For convenience we use the notation
$\varphi_{n}=\prod_{k=0}^{n-1}\varphi\circ\mathcal{F}^{k}$. Given a map
$\mathcal{F}:\mathcal{M}\to\mathcal{M}$ and transfer operator weight
$\varphi:\mathcal{M}\to\mathbb{C}$ as above, if $\gamma$ is a proper
discontinuity, we consider the following quantities:555In the appropriate
context, as here, $\left|\cdot\right|$ denotes the length of the curve.

(2.3)
$\displaystyle\Lambda_{\operatorname{BV}}\left(\mathcal{F},\varphi,\gamma\right)$
$\displaystyle=\liminf_{k\rightarrow\infty}\left(\inf_{x\in\gamma}\left|\varphi_{k}(x)\cdot\operatorname{Jac}({\mathcal{F}^{k}|}_{\gamma})(x)\right|\right)^{\frac{1}{k}},$
$\displaystyle\Lambda_{L^{\infty},1}\left(\mathcal{F},\varphi,\gamma\right)$
$\displaystyle=\liminf_{k\rightarrow\infty}\left(\inf_{x\in\gamma}\left|\varphi_{k}(x)\right|\right)^{\frac{1}{k}},$
$\displaystyle\Lambda_{L^{\infty},2}\left(\mathcal{F},\varphi,\gamma\right)$
$\displaystyle=\liminf_{k\rightarrow\infty}\left(\inf_{x\in\gamma}\left|\varphi_{k}(x)\cdot\operatorname{Jac}({\mathcal{F}^{k}|}_{\gamma})(x)\right|\left|\gamma_{k+1}\right|^{-1}\right)^{\frac{1}{k}},$
$\displaystyle\Lambda_{L^{\infty}}$
$\displaystyle=\max\left(\Lambda_{L^{\infty},1},\Lambda_{L^{\infty},2}\right).$

In different settings one or the other will provide a better bound (see Remark
2.7).

Our main result is the following.

######  Theorem 1.

Let $\mathcal{L}$ be the transfer operator associated to a piecewise non-
singular map $\mathcal{F}:\mathcal{M}\to\mathcal{M}$ and weight $\varphi$ as
above. Suppose that $\mathcal{F}$ admits a proper discontinuity $\gamma$ and
that the Banach space $(\mathfrak{B},{{\left\|\cdot\right\|}}_{\mathfrak{B}})$
satisfies

  * •

$\mathcal{C}^{\infty}(\mathcal{M})\subset\mathfrak{B}$;

  * •

$\mathcal{L}$ extends to a continuous operator on $\mathfrak{B}$.

Then the essential spectral radius of
$\mathcal{L}:\mathfrak{B}\to\mathfrak{B}$ is at least

(2.4) $\displaystyle\Lambda_{BV}(\mathcal{F},\varphi,\gamma)$
$\displaystyle\text{if ${\|h\|}_{BV}\leq{\|h\|}_{\mathfrak{B}}$, $\forall
h\in\mathfrak{B}$},$
$\displaystyle\Lambda_{L^{\infty}}(\mathcal{F},\varphi,\gamma)$
$\displaystyle\text{if ${\|h\|}_{L^{\infty}}\leq{\|h\|}_{\mathfrak{B}}$,
$\forall h\in\mathfrak{B}$}.$

The above theorem follows from the following two propositions. The first is an
abstract functional analytic result which uses the existence of a family of
linear functionals to obtain a lower bound of the essential spectral radius
for a linear operator.

######  Proposition 2.2.

Let $L$ be a bounded linear operator on a Banach space
$(\mathfrak{B},{\left\|\cdot\right\|})$. Assume that there exist
$h_{0}\in\mathfrak{B}$, $\Lambda>0$ and a sequence of linear functionals
${\\{\ell_{k}\\}}_{k=1}^{\infty}$, satisfying the following:666We use the
standard notation for the dual: if $h\in\mathfrak{B}$ and
$\ell\in\mathfrak{B}^{*}$ is an element of the dual, i.e., a linear
functional, the dual operator is $\left(L^{*}\ell\right)(h)=\ell(Lh)$ and the
dual norm is
${\left\|\ell\right\|}=\sup\left\\{\left|\ell(h)\right|:h\in\mathfrak{B},{\left\|h\right\|}\leq
1\right\\}$.

  1. (B-1)

$\limsup_{k\to\infty}{\left\|\ell_{k}\right\|}^{\frac{1}{k}}\leq\Lambda^{-1}$;

  2. (B-2)

$\ell_{1}(h_{0})=1$ and, for all $k\in\mathbb{N}$, $\ell_{k+1}(h_{0})=0$;

  3. (B-3)

For all $k\in\mathbb{N}$, $L^{*}\ell_{k+1}=\ell_{k}$.

Then the essential spectral radius of $L:\mathfrak{B}\to\mathfrak{B}$ is at
least $\Lambda$.

Section 4 is devoted to the proof of the above.

The following result uses the existence of a proper discontinuity to construct
a family of linear maps which satisfy the assumptions of the above
proposition.

######  Proposition 2.3.

Let $\mathcal{L}$ be the transfer operator associated to a piecewise non-
singular map $\mathcal{F}:\mathcal{M}\to\mathcal{M}$ and weight $\varphi$ as
above. Suppose that $\mathcal{F}$ admits a proper discontinuity $\gamma$. Then
there exists a sequence of linear maps
${\\{\ell_{k}:\mathfrak{A}\to\mathbb{C}\\}}_{k\in\mathbb{N}}$ and
$h_{0}\in\mathfrak{A}$ which satisfy:777Where
$\Lambda_{L^{\infty}}=\Lambda_{L^{\infty}}(\mathcal{F},\varphi,\gamma)$ and
$\Lambda_{BV}=\Lambda_{BV}(\mathcal{F},\varphi,\gamma)$ are as defined in
(2.3). When this does not create confusion, subsequently we may suppress the
dependency on $(\mathcal{F},\varphi,\gamma)$.

(2.5)
$\displaystyle\limsup_{k\to\infty}\Big{(}\sup\left\\{\left|\ell_{k}(h)\right|:{h\in\mathfrak{A},{\left\|h\right\|}_{BV}\leq
1}\right\\}\Big{)}^{\frac{1}{k}}$ $\displaystyle\leq\Lambda_{BV}^{-1},$
$\displaystyle\limsup_{k\to\infty}\Big{(}\sup\left\\{\left|\ell_{k}(h)\right|:{h\in\mathfrak{A},{\left\|h\right\|}_{L^{\infty}}\leq
1}\right\\}\Big{)}^{\frac{1}{k}}$
$\displaystyle\leq\Lambda_{L^{\infty}}^{-1};$

And also satisfy:

  * •

$\ell_{1}(h_{0})=1$ and, for all $k\in\mathbb{N}$, $\ell_{k+1}(h_{0})=0$;

  * •

For all $k\in\mathbb{N}$, $h\in\mathfrak{A}$,
$\ell_{k+1}(\mathcal{L}h)=\ell_{k}(h)$.

Section 5 is devoted to the proof of the above.

###### Proof of Theorem 1.

The assumptions of the theorem imply that Proposition 2.3 holds. Consider
first the case in which
${{\left\|\cdot\right\|}}_{BV}\leq{{\left\|\cdot\right\|}}_{\mathfrak{B}}$.
Let $\overline{\mathfrak{A}}$ denote the completion of $\mathfrak{A}$ with
respect to ${\left\|\cdot\right\|}_{BV}$. By assumption,
$\mathcal{C}^{\infty}(\mathcal{M})\subset\mathfrak{B}$ and $\mathcal{L}$
extends to a continuous operator on $\mathfrak{B}$, and so
$\mathfrak{A}\subset\mathfrak{B}$. Every complete subspace of a Banach space
is a Banach space and so
$(\overline{\mathfrak{A}},{\left\|\cdot\right\|}_{BV})$ is a Banach space.
Furthermore, each of the statements of Proposition 2.3 concerning $\ell_{k}$
extend to $\overline{\mathfrak{A}}$ by continuity. In particular the estimate
(2.5) implies (B-1) for $\Lambda=\Lambda_{BV}$. This means that the
assumptions of Proposition 2.2 are satisfied with respect to
$(\overline{\mathfrak{A}},{\left\|\cdot\right\|}_{BV})$ and so the essential
spectral radius of
$\mathcal{L}:\overline{\mathfrak{A}}\to\overline{\mathfrak{A}}$ is at least
$\Lambda_{BV}$. Finally, since $\overline{\mathfrak{A}}\subset\mathfrak{B}$,
we use the fact that the essential spectral radius of
$\mathcal{L}:\overline{\mathfrak{A}}\rightarrow\overline{\mathfrak{A}}$ is not
greater than the essential spectral radius of
$\mathcal{L}:\mathfrak{B}\rightarrow\mathfrak{B}$ (see [5, §4]). Consequently
we conclude that the essential spectral radius of
$\mathcal{L}:\mathfrak{B}\rightarrow\mathfrak{B}$ is at least $\Lambda_{BV}$.

Alternatively, in the case in which
${{\left\|\cdot\right\|}}_{L^{\infty}}\leq{{\left\|\cdot\right\|}}_{\mathfrak{B}}$,
we let $\overline{\mathfrak{A}}$ denote the completion of $\mathfrak{A}$ with
respect to ${\left\|\cdot\right\|}_{L^{\infty}}$ and proceed identically to
obtain the analogous conclusion. ∎

######  Remark 2.4 (observables).

Understanding the statistical properties of $\mathcal{C}^{\infty}$ observables
typically suffices, via approximation arguments, to determine the behaviour of
less regular space of observables. For example, one could consider
$\mathcal{C}^{r}$, Hölder, or indeed any spaced defined by fixing some modulus
of continuity. In such a case the general results can be obtained by arguing
about how “rapidly” the observables can be approximated by
$\mathcal{C}^{\infty}$ ones. The reference metric is the crucial detail we are
fixing when we consider $\mathcal{C}^{\infty}(\mathcal{M})$ rather than the
specific regularity.

######  Remark 2.5 (disc of spectrum).

The proof of Theorem 1, together with the arguments in [5, §4], implies a
slightly stronger statement, namely that the complement of the unbounded
component of the resolvent set contains a disc of radius $\Lambda_{BV}$ or
$\Lambda_{L^{\infty}}$.

######  Remark 2.6 (partitions and artificial discontinuities).

Often a convenient way to work with systems like we have here is to assume, by
definition, that there exists a finite partition $\Omega$ of a full measure
subset of $\mathcal{M}$ such that, for each $\omega\in\Omega$, $\mathcal{F}$
restricted to $\omega$ has no critical points, is injective and admits a
$\mathcal{C}^{r}$ extension to $\overline{\omega}$. We know that
$\Gamma\subset\bigcup_{\omega\in\Omega}\partial\omega\subset M$ but this might
easily be a proper subset. We would like to assume that $\Omega$ denotes the
_minimal partition_ of $\mathcal{M}$ with the aforementioned properties.
Unfortunately, for some systems, there might not be a unique such partition,
e.g., a system which is smooth apart from a short “slit” as introduced in
Section 6.3. In other words, working with such a definition can force
artificial discontinuities to be introduced into the setting, an issue which
we wish to avoid.

######  Remark 2.7 (comparison of estimates).

The different bounds (2.3) are more or less useful in different settings, as
illustrated by the following: If $\left|\gamma_{k}\right|$ is uniformly
bounded, as is the case for the cocycle examples of Section 6.2, then
$\Lambda_{BV}=\Lambda_{L^{\infty},2}$; If
$\operatorname{Jac}({\mathcal{F}^{k}|}_{\gamma})\equiv 1$, as could be
arranged for a cocycle example of Section 6.2 by choosing the fibre map
$S(x,y)$ to be a circle rotation, then $\Lambda_{BV}=\Lambda_{L^{\infty},1}$;
On the other hand, if there is contraction along $\gamma_{k}$, for example a
suitable piecewise smooth Anosov endomorphism, then $\Lambda_{BV}$ will be
strictly smaller than $\Lambda_{L^{\infty}}$. However this framework is
unsuitable for studying general systems with a contracting direction since one
would need to consider the connection with anisotropic Banach spaces which
permit distribution-like behaviour in the contracting direction.

##  3\. Discussion of the proper discontinuity assumptions

Differently from the one-dimensional case, where failing to be Markov (see [5]
for further details) was sufficient to prove the lower bound on the essential
spectral radius, in the present two-dimensional setting the geometry of the
discontinuities led us to impose some extra assumptions in order to construct
the suitable linear functionals in Section 5. Nevertheless the assumptions are
more than satisfactory for the systems of primary interest. Here we discuss
the distance between the proper discontinuity assumption (Definition 2.1) and
failing to be Markov. Let $\mathcal{M}$, $\mathcal{F}$ and $\Gamma$ be as
specified in Section 2 but without requiring the existence of a proper
discontinuity.

In the following we see that a combination of certain assumptions of a proper
discontinuity implies that the $\gamma_{k}$ curves are essentially pairwise
disjoint.

######  Lemma 3.1.

Suppose that, with respect to a curve $\gamma\subset\Gamma$, (A-2) and (A-3)
are satisfied. Then $\gamma_{j}\cap\gamma_{k}\approxeq\emptyset$ whenever
$j,k\in\mathbb{N}$, $j\neq k$.

###### Proof.

We first show, as a consequence of (A-3), that for all $j,k\in\mathbb{N}$,

(3.1) $\gamma_{j}\cap\gamma_{k}\approxeq\emptyset\quad\text{implies
that}\quad\gamma_{j+1}\cap\gamma_{k+1}\approxeq\emptyset.$

According to (A-3),
$\mathcal{F}^{-1}\gamma_{j+1}\cap\widehat{\Gamma}\approxeq\gamma_{j}$. Let
$\tilde{\gamma}_{j+1}\subset\gamma_{j+1}$ satisfy
$\tilde{\gamma}_{j+1}\approxeq\gamma_{j+1}$ and
$\mathcal{F}^{-1}\tilde{\gamma}_{j+1}\cap\widehat{\Gamma}\subset\gamma_{j}$.
In order to prove the above (3.1), it suffices to show that
$\gamma_{k+1}\cap\tilde{\gamma}_{j+1}\subset\mathcal{F}(\gamma_{k}\cap\gamma_{j})$.
Let $y\in\gamma_{k+1}\cap\tilde{\gamma}_{j+1}$. Since
$\gamma_{k+1}=\mathcal{F}\gamma_{k}$ there exists $x\in\gamma_{k}$ such that
$y=\mathcal{F}x$. However, as observed above,
$\mathcal{F}^{-1}\tilde{\gamma}_{j+1}\cap\gamma_{k}\subset\gamma_{j}$. We know
that $x\in\mathcal{F}^{-1}\tilde{\gamma}_{j+1}\cap\gamma_{k}$ and so this
shows that $x\in\gamma_{j}$. This completes the proof of the above statement
(3.1).

We now take advantage of (A-2). Without loss of generality, suppose that
$j<k$. Iterating what we just proved (3.1), we know that,

$\gamma_{j}\cap\gamma_{k}\approxeq\emptyset\quad\text{whenever}\quad\gamma_{1}\cap\gamma_{k-j+1}\approxeq\emptyset.$

Observe that $k-j+1\geq 2$ and that $\gamma_{1}\subset\Gamma_{1}$ and so, by
(A-2), $\gamma_{1}\cap\gamma_{k-j+1}\approxeq\emptyset$. Consequently we have
shown that $\gamma_{j}\cap\gamma_{k}\approxeq\emptyset$. ∎

We recall that the map $\mathcal{F}$ is said to be Markov if it admits a
Markov partition. Such a partition $\Omega$ is a finite set of subsets of
$\mathcal{M}$ which are each open, connected, have boundary consisting of a
finite union of finite length $\mathcal{C}^{1}$ curves and together form a
partition of a full measure subset of $\mathcal{M}$. Moreover $\mathcal{F}$ is
$\mathcal{C}^{r}$ and injective on each $\omega\in\Omega$ and
$\omega\cap\omega^{\prime}\neq\emptyset$ implies
$\mathcal{F}(\omega)\supset\omega^{\prime}$, for each
$\omega,\omega^{\prime}\in\Omega$.

######  Lemma 3.2.

Suppose that, with respect to $\gamma\subset\Gamma$, (A-0), (A-2) and (A-3)
are satisfied and $\left|\gamma_{k}\right|$ is uniformly bounded from below.
Then $\mathcal{F}$ is not Markov.

###### Proof.

Suppose, for sake of a contradiction, that $\mathcal{F}$ is Markov and let
$\Omega$ denote the associated partition. For convenience let
$\partial\Omega=\bigcup_{\omega\in\Omega}\partial\omega$. We know that
$\gamma\subset\Gamma\subset\partial\Omega$ since $\mathcal{F}$ is
$\mathcal{C}^{r}$ on each $\omega\in\Omega$. The Markov partition has the
property that $\mathcal{F}(\partial\Omega)\subset\partial\Omega$ and so
$\gamma_{k}\subset\partial\Omega$ for all $k\in\mathbb{N}$. By Lemma 3.1 we
know that $\gamma_{j}\cap\gamma_{k}\approxeq\emptyset$ whenever $j\neq k$.
Since we assumed a uniform lower length for the $\gamma_{k}$ we obtain a
contradiction. ∎

Observe that assuming $\mathcal{F}$ to be expanding might not guarantee that
$\left|\gamma_{k}\right|$ is uniformly bounded from below since
${\left.\mathcal{F}\right|_{\gamma_{k}}}$ need not be injective and indeed
won’t be in many basic examples (see Section 6.2 for such examples and [19]
for an illustration of the singular possibilities of piecewise expanding
maps). Without modifying the argument, the assumption of the lemma can be
weakened to requiring that $\sum_{k}\left|\gamma_{k}\right|$ is a divergent
series. However this is still not sufficient to deal with all cases.

### Assumption (A-0)

In general, if the map has no discontinuity (or is Markov) and is
$\mathcal{C}^{r}$ then the essential spectral radius can be made rather small
[13, 15]. Consequently some assumption like (A-0) is required.

Assumption (A-0) could be weakened to permit the higher dimensional version of
the over-lapping mechanism which is seen in “W-maps” [16, §6] and unimodal
expanding maps [4]. These exhibit behaviour typically seen with
discontinuities whilst being continuous. It appears that the results obtained
in this work would continue to hold for such a weakened assumption.

### Assumption (A-1)

Suppose that $\gamma\subset\Gamma$ is a curve and that
$\mathcal{F}:\gamma_{k-1}\to\gamma_{k}$ is injective, with the exception of a
countable set of points, for every $k\in\mathbb{N}$. Assumption (A-1) can
easily be satisfied by choosing $\alpha_{0}\equiv 1$ and then defining the
other $\alpha_{k}$ inductively. On the other hand, when
$\mathcal{F}:\gamma_{k-1}\to\gamma_{k}$ is many-to-one, then we require the
correct relationship between $\varphi$ and
$\operatorname{Jac}({\mathcal{F}|}_{\gamma_{k}})$ at preimages else it would
be impossible to satisfy the required relationship (2.2). It is unreasonable
to imagine that such a problem won’t exist for some map which fails to be
Markov and yet satisfies all other properties.

### Assumptions (A-2), (A-3)

Recall that (A-2) specified that $\gamma_{k}\cap\Gamma_{1}\approxeq\emptyset$
for all $k\geq 2$ and that (A-3) specified that
$\mathcal{F}^{-1}\gamma_{k+1}\cap\widehat{\Gamma}\approxeq\gamma_{k}$ for all
$k\in\mathbb{N}$. Lemma 3.2 tells us that this, together with (A-0) and a
lower bound on $\left|\gamma_{k}\right|$, these assumptions suffice to imply
that $\mathcal{F}$ is not Markov. These assumptions are not as tight as
desired, as illustrated with the following example. This is the same
possibility that already exists in the one-dimensional case [5]. E.g., suppose
that $f:[0,1]\to[0,1]$ is an expanding interval map and that $f$ is
discontinuous at two distinct points $a,b\in[0,1]$ which satisfy
$f(a^{+})=f(b^{+})$ but that $\\{f^{n}(a^{+})\\}_{n\in\mathbb{N}}$ is an
infinite set of points. In such a way the required assumption holds for the
second iterate, although not from the very first. A two-dimensional example
can then be built from this as a product or cocycle (as per Section 6.2).

##  4\. Essential spectrum: an abstract result

In this section we prove Proposition 2.2. Suppose that $\lambda\in\mathbb{C}$,
$\left|\lambda\right|<\Lambda$ and define,

$\Xi_{\lambda}=\sum_{k=1}^{\infty}\lambda^{k}\ell_{k}.$

Take $\underline{\Lambda}\in(|\lambda|,\Lambda)$. Assumption (B-1) implies
that there exists $C>0$ such that $\left|\ell_{k}(h)\right|\leq
C\underline{\Lambda}^{-k}{\left\|h\right\|}$ for all $k\in\mathbb{N}$,
$h\in\mathfrak{B}$. Consequently

$\left|\Xi_{\lambda}(h)\right|\leq
C\sum_{k=1}^{\infty}\left|\lambda\right|^{k}\underline{\Lambda}^{-k}{\left\|h\right\|}$

and so $\Xi_{\lambda}$ defines a linear functional on $\mathfrak{B}$. By
(B-2), $\Xi_{\lambda}(h_{0})=\lambda\ell_{1}(h_{0})=\lambda$ and so
$\Xi_{\lambda}$ is non-zero. We also define the rank-one operator
$\mathcal{K}:\mathfrak{B}\to\mathfrak{B}$,

(4.1) $\mathcal{K}h=\ell_{1}(Lh)\,h_{0}.$

Using assumption (B-3) we calculate that,

$\displaystyle L^{*}\Xi_{\lambda}$
$\displaystyle=\sum_{k=1}^{\infty}\lambda^{k}L^{*}\ell_{k}=\lambda
L^{*}\ell_{1}+\sum_{k=2}^{\infty}\lambda^{k}\ell_{k-1}$ $\displaystyle=\lambda
L^{*}\ell_{1}+\lambda\sum_{k=1}^{\infty}\lambda^{k}\ell_{k}=\lambda
L^{*}\ell_{1}+\lambda\ \Xi_{\lambda}.$

Since $\Xi_{\lambda}(h_{0})=\lambda$, we know that, for all
$h\in\mathfrak{B}$, $\Xi_{\lambda}(\mathcal{K}h)=\lambda\ell_{1}(Lh)$. I.e.,
$\mathcal{K}^{*}\Xi_{\lambda}=\lambda L^{*}\ell_{1}$. Consequently, we have
shown that,

$(L-\mathcal{K})^{*}\Xi_{\lambda}=\lambda\ \Xi_{\lambda},$

whenever $\left|\lambda\right|<\Lambda$. This means that the essential
spectral radius of $({L}-\mathcal{K}):\mathfrak{B}\to\mathfrak{B}$ is at least
$\Lambda$ and, since compact perturbations don’t change the essential spectral
radius, the same holds for $L$. ∎

##  5\. Linear functionals

In this section we construct a sequence of linear functionals and hence prove
Proposition 2.3. Let $\mathcal{F}:\mathcal{M}\to\mathcal{M}$,
$\varphi:\mathcal{M}\to\mathbb{C}$ be fixed for the remainder of this section,
satisfying the assumptions of the proposition. In particular this means that
$\mathcal{F}$ has a proper discontinuity $\gamma$ such that (A-0), (A-1),
(A-2) and (A-3) are each satisfied with respect to the curves
$\\{\gamma_{k}\\}$ and associated $\\{\alpha_{k}\\}$. The proof will consist
of defining the linear functionals and then proving that the required
properties are satisfied.

For each $h\in\mathfrak{A}$, let us define the “jump” of $h$ at a point
$x\in\mathcal{M}$ in the direction $v\in\mathcal{T}_{x}\mathcal{M}$
as,888Here, $\exp_{x}(v)$ denotes the exponential map based at $x$, i.e., the
application that sends $v$ to $\eta(1)$, where $\eta$ is the geodesic with
$\eta^{\prime}(x)=v$.

$J(h,x,v)=\lim_{\epsilon\rightarrow 0}\left[h\left(\exp_{x}(\epsilon
v)\right)-h\left(\exp_{x}(-\epsilon v)\right)\right].$

Recall that $\gamma\subset\Gamma\subset M$ is part of the boundary of the
manifold $M$. For $x\in\gamma$ let $v(x)\in\mathcal{T}_{x}M$ denote the unit
vector which is orthogonal to $\gamma$ and such that $\exp_{x}(\epsilon v)$ is
in $M$ for small $\epsilon>0$. For each $k\in\mathbb{N}$, $y\in\gamma_{k}$ let
$v_{k}(y)\in\mathcal{T}_{y}\mathcal{M}$ be defined as
$v_{k}(y)=D\mathcal{F}^{k}(x)\ v(x)$ where $y=\mathcal{F}^{k}x$. For each
$k\in\mathbb{N}$, $h\in\mathfrak{A}$, we define999If $h:\gamma\to\mathbb{C}$
and $\gamma\subset\mathcal{M}$ is a $\mathcal{C}^{1}$ curve, we systematically
write $\int_{\gamma}A(x)\ dx$ as shorthand for
$\smash{\int_{0}^{\left|\gamma\right|}}A(x(t))\ dt$ with the understanding
that $x:[0,\left|\gamma\right|]\to\gamma$ is a parametric representation of
the curve $\gamma$ which respects arc length. the linear functional
$\ell_{k}:\mathfrak{A}\rightarrow\mathbb{R}$,

(5.1) $\ell_{k}(h)=\int_{\gamma_{k}}\alpha_{k}(x)J(h,x,v_{k}(x))\ dx.$

In the remainder of this section we prove the required properties of these
linear functionals.

###### Proof of Proposition 2.3, part 1.

Let $x\in\gamma_{1}$, $k\in\mathbb{N}$ and
$y=\mathcal{F}^{k}x\in\gamma_{k+1}$. Iterating the assumed relationship (A-1)
and recalling that $\alpha_{1}\equiv 1$ implies that

(5.2)
$\alpha_{k+1}(y)=\varphi_{k}(x)^{-1}\operatorname{Jac}({\mathcal{F}^{k}|}_{\gamma_{1}})(x)^{-1}.$

It may happen that a given $y\in\gamma_{k+1}$ has more than one preimage in
$\gamma_{1}$. However, the assumed relationship (A-1) means that we are
working in the case where the term
$\varphi_{k}(\cdot)\operatorname{Jac}({\mathcal{F}^{k}|}_{\gamma_{1}})(\cdot)$
is equal at each such preimage, as implied by this calculation. The above
(5.2) implies that,

(5.3)
$\sup_{y\in\gamma_{k+1}}\left|\alpha_{k+1}\right|(y)\leq\left(\inf_{x\in\gamma_{1}}\left|\varphi_{k}(x)\cdot\operatorname{Jac}({\mathcal{F}^{k}|}_{\gamma_{1}})(x)\right|\right)^{-1}.$

For convenience, let $\tilde{\gamma}_{1}\subset\gamma_{1}$ be such that
$\mathcal{F}^{k}$ is injective from $\tilde{\gamma}_{1}$ to $\gamma_{k+1}$.
Using again the above formula (5.2),

$\displaystyle\int_{\gamma_{k+1}}\left|\alpha_{k+1}(y)\right|\ dy$
$\displaystyle=\int_{\tilde{\gamma}_{1}}\left|\alpha_{k+1}(\mathcal{F}^{k}x)\right|\operatorname{Jac}({\mathcal{F}^{k}|}_{\gamma_{1}})(x)\
dx$
$\displaystyle\leq\left(\inf_{x\in\gamma_{1}}\left|\varphi_{k}(x)\right|\right)^{-1}\int_{\tilde{\gamma}_{1}}\
dx.$

Consequently, observing that
$\left|\tilde{\gamma}_{1}\right|\leq\left|\gamma_{1}\right|$, we have shown
that

(5.4) $\int_{\gamma_{k+1}}\left|\alpha_{k+1}(y)\right|\
dy\leq\left(\inf_{x\in\gamma_{1}}\left|\varphi_{k}(x)\right|\right)^{-1}\left|\gamma_{1}\right|.$

Having prepared the above estimates, we now divide the argument into the three
cases (2.3).

Case 1: Suppose that ${\|h\|}_{BV}\leq{\|h\|}_{\mathfrak{B}}$, $\forall
h\in\mathfrak{A}$. The control given by the $BV$ norm implies that there
exists101010See e.g., [1, Definition 3.67, Theorem 3.86]. For this it is
essential that the linear function is defined by integrating along a curve
rather than considering jumps at points. $C>0$ such that, for each
$k\in\mathbb{N}$, $h\in BV(\mathcal{M})$,

(5.5) $\int_{\gamma_{k}}\left|J(h,x,v_{k}(x))\right|\ dx\leq
C{\left\|h\right\|}_{BV}\leq C{\left\|h\right\|}_{\mathfrak{B}}.$

In particular, the bound does not depend on the length of $\gamma_{k}$. Since,

$\displaystyle\left|\ell_{k+1}(h)\right|$
$\displaystyle=\left|\int_{\gamma_{k+1}}\alpha_{k+1}(x)J(h,x,v_{k+1}(x))\
dx\right|$
$\displaystyle\leq\sup_{y\in\gamma_{k+1}}\left|\alpha_{k+1}\right|(y)\int_{\gamma_{k}}\left|J(h,x,v_{k}(x))\right|\
dx,$

the above estimates (5.5), (5.3) imply that

$\left|\ell_{k+1}(h)\right|\leq
C\left(\inf_{x\in\gamma_{1}}\left|\varphi_{k}(x)\cdot\operatorname{Jac}({\mathcal{F}^{k}|}_{\gamma_{1}})(x)\right|\right)^{-1}{\left\|h\right\|}_{\mathfrak{B}}.$

Case 2: Suppose that ${\|h\|}_{L^{\infty}}\leq{\|h\|}_{\mathfrak{B}}$,
$\forall h\in\mathfrak{A}$. We take advantage of the trivial estimate,

(5.6) $\left|J(h,x,v_{k}(x))\right|\leq 2{\|h\|}_{L^{\infty}}\leq
2{\left\|h\right\|}_{\mathfrak{B}}.$

Since

$\displaystyle\left|\ell_{k+1}(h)\right|$
$\displaystyle=\left|\int_{\gamma_{k+1}}\alpha_{k+1}(x)J(h,x,v_{k+1}(x))\
dx\right|$ $\displaystyle\leq\int_{\gamma_{k+1}}\left|\alpha_{k+1}(y)\right|\
dy\ \ \sup_{x\in\gamma_{k+1}}\left|J(h,x,v_{k+1}(x))\right|,$

we use the above (5.4) and obtain the estimate,

$\left|\ell_{k+1}(h)\right|\leq
2\left|\gamma_{1}\right|\left(\inf_{x\in\gamma_{1}}\left|\varphi_{k}(x)\right|\right)^{-1}{\left\|h\right\|}_{\mathfrak{B}}.$

This corresponds to $\Lambda_{L^{\infty},1}(\mathcal{F},\varphi,\gamma)$ in
(2.3).

Case 3: Again suppose that ${\|h\|}_{L^{\infty}}\leq{\|h\|}_{\mathfrak{B}}$,
$\forall h\in\mathfrak{A}$. Estimating as above (5.6),

$\displaystyle\left|\ell_{k+1}(h)\right|$
$\displaystyle=\left|\int_{\gamma_{k+1}}\alpha_{k+1}(x)J(h,x,v_{k+1}(x))\
dx\right|$ $\displaystyle\leq
2\int_{\gamma_{k+1}}\left|\alpha_{k+1}(y)\right|\ dy\ {\|h\|}_{\mathfrak{B}}.$

Using also the prior estimate (5.3),

$\left|\ell_{k+1}(h)\right|\leq
2\left(\inf_{x\in\gamma_{1}}\left|\varphi_{k}(x)\cdot\operatorname{Jac}({\mathcal{F}^{k}|}_{\gamma_{1}})(x)\right|\right)^{-1}\
\left|\gamma_{k+1}\right|{\|h\|}_{\mathfrak{B}}.$

This corresponds to $\Lambda_{L^{\infty},2}(\mathcal{F},\varphi,\gamma)$ in
(2.3). ∎

Before proving the next claim of Proposition 2.3 we will introduce a few
further lemmas.

######  Lemma 5.1.

The cardinality of $\mathcal{F}^{-1}y$ is bounded, uniformly for
$y\in\mathcal{M}$.

###### Proof.

Let $\\{U_{j}\\}_{j}$ be the connected components of
$\mathcal{M}\setminus\Gamma$. This is a finite set since $\Gamma$ is the union
of a finite number of smooth curves of a finite length. Let $\delta>0$. For
each $j$, let $\\{V_{j,k}\\}_{k}$ be a finite cover of $U_{j}$ comprising of
subsets of $U_{j}$ of diameter less than $\delta$. Since the derivative of
$\mathcal{F}$ is bounded we may choose $\delta$ sufficiently small such that
$\left.\mathcal{F}\right|_{V_{j,k}}$ is 1-to-1. Consequently the cardinality
of $\mathcal{F}^{-1}y$ is bounded by the cardinality of $\\{V_{j,k}\\}_{j,k}$.
∎

Let $\sigma\subset\Gamma\subset\mathcal{M}$ denote the finite set of points of
intersection of the curves in $\Gamma$. Consequently, for any point in
$\Gamma\setminus\sigma$, there exists a neighbourhood such that the
intersection of this neighbourhood with $\Gamma$ is a curve.

######  Lemma 5.2.

Let $y\in\mathcal{M}$ and set111111The finite set of preimages of $y$, divided
into those in $\partial M$ and those in the interior.
$\mathcal{A}_{1}(y)=\partial M\cap F^{-1}y$ and
$\mathcal{A}_{2}(y)=F^{-1}y\setminus\mathcal{A}_{1}(y)$. There exists $U$, a
neighbourhood of $y$, and, for $x\in\mathcal{F}^{-1}y$, there exist mutually
disjoint neighbourhoods121212Although $V_{x}\subset M$, writing
${\left.\mathcal{F}\right|}_{V_{x}}$ we consider $V_{x}$ as a subset of
$\mathcal{M}$, discarding $\partial M$. $V_{x}$ of $x$, $\mathcal{F}:V_{x}\to
U$ injective, such that, for all $h\in\mathfrak{A}$, on
$U\setminus\Gamma_{1}$,

(5.7) $\mathcal{L}h=\sum_{x\in\mathcal{A}_{2}(y)}\left(\varphi\cdot
h\right)\circ{\left.\mathcal{F}\right|}_{V_{x}}^{-1}+\sum_{x\in\mathcal{A}_{1}(y)}\left(\varphi\cdot
h\right)\circ{\left.\mathcal{F}\right|}_{V_{x}}^{-1}\cdot\mathbf{1}_{\mathcal{F}V_{x}}.$

Moreover, if $y\in\gamma_{1}\setminus F(\iota^{-1}\sigma)$, there exists
$x_{0}\in\mathcal{A}_{1}(y)$ such that $\tilde{x}_{0}\notin\mathcal{A}_{1}(y)$
where $\tilde{x}_{0}\in\partial M$ denotes131313I.e., $\tilde{x}_{0}\in M$
denotes the point that is equal to $x_{0}$ as points in $\mathcal{M}$ yet
distinct in $M$. the point such that $x_{0}\neq\tilde{x}_{0}$ yet
$\iota(x_{0})=\iota(\tilde{x}_{0})$.

###### Proof.

The form of the transfer operator follows from the definition (2.1), the only
thing to note is that the term $\mathbf{1}_{\mathcal{F}V_{x}}$ can be removed
from the first sum. This is because $x\in\mathcal{A}_{2}(y)$ means that
$\mathcal{F}V_{x}$ is a neighbourhood of $y$ (as a point in $\mathcal{M}$).
Consequently, since $\mathcal{A}_{2}(y)$ is a finite set, $U$ can be chosen so
that $\mathcal{F}V_{x}$ covers $U$ for all $x\in\mathcal{A}_{2}(y)$.

The final statement follows from assumption (A-0). Suppose this statement were
false, then, for any $x_{0}\in\mathcal{A}_{1}(y)$, there exists
$\tilde{x}_{0}\in\mathcal{A}_{1}(y)$ such that $x_{0}$ and $\tilde{x}_{0}$ are
equal as points in $\mathcal{M}$. Let
$x=\iota(x_{0})=\iota(\tilde{x}_{0})\in\mathcal{M}$ and let
$W_{y}\subset\mathcal{M}$ be a neighbourhood of $y=\mathcal{F}x$. Notice that
any neighbourhood of $x$ is covered by $\iota(W_{x_{0}}\cup
W_{\tilde{x}_{0}})$ where $W_{x_{0}}\subset M$, $W_{\tilde{x}_{0}}\subset M$
are neighbourhoods of $x_{0}$, $\tilde{x}_{0}$ respectively. Shrinking the
neighbourhoods we can guarantee that $\mathcal{F}(\iota(W_{x_{0}}\cup
W_{\tilde{x}_{0}}))\subset W_{y}$. Consequently $\mathcal{F}$ is continuous at
$x$ in contradiction of (A-0). ∎

The following result concerns the structure of elements of $\mathfrak{A}$.

######  Lemma 5.3.

$\mathcal{L}^{n}g$ is uniformly $\mathcal{C}^{1}$ on
$\mathcal{M}\setminus\bigcup_{p=1}^{n}\Gamma_{p}$ for all
$n\in\mathbb{N}_{0}$, $g\in\mathcal{C}^{\infty}(\mathcal{M})$.

###### Proof.

The case $n=0$ is immediate. Let us now suppose the statement is true for $n$
and consider the $n+1$ case. Let $h=\mathcal{L}^{n}g$ and so
$\mathcal{L}^{n+1}g=\mathcal{L}h$. By the inductive assumption, we know that
$h$ is uniformly $\mathcal{C}^{1}$ on
$\mathcal{M}\setminus\bigcup_{p=1}^{n}\Gamma_{p}$. Consider
$y\notin\bigcup_{p=1}^{n+1}\Gamma_{p}$. To complete the proof it suffices to
show that $\mathcal{L}h$ is uniformly $\mathcal{C}^{1}$ in a neighbourhood of
any such $y$. Using Lemma 5.2, there exists $U$, a neighbourhood of $y$ and,
for each $x\in\mathcal{F}^{-1}y$, a neighbourhood $V_{x}$ such that, on $U$,

$\mathcal{L}h=\sum_{x\in\mathcal{F}^{-1}y}\left(\varphi\cdot
h\right)\circ{\left.\mathcal{F}\right|}_{V_{x}}^{-1}$

where $\mathcal{F}:V_{x}\to U$ is bijective for each $x$. We can consider this
reduced version of the formula (5.7) because of the following. Since
$y\notin\Gamma_{1}$ the second sum from the lemma, the one corresponding to
$\mathcal{A}_{1}(y)$, can be omitted and we can assume that
$U\subset\mathcal{M}\setminus\Gamma_{1}$. Moreover, shrinking the
neighbourhoods as required, we may choose
$V_{x}\subset\mathcal{M}\setminus\bigcup_{p=1}^{n}\Gamma_{p}$ since
$y\notin\mathcal{F}(\bigcup_{p=1}^{n}\Gamma_{p})=\bigcup_{p=2}^{n+1}\Gamma_{p}$.
As such, $\varphi\cdot h$ is uniformly $\mathcal{C}^{1}$ on each $V_{x}$.
Finally observe that $\mathcal{F}^{-1}:U\to V_{x}$ is uniformly
$\mathcal{C}^{1}$ since $\mathcal{F}$ is $\mathcal{C}^{r}$ and uniformly non-
singular. ∎

######  Remark 5.4.

Since $M$ is a manifold with boundary, the boundary is locally homeomorphic to
the half-space $\\{(x,y)\in\mathbb{R}^{2}:y\geq 0\\}$. We also know that
$F:M\to\mathcal{M}$ is $\mathcal{C}^{r}$. If $V\subset M$ is a small
neighbourhood of an interior point $x\in M$ then $FV\subset\mathcal{M}$ will
be a neighbourhood of $Fx$. On the other hand, if $W\subset M$ is a small
neighbourhood of a boundary point $x\in M$ then $FW\subset\mathcal{M}$ won’t
be a neighbourhood of $Fx$, it will only be “half”.

###### Proof of Proposition 2.3, part 2.

We will construct $h_{0}\in\mathfrak{A}$ such that

(5.8) $\ell_{1}(h_{0})=1\quad\text{and}\quad\ell_{k}(h_{0})=0\text{ for all
$k\geq 2$}.$

Let $h\in\mathcal{C}^{\infty}(\mathcal{M})$, to be chosen shortly, and let
$h_{0}=\mathcal{L}h$. By Lemma 5.3, $\mathcal{L}h$ is $\mathcal{C}^{1}$ on
$\mathcal{M}\setminus\Gamma_{1}$. Whenever $k\geq 2$, assumption (A-2) tells
that $\gamma_{k}\cap\Gamma_{1}\approxeq\emptyset$. Consequently,
$J(\mathcal{L}h,x,v_{k}(x))=0$ for all $x$ in a full measure subset of
$\gamma_{k}$. We have thereby shown that $\ell_{k}(\mathcal{L}h)=0$ whenever
$k\geq 2$. It remains to fix $h_{0}$ and show that $\ell_{1}(h_{0})=1$.

Fix $y\in\gamma_{1}\setminus F(\iota^{-1}\sigma)$. Lemma 5.2 tells us that, in
$U\setminus\Gamma_{1}$, where $U$ is a neighbourhood of a point
$y_{0}\in\gamma_{1}$,

$\mathcal{L}h=\sum_{x\in\mathcal{A}_{2}}\left(\varphi\cdot
h\right)\circ{\left.\mathcal{F}\right|}_{V_{x}}^{-1}+\sum_{x\in\mathcal{A}_{1}}\left(\varphi\cdot
h\right)\circ{\left.\mathcal{F}\right|}_{V_{x}}^{-1}\cdot\mathbf{1}_{\mathcal{F}V_{x}}.$

The first sum of terms contributes nothing to $J(\mathcal{L}h,y,v_{1}(y))$
because they are all continuous in $U$ (recall that
$h\in\mathcal{C}^{\infty}(\mathcal{M})$). Let $x_{0}\in\mathcal{A}_{1}\subset
M$, $x_{0}\in\gamma$, be as given by Lemma 5.2. We ensure that $y_{0}$ was
chosen such that the intersection of $V_{x_{0}}$ with $\Gamma$ is a simple
curve, a subset of $\iota\gamma$. We choose
$h\in\mathcal{C}^{\infty}(\mathcal{M})$ such that $\varphi\cdot h$ is positive
in $V_{x_{0}}$, the neighbourhood of $\iota x_{0}\in\mathcal{M}$, and zero
elsewhere. We may assume that this neighbourhood is sufficiently small that it
does not intersect any other curves in $\Gamma$ except for $\gamma$. This
means that
${J(\mathcal{L}h,\mathcal{F}x_{0},v_{1}(\mathcal{F}x_{0}))}={\varphi(x_{0})h(x_{0})}$
(or $-\varphi(x_{0})h(x_{0})$). Consequently $J(\mathcal{L}h,y,v_{1}(y))$ is
positive (or negative, but say positive) for a positive measure subset of
$y\in\gamma_{1}$, given by $\gamma_{1}\cap\mathcal{F}V_{x_{0}}$. Furthermore,
we can even ensure that $J(\mathcal{L}h,y,v_{1}(y))$ is everywhere non-
negative in $\gamma_{1}$ since, by eventually making $\varphi\cdot h$ positive
in even a smaller neighbourhood of $x_{0}$ and zero elsewhere, we can consider
points in $\gamma_{1}$ coming from just one side of $\Gamma$ (so that
${J(\mathcal{L}h,y,v_{1}(y))}$ does not change sign varying $y\in\gamma_{1}$).
This in turn implies that $\ell_{1}(\mathcal{L}h)$ is non-zero. Scaling
appropriately, we insure that $\ell_{1}(h_{0})=1$. ∎

It remains to prove the final claim of Proposition 2.3 which, together with
Lemma 5.2, makes use of the following two results.

######  Lemma 5.5.

Let $h\in\mathfrak{A}$. For any $k\in\mathbb{N}$, there exists
$\widetilde{\gamma}_{k}$ with $\widetilde{\gamma}_{k}\approxeq\gamma_{k}$ such
that for all $x\in\widetilde{\gamma}_{k}$ and any vector
$v\in\mathcal{T}_{x}\mathcal{M}$, $v$ not parallel to $\gamma_{k}^{\prime}(x)$
and $v\neq 0$, we have

(5.9) $\lim_{\epsilon\to 0^{+}}h\circ\mathcal{F}\left(\exp_{x}(\epsilon
v)\right)=\lim_{\epsilon\to
0^{+}}h\left(\exp_{\mathcal{F}x}(\epsilon\overline{v})\right),$

where
$\overline{v}=D_{x}\mathcal{F}v\in\mathcal{T}_{\mathcal{F}x}\mathcal{M}$.

###### Proof.

Suppose that $h=\mathcal{L}^{n}g$ for some
$g\in\mathcal{C}^{\infty}(\mathcal{M})$. Consider the two paths,

$\rho(\epsilon)=\mathcal{F}\left(\exp_{x}(\epsilon
v)\right),\quad\quad\tilde{\rho}(\epsilon)=\exp_{\mathcal{F}x}(\epsilon\overline{v}).$

We have to show that $\lim_{\epsilon\rightarrow 0}h\circ\rho(\epsilon)$ is
equal to $\lim_{\epsilon\rightarrow 0}h\circ\tilde{\rho}(\epsilon)$. Observe
that $\rho(0)=\tilde{\rho}(0)=\mathcal{F}x$ and the two paths are tangent at
$0$. We still need to work a little because $h$ might fail to be continuous at
any point in $\bigcup_{j=1}^{n}\Gamma_{j}$. For the purpose of this proof, we
say that a point $y\in\gamma_{k+1}$ is a _regular_ point if there exists
$\eta>0$ such that the $\eta$-size neighbourhood of $y$,
$U_{\eta}(y)\subset\mathcal{M}$ satisfies
$U_{\eta}(y)\cap\bigcup_{j=1}^{n}\Gamma_{j}=\gamma_{k+1}$. Since
$\bigcup_{j=1}^{n}\Gamma_{j}$ is composed of a finite number of finite length
curves, there can be at most a finite number of points which fail to be
regular. Let $\widetilde{\gamma}_{k}\subset\gamma_{k}$ be the set of all
points $x$ such that $y=\mathcal{F}x\in\gamma_{k+1}$ is regular. By the above
considerations, $\widetilde{\gamma}_{k}\approxeq\gamma_{k}$. Observe that
$U_{\eta}(\mathcal{F}x)\setminus\gamma_{k+1}$ consists of two connected
components, the boundary between the two given by the curve $\gamma_{k+1}$. By
Lemma 5.3, $h$ is continuous in each of these two components. Now, the fact
that $v\in\mathcal{T}_{x}\mathcal{M}$ is not parallel to
$\gamma_{k}^{\prime}(x)$, implies that $\overline{v}$ is not parallel to
$\gamma_{k+1}^{\prime}(\mathcal{F}x)$ and the ranges of $\rho(\epsilon)$ and,
for all $\epsilon$ small enough, $\widetilde{\rho}(\epsilon)$ belong to the
same connected component of $U_{\eta}(\mathcal{F}x)$ on which $h$ is
continuous. Consequently the two limits coincide. ∎

######  Lemma 5.6.

For all $k\in\mathbb{N}$ there exists
$\widetilde{\gamma}_{k+1}\approxeq\gamma_{k+1}$ such that for all
$y\in\widetilde{\gamma}_{k+1}$, $h\in\mathfrak{A}$,

$J(\mathcal{L}h,y,v_{k+1}(y))=\sum_{\begin{subarray}{c}x\in\gamma_{k},\\\
\mathcal{F}x=y\end{subarray}}\varphi(x)J(h,x,v_{k}(x)).$

###### Proof.

Let $h=\mathcal{L}^{n}g$ for some $g\in\mathcal{C}^{\infty}(\mathcal{M})$. By
Lemma 5.3, $h$ is uniformly $\mathcal{C}^{1}$ on
$\mathcal{M}\setminus\bigcup_{p=1}^{n}\Gamma_{p}$. By (A-3),
$\mathcal{F}^{-1}\gamma_{k+1}\cap\widehat{\Gamma}\approxeq\gamma_{k}$.
Additionally, by (A-2), $\gamma_{k+1}\cap\Gamma_{1}\approxeq\emptyset$.
Consequently, let $\widetilde{\gamma}_{k+1}\approxeq\gamma_{k+1}$ denote the
subset of $\gamma_{k+1}$ such
$\mathcal{F}^{-1}{\widetilde{\gamma}}_{k+1}\cap\widehat{\Gamma}\subset\gamma_{k}$
and $\widetilde{\gamma}_{k+1}\cap\Gamma_{1}=\emptyset$.

Applying Lemma 5.2, for a point
$y\in\widetilde{\gamma}_{k+1}\subset\gamma_{k+1}$, we obtain neighbourhoods
which, following the notation of the lemma, are denoted $U$, $\\{V_{x}\\}$,
such that $J(\mathcal{L}h,y,v_{k+1}(y))$ is equal to the limit as $\epsilon\to
0$ of

(5.10) $\sum_{x\in\mathcal{F}^{-1}y}\Bigl{[}\left(\varphi\cdot
h\right)\circ{\left.\mathcal{F}\right|}_{V_{x}}^{-1}\left(\exp_{y}(\epsilon
v_{k+1}(y))\right)\cdot\mathbf{1}_{\mathcal{F}V_{x}}\left(\exp_{y}(\epsilon
v_{k+1}(y))\right)\\\ -\left(\varphi\cdot
h\right)\circ{\left.\mathcal{F}\right|}_{V_{x}}^{-1}\left(\exp_{y}(-\epsilon
v_{k+1}(y))\right)\cdot\mathbf{1}_{\mathcal{F}V_{x}}\left(\exp_{y}(-\epsilon
v_{k+1}(y))\right)\Bigr{]}.$

Recall the notation from before (Lemma 5.2), that $\mathcal{A}_{1}=\partial
M\cap\mathcal{F}^{-1}y$ and
$\mathcal{A}_{2}=\mathcal{F}^{-1}y\setminus\mathcal{A}_{1}$. As mentioned
above, by (A-2) we know that
$\widetilde{\gamma}_{k+1}\cap\Gamma_{1}\approxeq\emptyset$ and so, in this
case, $\mathcal{A}_{1}$ is an empty set and we need only to consider
$\mathcal{A}_{2}$. This means that, for all $x\in\mathcal{F}^{-1}y$,
$\mathcal{F}V_{x}$ is a neighbourhood of $y$ so the indicator functions in the
above sum can be discarded.

Since
$\mathcal{F}^{-1}{\widetilde{\gamma}}_{k+1}\cap\widehat{\Gamma}\subset\gamma_{k}$,
if $x\in\mathcal{F}^{-1}\widetilde{\gamma}_{k+1}$ then $x\in\gamma_{k}$ or
$x\notin\widehat{\Gamma}$. In the above (5.10) all the summands are zero,
except the ones corresponding to $x\in\gamma_{k}$. This is because, if
$x\notin\widehat{\Gamma}$, then $\varphi\cdot h$ is $\mathcal{C}^{1}$ in a
neighbourhood of $x$ and so the positive and negative part of the summand
cancel out. Here we have used the fact that $h$ is $\mathcal{C}^{1}$ on
$\mathcal{M}\setminus\bigcup_{p=1}^{n}\Gamma_{p}$ and
$x\notin\bigcup_{p=1}^{n}\Gamma_{p}$.

Taking advantage of Lemma 5.5, $J(\mathcal{L}h,y,v_{k+1}(y))$ is equal to, for
all $y\in\widetilde{\gamma}_{k+1}$,

(5.11) $\sum_{\begin{subarray}{c}x\in\gamma_{k}\\\
\mathcal{F}x=y\end{subarray}}\lim_{\epsilon\rightarrow
0^{+}}\left[(\varphi\cdot h)\left(\exp_{x}(\epsilon
v_{k}(x))\right)-(\varphi\cdot h)\left(\exp_{x}(-\epsilon
v_{k}(x))\right)\right].$

We conclude since $\varphi$ is continuous in $x$ (with the exception of at
most a finite set of points, which we can always remove by redefining
$\widetilde{\gamma}_{k+1}$). Indeed, since
$\tilde{\gamma}_{k+1}\cap\Gamma_{1}=\emptyset$, for all
$y\in\widetilde{\gamma}_{k+1}$, the above (5.11) is equal to

$\displaystyle\sum_{\begin{subarray}{c}x\in\gamma_{k}\\\
\mathcal{F}x=y\end{subarray}}\varphi(x)\lim_{\epsilon\rightarrow
0^{+}}\biggl{(}h\left(\exp_{x}(\epsilon
v_{k}(x))\right)-h\left(\exp_{x}(-\epsilon v_{k}(x))\right)\biggr{)}$
$\displaystyle\quad\quad\quad\quad\quad=\sum_{\begin{subarray}{c}x\in\gamma_{k}\\\
\mathcal{F}x=y\end{subarray}}\varphi(x)J(h,x,v_{k}(x))$

as required by the statement of the lemma. ∎

###### Proof of Proposition 2.3, part 3.

Recalling the definition (5.1) of the linear functionals $\ell_{k}$, Lemma 5.6
implies that

$\displaystyle\ell_{k+1}(\mathcal{L}h)$
$\displaystyle=\int_{\gamma_{k+1}}\alpha_{k+1}(y)J(\mathcal{L}h,y,v_{k+1}(y))\
dy$
$\displaystyle=\int_{\gamma_{k+1}}\alpha_{k+1}(y)\sum_{\begin{subarray}{c}x\in\gamma_{k},\\\
\mathcal{F}x=y\end{subarray}}\ \varphi(x)\ J(h,x,v_{k}(x))\ dy.$

Changing variables, the above is equal to

$\int_{\gamma_{k}}\alpha_{k+1}(\mathcal{F}x)\ \varphi(x)\ J(h,x,v_{k}(x))\
\operatorname{Jac}({\mathcal{F}|}_{\gamma_{k}})(x)\ dx.$

The relationship (2.2) for the $\alpha_{k}$ (A-1) means that,

$\ell_{k+1}(\mathcal{L}h)=\int_{\gamma_{k}}\alpha_{k}(x)\ J(h,x,v_{k}(x))\
dx=\ell_{k}(h),$

as required for the final claim of Proposition 2.3. ∎

##  6\. Examples

In this section we discuss some examples for which our results apply, namely
we exhibit maps admitting a proper discontinuity for which Theorem 1 holds. In
the first example, concerning two-dimensional piecewise affine expanding maps,
we can say more: our lower bound for the essential spectral radius coincides
with the upper bound known in the literature.

###  6.1. Piecewise affine expanding maps

Let $\beta\in(1,\infty)$ be non-algebraic and $\mathcal{M}=\mathbb{T}^{2}$.
Let $\mathcal{F}_{A}:\mathcal{M}\rightarrow\mathcal{M}$ be defined through its
action on the fundamental domain $[0,1)^{2}$,

(6.1) $\mathcal{F}_{A}\left(x,y\right)=\left([\beta x+y],[2y]\right),$

where we write $[x]=x-\lfloor x\rfloor$ for the fractional part of any real
number $x$. Let $\Gamma=\\{0\\}\times\mathbb{T}^{1}$. Let
$\varphi:\mathcal{M}\rightarrow\mathbb{C}$ be as per Theorem 1, i.e.,
$\mathcal{C}^{1}$ on $\mathcal{M}\setminus\Gamma$, uniformly bounded and
uniformly bounded away from zero. Note that $\mathcal{F}_{A}$ is
$\mathcal{C}^{\infty}$ on $\mathcal{M}\setminus\Gamma$, but for any
$y\in\mathbb{T}^{1}$,

(6.2) $\lim_{x\rightarrow 1^{-}}\mathcal{F}_{A}(x,y)\neq\lim_{x\rightarrow
0^{+}}\mathcal{F}_{A}(x,y).$

According to Section 2, we consider the extension of $\mathcal{F}_{A}$,
denoted $F_{A}:M\rightarrow\mathcal{M}$, where $M=[0,1]\times\mathbb{T}^{1}$.
Furthermore, $M$ is a smooth manifold whose boundary is $\partial
M=\gamma^{\prime}\cup\gamma$, where

$\gamma^{\prime}=\left\\{0\right\\}\times\mathbb{T}^{1},\quad\gamma=\left\\{1\right\\}\times\mathbb{T}^{1}.$

Note that, doing so, we have “doubled” the discontinuity set such that the
extension is well defined. For all $k\geq 1$ we set
$\widetilde{\beta}_{k}=\frac{1}{2}\sum_{\ell=0}^{k-1}\big{(}\frac{\beta}{2}\big{)}^{\ell}$
and $\widetilde{\beta}_{0}=0$. Observe that $\\{\widetilde{\beta}_{k}\\}_{k}$
is a strictly increasing sequence and, in particular, that
$\widetilde{\beta}_{k}\neq\widetilde{\beta}_{k^{\prime}}$ whenever $k\neq
k^{\prime}$. Next, let $v=\left(\begin{smallmatrix}0\\\
1\end{smallmatrix}\right)$ be the unit tangent vector to the curve $\gamma$
(or $\gamma^{\prime}$). One has that141414Here we systematically write
$D\mathcal{F}_{A}^{k}v$ to mean $D\mathcal{F}_{A}^{k-1}DF_{A}v$,
$\mathcal{F}_{A}^{k}\gamma$ to mean $\mathcal{F}_{A}^{k-1}F_{A}\gamma$ and
similar for $\gamma^{\prime}$.

$w_{k}=D\mathcal{F}_{A}^{k}v=\begin{pmatrix}\beta^{k}&2^{k}\tilde{\beta}_{k}\\\
0&2^{k}\end{pmatrix}\begin{pmatrix}0\\\
1\end{pmatrix}=\begin{pmatrix}2^{k}\widetilde{\beta}_{k}\\\
2^{k}\end{pmatrix},$

is tangent to $\mathcal{F}_{A}^{k}\gamma$ (or
$\mathcal{F}_{A}^{k}\gamma^{\prime}$). This shows that both
$\mathcal{F}_{A}^{k}\gamma$ and $\mathcal{F}_{A}^{k}\gamma^{\prime}$ are
unions of straight lines with slopes $\widetilde{\beta}^{-1}_{k}$. Note also
that, for all $x\in\mathcal{F}^{k}_{A}\gamma$,

(6.3)
$\operatorname{Jac}({\mathcal{F}|}_{\mathcal{F}^{k}_{A}\gamma})(x)=\frac{\|D\mathcal{F}_{A}w_{k}\|}{\|w_{k}\|}.$

######  Lemma 6.1.

(i) For each $k\in\mathbb{N}$ and $a\in\mathcal{F}_{A}^{k}\gamma$, there exist
$t\in[0,1],\\{n_{j}\\}_{j=0}^{k-1},m\in\mathbb{Z}$ such that

$a=\left(\beta^{k}+\textstyle\sum_{j=0}^{k-1}\beta^{j}2^{k-1-j}t+\sum_{j=0}^{k-1}\beta^{j}n_{j},\
2^{k}t+m\right).$

(ii) For each $k\in\mathbb{N}$ and $b\in\mathcal{F}_{A}^{k}\gamma^{\prime}$,
there exist $\bar{t}\in[0,1]$,
$\\{\bar{n}_{j}\\}_{j=0}^{k-1},\bar{m}\in\mathbb{Z}$ such that

$b=\left(\textstyle\sum_{j=0}^{k-1}\beta^{j}2^{k-1-j}\bar{t}+\sum_{j=0}^{k-1}\beta^{j}\bar{n}_{j},\
2^{k}\bar{t}+\bar{m}\right).$

###### Proof.

(i) If $k=1$, a direct computation yields

$\mathcal{F}_{A}\gamma=\bigl{\\{}\left([\beta+t],[2t]\right)\bigr{\\}}_{t\in[0,1]},$

which is in agreement with the statement choosing $n_{0}\in\mathbb{Z}$,
$m\in\\{-1,0,1\\}$ depending on $t$. Let us assume that it is true for $k\geq
2$. Then any $a\in\mathcal{F}_{A}^{k+1}\gamma$ is of the form

$\begin{split}&\Biggl{[}\begin{pmatrix}\beta&1\\\
0&2\end{pmatrix}\begin{pmatrix}\beta^{k}+\sum_{j=0}^{k-1}\beta^{j}2^{k-1-j}t+\sum_{j=0}^{k-1}\beta^{j}n_{j}\\\
2^{k}t+m\end{pmatrix}\Biggr{]}\\\
&=\begin{pmatrix}\left[\beta^{k+1}+\sum_{j=0}^{k-1}\beta^{j+1}2^{k-1-j}t+\sum_{j=0}^{k-1}\beta^{j+1}n_{j}+2^{k}t+m\right]\\\
\left[2^{k+1}t+2m\right]\end{pmatrix},\end{split}$

for some $t\in[0,1]$. Therefore, there exist $t\in[0,1]$,
$\widetilde{n},\widetilde{m}\in\mathbb{Z}$ such that the coordinates of $a$
are

$\begin{pmatrix}\beta^{k+1}+\sum_{j=0}^{k}\beta^{j}2^{k-j}t+\sum_{j=1}^{k}\beta^{j}n_{j-1}+\widetilde{n}\\\
2^{k+1}t+\widetilde{m}\end{pmatrix}.$

Renaming the integers $\widetilde{n},\widetilde{m}$ and $n_{j}$ properly, we
have proved the statement for $k+1$ and conclude by induction.

(ii) If $k=1$,

$\mathcal{F}_{A}\gamma^{\prime}=\bigl{\\{}\left(\left[\bar{t}\right],\left[2\bar{t}\right]\right)\bigr{\\}}_{\bar{t}\in[0,1]},$

which is in agreement with the statement choosing $\bar{n}_{0},$
$\bar{m}\in\\{-1,0,1\\}$ depending on $t$. Assuming that is true for $k\geq
2$, any $b\in\mathcal{F}_{A}^{k+1}\gamma^{\prime}$ is of the form

$\begin{split}&\left[\begin{pmatrix}\beta&1\\\
0&2\end{pmatrix}\begin{pmatrix}\sum_{j=0}^{k-1}\beta^{j}2^{k-1-j}\bar{t}+\sum_{j=0}^{k-1}\beta^{j}\bar{n}_{j}\\\
2^{k}\bar{t}+\bar{m}\end{pmatrix}\right]\\\
&=\begin{pmatrix}&\left[\sum_{j=0}^{k-1}\beta^{j+1}2^{k-1-j}\bar{t}+\sum_{j=0}^{k-1}\beta^{j+1}\bar{n}_{j}+2^{k}\bar{t}+\bar{m}\right]\\\
&\left[2^{k+1}\bar{t}+2\bar{m}\right]\end{pmatrix},\end{split}$

for some $\bar{t}\in[0,1]$. Therefore, there are integers $\hat{n}$, $\hat{m}$
such that the coordinates of $b$ are

$\begin{pmatrix}&\sum_{j=0}^{k}\beta^{j}2^{k-j}\bar{t}+\sum_{j=1}^{k}\beta^{j}\bar{n}_{j-1}+\hat{n}\\\
&2^{k+1}\bar{t}+\hat{m}\end{pmatrix},$

which implies the statement for $k+1$. ∎

######  Lemma 6.2.

For any $k\geq 1$,

$\mathcal{F}_{A}^{k}\gamma\cap\mathcal{F}^{k}_{A}\gamma^{\prime}=\emptyset.$

###### Proof.

By Lemma 6.1, if
$a\in\mathcal{F}_{A}^{k}\gamma\cap\mathcal{F}^{k}_{A}\gamma^{\prime}$ for some
$k$ then there exist $t,\bar{t}\in[0,1]$,
$n_{j},\bar{n}_{j},m,\bar{m}\in\mathbb{Z}$, such that

$\begin{pmatrix}&\beta^{k}+\sum_{j=0}^{k-1}\beta^{j}2^{k-1-j}t+\sum_{j=0}^{k-1}\beta^{j}n_{j}\\\
&2^{k}t+m\end{pmatrix}\\\
=\begin{pmatrix}&\sum_{j=0}^{k-1}\beta^{j}2^{k-1-j}\bar{t}+\sum_{j=0}^{k-1}\beta^{j}\bar{n}_{j}\\\
&2^{k}\bar{t}+\bar{m}\end{pmatrix}.$

The equation for the $y$-coordinate implies that, for some $p\in\mathbb{Z}$,

$\bar{t}=t+2^{-k}p.$

Substituting this in the first equation we have

$\beta^{k}+\sum_{j=0}^{k-1}\beta^{j}\left(n_{j}-\bar{n}_{j}-2^{-1-j}p\right)=0.$

Since the the function
$\mathcal{P}(\beta)=\beta^{k}+\sum_{j=0}^{k-1}\beta^{j}\left(n_{j}-\bar{n}_{j}-2^{-1-j}p\right)$
is a polynomial with rational coefficients and $\beta$ is non-algebraic by
assumption, the above equation is never satisfied and we conclude. ∎

In the next lemma we show that $\gamma$ is a proper discontinuity (Definition
2.1) of the map $\mathcal{F}_{A}$. First we construct the $\alpha_{k}$, the
coefficient functions. Let $\gamma_{1}=F\gamma$,
$\gamma_{k+1}=\mathcal{F}^{k}\gamma_{1}$, $k\in\mathbb{N}$. Let
$\alpha_{1}:\gamma_{1}\to\mathbb{C}$ be defined as $\alpha_{1}\equiv 1$. We
then define inductively, for all $k\in\mathbb{N}$,
$\alpha_{k}:\gamma_{k}\to\mathbb{C}$ such that, for a.e. $x\in\gamma_{k}$,

(6.4)
$\alpha_{k+1}(\mathcal{F}x)=\alpha_{k}(x)\varphi(x)^{-1}\operatorname{Jac}({\mathcal{F}|}_{\gamma_{k}})(x)^{-1}.$

This is well defined since $\mathcal{F}:\gamma_{k}\to\gamma_{k+1}$ is
injective, at least a.e. $x\in\gamma_{k}$. That it is injective is due to the
fact that $\gamma_{k}$ is the union of straight lines with slope
$\widetilde{\beta}^{-1}_{k}$ and
$\widetilde{\beta}_{k}\neq\widetilde{\beta}_{k+1}$.

######  Lemma 6.3.

The curve $\gamma$ is a proper discontinuity.

###### Proof.

Assumption (A-0) is immediate as observed above (6.2).

(A-1): By construction (6.4), the required property (2.2) holds.

(A-2): Suppose that, for some $k\geq 2$, $\gamma_{k}\cap(F\gamma\cup
F\gamma^{\prime})$ contains some curve $\widetilde{\gamma}$. This would imply
that $\widetilde{\gamma}$ is a line of slope $\widetilde{\beta}_{k}^{-1}$
since $\gamma_{k}$ is the union of lines of slope
$\widetilde{\beta}_{k}^{-1}$. On the other hand it also implies that
$\widetilde{\gamma}$ is a line of slope $\widetilde{\beta}_{1}^{-1}$ since
$\widetilde{\gamma}\subset(F\gamma\cup F\gamma^{\prime})$. This is a
contradiction since $\widetilde{\beta}_{k}\neq\widetilde{\beta}_{1}$.

(A-3): Let $\widetilde{\gamma}\subseteq\widehat{\Gamma}$ be defined as

$\widetilde{\gamma}=\left(\mathcal{F}_{A}^{-1}\gamma_{k+1}\cap\widehat{\Gamma}\right)\setminus\mathcal{F}_{A}^{k}\gamma,$

and assume, for sake of contradiction, that $\widetilde{\gamma}$ has positive
1-dimensional Hausdorff measure. Then, for some $n\in\mathbb{N}$, the set
$\widetilde{\gamma}_{n}=\widetilde{\gamma}\cap\mathcal{F}_{A}^{n}(\gamma^{\prime}\cup\gamma)$
has positive 1-dimensional Hausdorff measure ($\widehat{\Gamma}$ is the
countable union of the $\mathcal{F}_{A}^{n}(\gamma\cup\gamma^{\prime})$,
$n\in\mathbb{N}_{0}$) and it is such that
$\mathcal{F}_{A}\widetilde{\gamma}_{n}\subseteq\mathcal{F}_{A}^{k+1}\gamma$.
Let us consider two separate cases. First, suppose that $n\neq k$. Note that
$\mathcal{F}_{A}\widetilde{\gamma}_{n}\subseteq\mathcal{F}_{A}^{n+1}(\gamma\cup\gamma^{\prime})$
is a union of lines with slope $\widetilde{\beta}_{n+1}^{-1}$ while
$\mathcal{F}_{A}^{k+1}\gamma$ is a union of lines with slope
$\widetilde{\beta}_{k+1}^{-1}$. Since $k\neq n$, we have a contradiction,
because the intersection of two lines with different slopes on the torus
cannot have positive 1-dimensional Hausdorff measure. Let us suppose that
$n=k$. In this case,

$\widetilde{\gamma}_{k}=\widetilde{\gamma}\cap\mathcal{F}_{A}^{n}(\gamma^{\prime}\cup\gamma)\subseteq\mathcal{F}_{A}^{k}\gamma^{\prime}\quad\text{and}\quad\mathcal{F}_{A}\widetilde{\gamma}_{k}\subseteq\mathcal{F}^{k+1}_{A}\gamma,$

which is not possible because of Lemma 6.2, concluding the proof. ∎

#### Comparison with upper bounds on the essential spectrum

Consider the transfer operator associated to a piecewise smooth expanding map
$\mathcal{F}$ with piecewise Hölder weight $\varphi$. It is known [3, 18] that
the essential spectral radius of this operator acting on $BV$ is bounded above
by

(6.5)
$\overline{\Lambda}(\mathcal{F},\varphi)=e^{h_{\mathrm{m}}(\mathcal{F})}\lim_{n\to\infty}\left\|\varphi_{n}\det
D\mathcal{F}^{n}\cdot\lambda_{n}^{-1}\right\|_{L^{\infty}}^{\frac{1}{n}},$

where
$\lambda_{n}(x)=\inf_{\omega\in\Omega^{n}}\inf_{v\in\mathcal{T}_{x}\mathcal{M}:\|v\|=1}\|D_{x}\mathcal{F}^{n}_{\omega}\|$
and151515Note that $h_{\mathrm{m}}$ depends also on the partition $\Omega$.

(6.6) $h_{\mathrm{m}}(\mathcal{F})=\lim_{n\to\infty}\frac{1}{n}\log\max_{x\in
M}\\#\left\\{\omega_{n}\in\Omega^{n}\mid\overline{\omega_{{n}}}\ni
x\right\\},$

$\Omega^{n}$ being the partition associated to $\mathcal{F}^{n}$ made of
cylinder sets. The quantity $h_{\mathrm{m}}(\mathcal{F})$ provides a measure
of the growth of the intersection multiplicities of a discontinuity in a
single point. Notice that
$\Lambda_{BV}(\mathcal{F},\varphi,\gamma)\leq\overline{\Lambda}(\mathcal{F},\varphi)$,
as it should be. In the affine case, such as $\mathcal{F}_{A}$, equality
holds, as shown in the following.

######  Lemma 6.4.

Let $\mathcal{F}_{A}$, $\beta$ and $\gamma$ be as above. Then

$\displaystyle\overline{\Lambda}(\mathcal{F}_{A},1)$
$\displaystyle=\Lambda_{BV}(\mathcal{F}_{A},1,\gamma)=\max\\{2,\beta\\},$
$\displaystyle\overline{\Lambda}(\mathcal{F}_{A},{\left|\det
D\mathcal{F}_{A}\right|}^{-1})$
$\displaystyle=\Lambda_{BV}(\mathcal{F}_{A},{\left|\det
D\mathcal{F}_{A}\right|}^{-1},\gamma)=\min\\{2,\beta\\}^{-1}.$

###### Proof.

Following the definition (2.3), to determine
$\Lambda_{{BV}}(\mathcal{F},1,\gamma)$ we calculate
$\smash{|\operatorname{Jac}({\mathcal{F}_{A}^{k}|}_{\gamma})|^{{1}/{k}}}$. For
all $\beta\in(1,\infty)$, the curve $\gamma$ is not aligned to the direction
of minimal expansion. Therefore, its slope approaches the direction of the
eigenvector corresponding to the maximal eigenvalue, i.e.,
$\max\\{2,\beta\\}$. In this way, one gets that
$\smash{\operatorname{Jac}({\mathcal{F}|}_{\gamma_{k}})}$ tends to
$\max\\{2,\beta\\}$ as $k\rightarrow\infty$ and we can conclude. Similarly, in
order to determine $\Lambda_{{BV}}(\mathcal{F},{|\det
D\mathcal{F}_{A}|}^{-1},\gamma)$ we calculate $({\left|\det
D\mathcal{F}_{A}\right|}^{-1}\operatorname{Jac}({\mathcal{F}_{A}^{k}|}_{\gamma}))^{\frac{1}{k}}$.
Since $\left|\det D\mathcal{F}_{A}\right|=2\beta$,
$\Lambda_{{BV}}(\mathcal{F},{\left|\det
D\mathcal{F}_{A}\right|}^{-1},\gamma)=2^{-1}\beta^{-1}\max\\{2,\beta\\}=\min\\{2,\beta\\}^{-1}$.

Let us then compute the upper bounds. By [8, Lemma 1] we have
$h_{\mathrm{m}}(\mathcal{F}_{A})=0$. Moreover,

$\det
D\mathcal{F}^{n}_{A}=(2\beta)^{n},\quad\lambda_{n}=\min\\{2,\beta\\}^{n}.$

Therefore, by (6.5), $\overline{\Lambda}(\mathcal{F}_{A},1)=\max\\{2,\beta\\}$
whereas $\overline{\Lambda}(\mathcal{F}_{A},{|\det
D\mathcal{F}_{A}|}^{-1})=\min\\{2,\beta\\}^{-1}$. ∎

######  Remark 6.5.

All indications suggest that a similar argument would imply the equality of
upper and lower bounds for a larger class of piecewise affine expanding maps,
whenever the discontinuity set is nowhere aligned to the direction of minimal
expansion, as the ones introduced in [8]. It remains to understand any
possible relations between $\Lambda_{BV}$ and the Lyapunov exponents for
general two-dimensional maps with discontinuities.

###  6.2. Cocycles with base which fails to be Markov

Let $\mathcal{M}=\mathbb{T}^{2}$ and let
$\mathcal{F}_{B}:\mathcal{M}\rightarrow\mathcal{M}$ be defined as

$\mathcal{F}_{B}(x,y)=\left(T(x),S(x,y)\right),$

where $T:\mathbb{T}^{1}\to\mathbb{T}^{1}$ is piecewise $\mathcal{C}^{2}$
without critical points and $S$ is
$\mathcal{C}^{2}(\mathbb{T}^{2},\mathbb{T}^{1})$ without critical points such
that $S(x,\cdot):\mathbb{T}^{1}\rightarrow\mathbb{T}^{1}$ is surjective for
all $x$. We assume that $T$ is $\mathcal{C}^{1}$ on
$\mathbb{T}^{1}\setminus\\{a\\}$ for some $a\in\mathbb{T}^{1}$. Let
$\varphi={|\det D\mathcal{F}_{B}|}^{-1}$, i.e., in this example we specialise
to the case where the transfer operator is the one corresponding to the SRB
measure. For convenience we use the notation $a_{+}$, $a_{-}$ to write
$T^{k}a_{+}=\lim_{\epsilon\rightarrow 0}T^{k}(a+\epsilon)$ and
$T^{k}a_{-}=\lim_{\epsilon\rightarrow 0}T^{k}(a-\epsilon)$. Finally we suppose
that $\\{T^{k}a_{+}\\}_{k=0}^{\infty}$ is a finite set of distinct points
whilst $\\{T^{k}a_{-}\\}_{k=0}^{\infty}$ is an infinite set of distinct
points.

The assumptions allow us to choose $T$ to be a $\beta$-map where $\beta$ is
chosen such that the map is non-Markov. However the class of maps we consider
here is much larger, there is no requirement for $T$ to be affine. The
assumptions imply that $T$ is non-Markov and

(6.7) $\begin{split}T^{k}a_{+}&\neq T^{j}a_{-},\quad\forall
j,k\in\mathbb{N}\\\ T^{k}a_{-}&\neq T^{j}a_{-},\quad j\neq k.\end{split}$

Let $\Delta=\bigcup_{j=0}^{\infty}\\{T^{j}a_{+},T^{j}a_{-}\\}$. In the present
case we can apply the result available for one-dimensional maps with
discontinuities ([5, Lemma 3.2] with $k_{0}=1$) and consequently

(6.8) $T^{-1}(T^{k+1}a_{-})\cap\Delta=T^{k}a_{-},\quad k\geq 1.$

Let $\Gamma=\\{a\\}\times\mathbb{T}^{1}$. The assumptions on $T$ and $S$ imply
that $\mathcal{F}_{B}$ is $\mathcal{C}^{2}$ on
$\mathbb{T}^{2}\setminus\Gamma$. Following the notation of Section 2 we
consider the smooth manifold $M$ with boundary $\partial
M=\gamma^{\prime}\cup\gamma$, where

$\quad\gamma=\\{a_{-}\\}\times\mathbb{T}^{1},\quad\gamma^{\prime}=\\{a_{+}\\}\times\mathbb{T}^{1}.$

Following the notations of Definition 2.1,
$\gamma_{1}=\\{Ta_{-}\\}\times\mathbb{T}^{1}$ and, for all $k\in\mathbb{N}$,
$\gamma_{k}=\\{T^{k}a_{-}\\}\times\mathbb{T}^{1}$ (since $S(x,\cdot)$ is
surjective). In this particular case we may choose the functions $\alpha_{k}$
to be constant on $\gamma_{k}$. Let, for $k\geq 2$,

$\alpha_{k}=(T^{k-1})^{\prime}(Ta_{-}).$

Note that, since $\varphi={|\det D\mathcal{F}_{B}|}^{-1}$, for all
$k\in\mathbb{N}$, $x\in\gamma_{k}$,

(6.9)
$\varphi(x)\operatorname{Jac}({\mathcal{F}_{B}|}_{\gamma_{k}})(x)={\left|\smash{T^{\prime}(T^{k}a_{-})}\right|}^{-1}.$

######  Lemma 6.6.

The curve $\gamma$ is a proper discontinuity.

###### Proof.

Assumption (A-0) is satisfied since $Ta_{+}\neq Ta_{-}$ (6.7).

(A-1): Using the above (6.9),

$\displaystyle\alpha_{k+1}$
$\displaystyle=(T^{(k+1)-1})^{\prime}(Ta_{-})=(T^{k-1})^{\prime}(Ta_{-})\cdot|T^{\prime}(T^{k}a_{-})|$
$\displaystyle=\alpha_{k}\cdot\varphi(x)^{-1}\cdot\operatorname{Jac}({\mathcal{F}_{B}|}_{\gamma_{k}})(x)^{-1},$

so proving (A-1).

(A-2): Again, following the notation of Definition 2.1,
$\Gamma_{1}=\\{Ta_{+},Ta_{-}\\}\times\mathbb{T}^{1}$ (using the fact that
$S(x,\cdot)$ is surjective). Since
$\gamma_{k}=\\{T^{k}a_{-}\\}\times\mathbb{T}^{1}$, the relations between the
images of the discontinuity (6.7) implies that, for all $k\geq 2$,
$\gamma_{k}\cap\Gamma_{1}=\emptyset$

(A-3): Following the notation of Section 2,
$\widehat{\Gamma}=\bigcup\nolimits_{j\in\mathbb{N}}\mathcal{F}^{j}\Gamma_{1}$
and so (again using the fact that $S(x,\cdot)$ is surjective),

$\widehat{\Gamma}=\left({\bigcup\nolimits_{j\in\mathbb{N}}}\left\\{T^{j}a_{-}\right\\}\times\mathbb{T}^{1}\right)\cup\left({\bigcup\nolimits_{j\in\mathbb{N}}}\left\\{T^{j}a_{+}\right\\}\times\mathbb{T}^{1}\right).$

For $k\in\mathbb{N}$,

$\begin{split}\mathcal{F}_{B}^{-1}\gamma_{k+1}&=\mathcal{F}_{B}^{-1}\left(\left\\{T^{k+1}a_{-}\right\\}\times\mathbb{T}^{1}\right)=\left\\{T^{-1}\left(T^{k+1}a_{-}\right)\right\\}\times\mathbb{T}^{1}.\end{split}$

However, using (6.8),

$\begin{split}\left(\mathcal{F}_{B}^{-1}\gamma_{k+1}\right)\cap\widehat{\Gamma}&=\bigcup_{j\in\mathbb{N}}\left(\left\\{T^{-1}(T^{k+1}a_{-})\cap
T^{j}(\\{a_{+}\\}\cup\\{a_{-}\\})\right\\}\times\mathbb{T}^{1}\right)\\\
&=\\{T^{k}a_{-}\\}\times\mathbb{T}^{1}=\gamma_{k},\end{split}$

showing (A-3) and concluding the proof. ∎

######  Remark 6.7.

Calculating the estimates (2.3), using the formula previously obtained (6.9),

$\Lambda_{BV}(\mathcal{F}_{B},\left|\det
D\mathcal{F}_{B}\right|^{-1},\gamma)=\liminf_{n\to\infty}\left|(T^{n})^{\prime}(a_{-})\right|^{-1/n}.$

Since, in this present setting, $\left|\gamma_{k}\right|$ is uniformly
bounded, $\Lambda_{L^{\infty},2}$ will be equal to $\Lambda_{BV}$. On the
other hand $\Lambda_{L^{\infty},1}$ would give a worse (if there is expansion
in the second coordinate) or equal (if neutral behaviour in second coordinate)
bound. The estimates on the essential spectral radius for this class of
examples are equal to the estimates obtained [5] for the one-dimensional map
$T$.

######  Remark 6.8.

A broad class of interesting examples fits with the above, in particular skew
products on $\mathbb{T}^{2}$ of the form

$(x,y)\mapsto(T(x),y+\sigma(x)).$

If $T$ is a piecewise smooth expanding non-Markov map satisfying the
assumptions above with $T^{\prime}\geq\lambda>1$ and $\sigma$ smooth (this is
a partially hyperbolic system with a neutral direction) we obtain, for the
transfer operator with weight as above, the estimates,
$\Lambda_{BV}=\Lambda_{L^{\infty}}\geq\lambda^{-1}$.161616See [11] for upper
bounds of the essential spectral radius (in the smooth case) and [6] for a
proof of exponential mixing in the presence of discontinuities.

###  6.3. Local discontinuity

Let $\mathcal{M}=\mathbb{T}^{2}$. This example, denoted
$\mathcal{F}_{C}:\mathcal{M}\rightarrow\mathcal{M}$, to be defined shortly, is
a perturbation of a smooth uniformly expanding map in an arbitrary small
neighbourhood, yet has large essential spectrum. The only discontinuity will
be an $\epsilon$-sized “slit” which is introduced by the perturbation.

Let $f:\mathbb{T}^{1}\rightarrow\mathbb{T}^{1}$ be the doubling map defined as
$f:x\mapsto 2x\mod 1$. For notational convenience, let
$x_{0}=y_{0}=\frac{1}{2}$. For $k\in\mathbb{N}$ let
$a_{k}=f^{k}(x_{0}-\epsilon)$ and choose, once and for all, $\epsilon>0$ small
and such that ${\\{a_{k}\\}}_{k\in\mathbb{N}}$ is an infinite set of distinct
points, disjoint from the interval $[x_{0},x_{0}+\epsilon]$. Let
$J=[x_{0},x_{0}+\epsilon]\times[y_{0}-\epsilon,y_{0}+\epsilon]\subset\mathbb{T}^{2}$.
Fix $\rho\in\mathcal{C}^{\infty}(\mathbb{R},\mathbb{R})$ such that $\rho(x)=1$
when $x\leq 0$, that $\rho(x)=0$ when $x\geq 1$ and that, for all
$x\in\mathbb{R}$, $\rho^{\prime}(x)\leq 0$. Let
$\rho_{\epsilon}(x)=\epsilon\rho(\epsilon^{-1}x)$. Let
$T,S:\mathbb{T}^{2}\rightarrow\mathbb{T}^{2}$ be defined as

$\begin{split}T&:(x,y)\mapsto(f(x),f(y)),\\\
S&:(x,y)\mapsto\begin{cases}(x,y)&\text{if $(x,y)\in\mathbb{T}^{2}\setminus
J$}\\\ (x-\rho_{\epsilon}(x-x_{0}),y)&\text{if $(x,y)\in
J$}\end{cases}\end{split}$

and hence let $\mathcal{F}_{C}:\mathcal{M}\rightarrow\mathcal{M}$ be defined
as

$\mathcal{F}_{C}=T\circ S.$

Note that $\Gamma$, the set of discontinuities for $\mathcal{F}_{C}$, is
composed of the vertical line segment
$\\{(x_{0},y_{0}+t):t\in[-\epsilon,\epsilon]\\}$ and the two horizontal line
segments, $[x_{0},x_{0}+\epsilon]\times\\{y_{0}-\epsilon\\}$ and
$[x_{0},x_{0}+\epsilon]\times\\{y_{0}+\epsilon\\}$. As per Section 2, we
consider the smooth manifold with boundary, which we denote $M$, such that
$\mathcal{F}_{C}:M\to\mathcal{M}$ is $\mathcal{C}^{\infty}$. We define

$\gamma=\\{x_{0}^{+}\\}\times[y_{0}-\epsilon,y_{0}+\epsilon],\quad\gamma^{\prime}=\\{x_{0}^{-}\\}\times[y_{0}-\epsilon,y_{0}+\epsilon],$

where, as per the previous example, $x_{0}^{+}$ and $x_{0}^{-}$ correspond to
the doubling of points at
$\\{x_{0}\\}\times[y_{0}-\epsilon,y_{0}+\epsilon]\subset\mathcal{M}$. Note
that $\gamma$ and $\gamma^{\prime}$ are the only vertical components of
$\partial M$. The definition of $a_{k}$ has the consequence that
$\mathcal{F}_{C}^{k}(x_{0}^{+},y)=(a_{k},f^{k}(y))$. Let $\varphi=\left|\det
D\mathcal{F}_{C}\right|^{-1}$ and consider, as specified in Theorem 1, the
associated transfer operator $\mathcal{L}$. Observe that
$\varphi(x,y)=\frac{1}{4}$ whenever $(x,y)\in\mathbb{T}^{2}\setminus J$.
Following the notation of Definition 2.1, $\gamma_{1}=F_{C}\gamma$ and,
$\gamma_{k+1}=\mathcal{F}_{C}^{k}\gamma_{1}$, $k\in\mathbb{N}$. As such,
$\gamma_{k}\subset\\{a_{k}\\}\times\mathbb{T}^{1}$.

######  Lemma 6.9.

The curve $\gamma$ is a proper discontinuity.

###### Proof.

We choose $\alpha_{k}\equiv 2^{k-1}$, $k\in\mathbb{N}$. Note that each
$\gamma_{k}$ is a vertical line whose first coordinate is equal to $a_{k}$.
The choice of $\epsilon$ means that $\gamma_{k}$ never intersects $J$. For
small $k$ these lines will have length $2^{k}\epsilon$ and for large $k$ they
are equal to $\\{a_{k}\\}\times\mathbb{T}^{1}$. We will, in turn, check each
of the required properties.

(A-0): This assumption is satisfied since
$\mathcal{F}_{C}(x_{0}^{+},y)=(f(x_{0}-\epsilon),f(y))$ whereas
$\mathcal{F}_{C}(x_{0}^{-},y)=(f(x_{0}),f(y))$ and $f(x_{0}-\epsilon)\neq
f(x_{0})$.

(A-1): We see that $\alpha_{1}\equiv 1$ as required and so it remains to show
that, for all $x\in\gamma_{k}$, $k\in\mathbb{N}$,

$\alpha_{k+1}(\mathcal{F}_{C}x)=\alpha_{k}(x)\varphi(x)^{-1}\operatorname{Jac}({\mathcal{F}_{C}|}_{\gamma_{k}})(x)^{-1}.$

For every $x\in\gamma_{k}$ we know that
$\operatorname{Jac}({\mathcal{F}_{C}|}_{\gamma_{k}})(x)=2$ and
$\varphi(x)=\frac{1}{4}$ so, consequently,
$\varphi(x)\operatorname{Jac}({\mathcal{F}_{C}|}_{\gamma_{k}})\equiv\frac{1}{2}$.
And so, since $\alpha_{k}\equiv 2^{k-1}$, (A-1) is satisfied.

(A-2): We must show that $\gamma_{k}\cap\Gamma_{1}\approxeq\emptyset$ for all
$k\geq 2$. We know that $\Gamma_{1}$ is made up of horizontal and vertical
line segments. However, since $\gamma_{k}$ consists only of a vertical
segment, we can avoid considering the horizontal segments. The vertical
segments of $\Gamma_{1}$ are contained within
$\\{f(x_{0}-\epsilon),f(x_{0})\\}\times\mathbb{T}^{1}=\\{a_{1},f(x_{0})\\}\times\mathbb{T}^{1}$.
On the other hand, $\gamma_{k}\subset\\{a_{k}\\}\times\mathbb{T}^{1}$. Since
$\\{a_{k}\\}_{k}$ was assumed to be an infinite set of points, we know that
$a_{n}\neq a_{k}$, if $n\neq k$, and $a_{k}\neq x^{+}_{0},x^{-}_{0},0$, for
$k\in\mathbb{N}$ (otherwise $a_{k}$ would be pre-periodic). Consequently the
required conclusion holds.

(A-3): We must show that
$\mathcal{F}_{C}^{-1}\gamma_{k+1}\cap\widehat{\Gamma}\approxeq\gamma_{k}$ for
all $k\in\mathbb{N}$. Again we can ignore the horizontal component of the
discontinuity. Observe that
$\widehat{\Gamma}\subset\left(\cup_{k=1}^{\infty}\\{a_{k}\\}\cup\\{0\\}\right)\times\mathbb{T}^{1}$.
Furthermore, the $a_{k}$ have the property (see [5, Lemma 3.2])

$\begin{split}&f^{-1}(a_{k+1})\cap\left(\cup_{j=1}^{\infty}\\{a_{j}\\}\cup\\{0\\}\right)=a_{k},\quad
k\geq 1,\end{split}$

which readily implies (A-3). ∎

######  Lemma 6.10.

$\Lambda_{BV}(\mathcal{F}_{C},\left|\det
D\mathcal{F}_{C}\right|^{-1},\gamma)=\Lambda_{L^{\infty}}(\mathcal{F}_{C},\left|\det
D\mathcal{F}_{C}\right|^{-1},\gamma)=\frac{1}{2}$.

###### Proof.

Recall that, for $x\in\gamma_{k}$,
$\varphi(x)\operatorname{Jac}({\mathcal{F}_{C}|}_{\gamma_{k}})(x)=\frac{1}{2}$.
Consequently (2.3),

$\Lambda_{\operatorname{BV}}=\liminf_{k\rightarrow\infty}\left(\smash{\inf_{x\in\gamma_{k}}}\left|\varphi_{k}(x)\cdot\operatorname{Jac}({\mathcal{F}_{C}^{k}|}_{\gamma})(x)\right|\right)^{1/k}=\tfrac{1}{2},$

Additionally, for all $k$ sufficiently large, $\left|\gamma_{k}\right|=1$ and
so $\Lambda_{L^{\infty}}=\Lambda_{L^{\infty},2}$. ∎

######  Remark 6.11.

In the second and third example (Section 6.2, Section 6.3), we considered the
case where the weight of transfer operator was chosen to be $\left|\det
D\mathcal{F}\right|^{-1}$. Arguably this is the most relevant of transfer
operators but other choices of weight are also of significant interest. In
both these examples the present choice was a great convenience and led to the
possibility of choosing $\alpha_{k}$ constant. In the second example we took
advantage of the fact that
$\varphi(x)\operatorname{Jac}({\mathcal{F}_{B}|}_{\gamma_{k}})(x)$ is
constant. This simplifies the choice of the $\alpha_{k}$ in cases like this
and it is not clear how to proceed in full generality (see the discussion of
(A-1) in Section 3). In the third example any weight which was constant on
$\mathbb{T}^{2}\setminus J$ would suffice without modifying the argument as
can be seen in the proof of Lemma 6.9.

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