# Discontinuities cause essential spectrum

Oliver Butterley  (Oliver Butterley) Dipartimento di Matematica, II Università
di Roma “Tor Vergata” – Via della Ricerca Scientifica 1 – 00133 Roma – Italy
<EMAIL_ADDRESS>,  Giovanni Canestrari  (Giovanni Canestrari)
Dipartimento di Matematica, II Università di Roma “Tor Vergata” – Via della
Ricerca Scientifica 1 – 00133 Roma – Italy<EMAIL_ADDRESS>and
Sakshi Jain  (Sakshi Jain) Dipartimento di Matematica, II Università di Roma
“Tor Vergata” – Via della Ricerca Scientifica 1 – 00133 Roma – Italy
<EMAIL_ADDRESS>
###### Abstract.

We study transfer operators associated to piecewise monotone interval
transformations and show that the essential spectrum is large whenever the
Banach space bounds $L^{\infty}$ and the transformation fails to be Markov.
Constructing a family of Banach spaces we show that the lower bound on the
essential spectral radius is optimal. Indeed, these Banach spaces realise an
essential spectral radius as close as desired to the theoretical best possible
case.

##  1\. Introduction

Describing the statistical properties of a hyperbolic dynamical system is an
important question and one powerful approach to this problem is to study the
associated transfer operator acting on a “good” choice of Banach space (see
[30] and references within). In recent decades these techniques have seen
massive development and many systems were studied in this way with valuable
results being obtained [26]. Results include statistical properties (invariant
measures, CLT, LLT, decay of correlations, large deviations, zeta functions,
etc.), stability of these properties under perturbations [27] and rigorous
numerics [29]. Different Banach spaces were used in various applications and
often it was beneficial to engineer appropriate Banach spaces. At present we
have, a “bestiary” [37] of available spaces, each with benefits for different
settings and purposes.

Arguably, particularly with physical applications in mind, discontinuities are
natural. For example, in billiard systems discontinuities arise from
tangential collisions [13] and in Lorenz-like flows discontinuities arise from
shearing close to the singularity [41]. A weight of evidence suggests that
discontinuities don’t worsen the statistical properties of the system even if
they cause technical difficulties (e.g., [2, 9]).

Let $\mathcal{L}:\mathfrak{B}\to\mathfrak{B}$ be a transfer operator
associated to some dynamical system. Since the Banach space must contain the
observables of interest, we take the point of view that, at the very least, it
includes $\mathcal{C}^{\infty}$. Consequently, it also contains
$\mathcal{L}^{n}(\mathcal{C}^{\infty})$ for all $n\in\mathbb{N}$ and so
discontinuities of the underlying system force the Banach space to contain
observables with discontinuities. On the other hand, if the space is big, the
spectrum can be huge and consequently useless (further details in the
following section). The aim is to choose a Banach space for which the
essential spectrum is as small as possible. Once this is done, one may proceed
to determine the point spectrum or “resonances” of the system (see, e.g., [12,
18] and references within). Moreover the point spectrum is essentially
independent of the choice of Banach space, it is inherent in the system
studied [7].

Our present focus is piecewise monotone interval transformations. Classically
these are studied using $\operatorname{BV}$ and the essential spectral radius
is equal to a value depending on the expansion [24]. However, for $C^{\infty}$
systems without discontinuities, the essential spectral radius can be made as
small as desired by choosing a Banach space of smoother observables [15] and
so we ask ourselves:

######  Question.

Does there exist a Banach space for studying piecewise monotone interval
transformations on which the essential spectrum is as small as desired?

Somewhat unfortunately, one success of this present work is to answer this
question in the negative when the transformation fails to be Markov.111A
piecewise monotone transformation $T:[0,1]\to[0,1]$ is said to be _Markov_ if
it admits a finite partition $\Omega$ which additionally has the property
that, for any $\omega,\omega^{\prime}\in\Omega$, if
$T\omega\cap\omega^{\prime}\neq\emptyset$ then
$T\omega\supseteq\omega^{\prime}$. In Theorem 1, we show the existence of
essential spectrum caused by the discontinuities of the transformation and
which cannot be avoided by choice of Banach space. The key idea is to
construct eigenvalues of the dual operator in a way that relies on the
discontinuities. Consequently this component of the spectrum has a different
origin compared to the essential spectrum studied previously [15] and which is
due to limited differentiability. This part of the work gives a lower estimate
on the essential spectral radius which holds for any Banach space. In Theorem
2 we construct an example of a piecewise expanding interval transformation for
which this theoretical lower estimate is strictly smaller than the essential
spectral radius studying the same system on $\operatorname{BV}$ [24].
Consequently the natural next question is:

######  Question.

Given a piecewise monotone interval transformation which fails to be Markov,
does there exist a Banach space on which the essential spectral radius is
smaller than for $\operatorname{BV}$, closer to the theoretical best?

In Theorem 3 we answer this question in the affirmative by demonstrating how,
for any given expanding interval transformation, it is possible to construct a
Banach space so that the essential spectral radius is arbitrarily close to the
theoretical best case we have established in Theorem 1.

Intimately connected to the present work and of physical relevance is the task
of understanding when the correlations of $C^{\infty}$ observables admit a
description in terms of resonances (see definition and discussion in [18,
§1]). It is known (see [36, 6, 21, 18] and citations within) that all
uniformly expanding and uniformly hyperbolic $\mathcal{C}^{\infty}$ maps (in
any dimension) admit Ruelle resonances to arbitrary precision. This includes
Markov expanding systems and pseudo Anosov diffeomorphisms, which are
$\mathcal{C}^{\infty}$ in the sense that, apart from a set of
discontinuities/singularities, they are $\mathcal{C}^{\infty}$ and this set is
both invariant under the dynamics and has zero measure. Essentially all proofs
of resonances rely on spectral considerations for the transfer operator. As
such, the present work demonstrates that discontinuities are an obstacle to
the precision with which the correlations of observables can be described by
resonances.

##  2\. Results

We say that $T:[0,1]\to[0,1]$ is a $\mathcal{C}^{r}$ _piecewise monotone
transformation_ if there exists a finite partition $\Omega$ of a full measure
subset of $[0,1]$ into open intervals such that $\left.T\right|_{\omega}$ is
strictly monotone and $r$-times differentiable with continuous derivative for
each $\omega\in\Omega$. Let $\mathcal{I}$ denote the disjoint union of
$\overline{\omega}$, $\omega\in\Omega$ 222In the rest of the article, it will
be convenient to see $T$ as a transformation from $\mathcal{I}$ to
$\mathcal{I}$ rather than from $I$ to $I$. For each $\omega\in\Omega$, $T$ has
a unique $\mathcal{C}^{0}$ extension to $\overline{\omega}$. In such a way,
$T$ can be considered as a transformation from $\mathcal{I}$ to itself.. We
suppose throughout that $\Omega$ denotes the maximal such partition. The
_transfer operator_ is defined333If $\varphi$ is constant, or if
$\varphi=1/{\left|T^{\prime}\right|}$ then $\mathcal{L}_{\varphi}$ is the
operator associated to the measure of maximal entropy or the SRB measure,
respectively (see e.g., [5]). pointwise for any $h:[0,1]\to\mathbb{C}$,

(1) $\mathcal{L}_{\varphi}h=\sum_{\omega\in\Omega}(\varphi\cdot
h)\circ{{\left.T\right|}_{\omega}^{-1}}\cdot\mathbf{1}_{T\omega}.$

Note that $\mathcal{L}_{\varphi}$ doesn’t necessarily
leave444$h\in\mathcal{C}^{\infty}(\mathcal{I})$ means that
$h:[0,1]\to\mathbb{C}$ and that
$\left.h\right|_{\omega}\in\mathcal{C}^{\infty}(\omega)$ for each
$\omega\in\Omega$. $\mathcal{C}^{\infty}(\mathcal{I})$ invariant, particularly
in the cases of present interest. We say that $a\in\mathcal{I}$ is a _non-
trivial_ discontinuity of $T$ if $a\in\partial\omega$, $\omega\in\Omega$, and
${\\{T^{k}a\\}}_{k=1}^{\infty}$ is an infinite set. For each such non-trivial
discontinuity of $T$ we consider the quantities

$\displaystyle\Lambda^{\inf}({T,\varphi,a})$
$\displaystyle=\liminf_{n\to\infty}\textstyle\left|\prod_{k=1}^{n}\varphi(T^{k}a)\right|^{\frac{1}{n}},$
$\displaystyle\Lambda^{\sup}({T,\varphi,a})$
$\displaystyle=\limsup_{n\to\infty}\textstyle\left|\prod_{k=1}^{n}\varphi(T^{k}a)\right|^{\frac{1}{n}}.$

A given transformation might have more than one non-trivial discontinuity but,
at most, a finite number. We define $\Lambda^{\inf}(T,\varphi)$ and
$\Lambda^{\sup}(T,\varphi)$ to be equal to the maximum of
$\Lambda^{\inf}({T,\varphi,a})$ and $\Lambda^{\sup}({T,\varphi,a})$, over all
non-trivial discontinuities, respectively. If $T$ has no non-trivial
discontinuities, i.e., is Markov,555Markov implies the set of all images of
discontinuities is a finite set and so there is no non-trivial discontinuity.
On the other hand, no non-trivial discontinuities means the set of all images
of discontinuities is a finite set and this set describes a Markov partition.
we set $\Lambda^{\inf}(T,\varphi)$ and $\Lambda^{\sup}(T,\varphi)$ equal to
zero. Whenever it happens that
$\Lambda^{\inf}(T,\varphi)=\Lambda^{\sup}(T,\varphi)$, we set
$\Lambda^{\inf}(T,\varphi)=\Lambda^{\sup}(T,\varphi)=\Lambda(T,\varphi)$.

Our first result gives a lower bound for the essential spectral radius.

######  Theorem 1.

Let $\mathcal{L}_{\varphi}$ be the transfer operator associated to a
$\mathcal{C}^{0}$ piecewise monotone transformation $T:[0,1]\to[0,1]$ and let
$\varphi:[0,1]\to\mathbb{C}$ be bounded666In particular, $\varphi$ admits a
continuous extension up to $\overline{\omega}$, for all $\omega\in\Omega$. and
$\mathcal{C}^{0}$ on each $\omega\in\Omega$. Suppose that
$(\mathfrak{B},\left\|\cdot\right\|_{\mathfrak{B}})$ is a Banach space
satisfying,

  1. (1)

$\mathcal{C}^{\infty}(\mathcal{I})\subseteq\mathfrak{B}$;

  2. (2)

$\mathcal{L}_{\varphi}$ extends to a continuous operator on $\mathfrak{B}$;

  3. (3)

${\left\|h\right\|}_{L^{\infty}}\leq\left\|h\right\|_{\mathfrak{B}}$ for all
$h\in\bigcup_{n=0}^{\infty}\mathcal{L}^{n}_{\varphi}(\mathcal{C}^{\infty}(\mathcal{I}))$.

If $T$ has a non-trivial discontinuity $a\in[0,1]$ then the essential spectral
radius of $\mathcal{L}_{\varphi}:\mathfrak{B}\rightarrow\mathfrak{B}$ is not
less than $\Lambda^{\inf}(T,\varphi,a)$.

The proof of this result777We prove the slightly stronger statement
$\operatorname{\underline{EssSpec}}(\mathcal{L}_{\varphi})\supseteq\\{\left|\lambda\right|\leq\Lambda^{\inf}\\}$
(see Section 4). is the content of Section 3 and takes advantage of some
details related to the essential spectrum which are described in Section 4.

In particular, this theorem means that, when a system fails to be Markov, even
increasing the regularity of observables by choosing suitable Banach spaces,
it is not possible to obtain a precision better that
$\Lambda^{\inf}(T,\varphi)$ in a Ruelle resonance description of correlations.

A particular case of the theorem is when $\varphi$ is a constant as happens
for the transfer operator associated to the measure of maximal entropy. In
this case Theorem 1 gives a lower estimate for the essential spectral radius
which is equal to the known [4] upper estimate for the essential spectral
radius when using $\operatorname{BV}$ (equal to the constant value of
$\varphi$). In other words, when studying the transfer operator associated to
the measure of maximal entropy, if there is a non-trivial discontinuity, then
it is impossible to get a smaller essential spectral radius than the one
obtained by using $\operatorname{BV}$.

The $\beta$-transformations defined as $x\mapsto\beta x\mod 1$ for
$\beta\in(1,2]$ (see, e.g., [14, 34]) fit the setting of this present study.
In this case, also for the transfer operator associated to the SRB measure,
the weight $\varphi$ is constant because the derivative is constant. Again we
see that, for all $\beta$ such that the transformation fails to be Markov, the
essential spectral radius can’t be smaller than $\frac{1}{\beta}$ (the
essential spectral radius in $\operatorname{BV}$).

In the situation where one studies the transfer operator acting on a Banach
space which is large, e.g., $L^{p}$ with $1\leq p<\infty$, then it is possible
to construct approximate eigenfunctions which are piecewise constant and hence
show that the spectrum of the transfer operator is the entire disc of radius
equal to the spectral radius 888Indeed, for all $\epsilon>0$ and
$n\in\mathbb{N}$ there exists a set $A$ such that the sets
$\\{T^{-k}A\\}_{k=0}^{n-1}$ are pairwise disjoint and their measure add up to
$1-\epsilon$ (this can be done using a version of the Rokhlin Lemma for non-
invertible transformations). Then, using this one can define a function which
satisfies pointwise $\mathcal{L}g(x)=\lambda g(x)$ on
$\\{T^{-k}A\\}_{k=0}^{n-2}$ and has unit $L^{p}$ norm, providing approximate
eigenvalues.. Similar results hold when the space is large, even if not as
large as $L^{2}$ (see [5, Remark 1.5] where a disc of eigenvalues is shown to
exist in the case of Hölder spaces).

In the case of $\mathcal{C}^{\infty}$ expanding circle transformations, with
the weight for the transfer operator again given by
$\varphi=1/\left|T^{\prime}\right|$, Collet & Isola [15] studied the essential
spectrum of the transfer operator acting on the space of $\mathcal{C}^{r}$
observables. They showed that the essential spectrum is a full disc which gets
arbitrarily small as $r$ increases.999If $m$ denotes the invariant measure,
the logarithm of the essential spectral radius on $\mathcal{C}^{r}$ is equal
to
$\lim_{n\to\infty}\frac{1}{n}\log\int_{0}^{1}\left|{(T^{n})}^{\prime}\right|^{-r}dm$.
(See [22] for the higher dimensional expanding case studied on
$\mathcal{C}^{r+\alpha}$ spaces.) Similarly, for $\mathcal{C}^{\infty}$
hyperbolic systems, the essential spectrum can be made as small as desired for
a specific choice of the Banach space [21]. In the case when the piecewise
monotone interval transformation is Markov, it is possible to make the
essential spectrum as small as desired (see e.g., [12]). For this reason, the
requirement in Theorem 1 that the transformation has a non-trivial
discontinuity, is essential. However, particularly as the set of
transformations with non-trivial discontinuity is dense amongst the set of all
piecewise monotone transformations,101010If the transformation doesn’t have a
non-trivial discontinuity then the orbit of the discontinuity is pre-periodic.
Iteratively making smaller and smaller perturbations guarantees that the orbit
isn’t pre-periodic and hence the transformation fails to be Markov. the
present work fills in much of the gap that had remained on this topic.

Piecewise monotone interval transformations, the setting of this current work,
have routinely been studied using $\operatorname{BV}$ [31, 35, 24, 26, 4] and
generalized bounded variation [25, 10, 11]. Keller [24, p.184] demonstrated
that previous estimates for the essential spectral radius of transfer
operators acting on $\operatorname{BV}$ are optimal, i.e., the essential
spectral radius is equal to
$\lim_{n}\left\|\smash{\prod_{k=0}^{n-1}\varphi\circ
T^{k}}\right\|_{L^{\infty}}^{1/n}$ in the case when
$\varphi=1/\left|T^{\prime}\right|$. The argument uses a sequence of indicator
functions supported on smaller and smaller intervals in order to produce the
lower bound on the essential spectral radius. Theorem 1 shows that we can’t
have a smaller essential spectrum than we would obtain using
$\operatorname{BV}$ if the transformation has a non-trivial discontinuity and
$\left\|\smash{\prod_{k=0}^{n-1}\varphi\circ
T^{k}}\right\|_{L^{\infty}}^{1/n}$ is comparable to
$\Lambda^{\inf}_{T,\varphi}$. The first quantity is never smaller than the
second quantity but this raises the question: Is there a transformation where
the first is strictly greater? Specializing to the case where
$\varphi=\frac{1}{\left|T^{\prime}\right|}$, we answer this question as
follows.

######  Theorem 2.

For any $c\in(0,1)$, there exists a piecewise monotone interval transformation
$T:[0,1]\to[0,1]$ with a non-trivial discontinuity such that

$\lim_{n\to\infty}\left\|\smash{\tfrac{1}{(T^{n})^{\prime}}}\right\|_{L^{\infty}}^{1/n}-\Lambda({T,\tfrac{1}{\left|T^{\prime}\right|}})>c.$

Section 5 is devoted to the explicit construction of these examples and hence
the proof of above result.

This theorem tells us that there are transformations where the theoretical
best case of Theorem 1 is substantially different from the essential spectral
radius obtained using $\operatorname{BV}$. Hence it could be possible to make
the essential spectral radius smaller by choosing a “better” Banach space. By
constructing specialized Banach spaces we are able to achieve exactly this as
per the following result.

######  Theorem 3.

Suppose that $T:[0,1]\to[0,1]$ is a $\mathcal{C}^{\infty}$ piecewise monotone
transformation (with associated partition $\Omega$) and that
$\varphi:[0,1]\to\mathbb{C}$ admits a $\mathcal{C}^{\infty}$ extension to
$\overline{\omega}$ for each $\omega\in\Omega$ and is uniformly bounded away
from zero. Further suppose that $T$ is uniformly expanding111111In the sense
that there exists $C>0$, $0<\lambda<1$ such that
$\left|(T^{n})^{\prime}\right|\geq C\lambda^{-n}$ and all $n$. and
${\|\varphi\cdot T^{\prime}\|}_{L^{\infty}}<\infty$.

Then, for all $\tilde{\Lambda}>\Lambda^{\sup}(T,\varphi)$, there exists a
Banach space $\mathfrak{B}\subset L^{\infty}([0,1])$ such that,

  1. (1)

$\mathcal{C}^{\infty}(\mathcal{I})\subseteq\mathfrak{B}$;

  2. (2)

$\mathcal{L}_{\varphi}$ extends to a continuous operator on $\mathfrak{B}$;

  3. (3)

The essential spectral radius of
$\mathcal{L}_{\varphi}:\mathfrak{B}\to\mathfrak{B}$ is at most
$\tilde{\Lambda}$.

The proof of the above result is the content of Section 6 where we explicitly
construct the family of Banach spaces. The required distortion result is the
content of Section 7.

This theorem tells that, when
$\Lambda^{\sup}(T,\varphi)=\Lambda^{\inf}(T,\varphi)$, there exists a
particular choice of Banach space such that the essential spectral radius is
as close to the theoretical minimum as desired. However we have to pay a price
for this and the major downside is that the Banach space is very closely tied
to the given system, even similar transformations couldn’t be studied on the
same Banach space which unfortunately would make delicate study of
perturbations impossible (see, e.g., [27]).

The present results don’t say anything about the point spectrum and a high
degree of smoothness of the system doesn’t prevent the existence of plenty of
it [28]. Sometimes it is possible to use other arguments in order to identify
the point spectrum [18, 12] but control on the essential spectrum, as obtained
here, is the key prerequisite to further investigation.

######  Remark (Fractional Sobolev spaces).

An important class of Banach spaces which have been useful for expanding
transformations with discontinuities are the “fractional” Sobolev spaces (see,
e.g., [40, 3, 37]). They have the advantage that they allow the study of
unbounded observables. On the other hand, assumption (3) of Theorem 1 is not
satisfied. Let $p\in\mathbb{N}$, $t\in(0,\frac{1}{p})$ and consider the
factional Sobolev space $H_{p}^{t}$ [40]. The norm of the characteristic
function of an interval is bounded by a constant independent of the interval
[38, Corollary 3.7]. For simplicity consider the case of $T:x\mapsto kx\mod
1$, $k\in\\{2,3,\ldots\\}$ and the transfer operator associated to the SRB
measure acting on $H_{p}^{t}$. One may search for eigenfunctions of the form
${h_{z}=\sum_{\ell=0}^{\infty}z^{\ell}h\circ T^{\ell}}$ where $h$ is in the
kernel of the transfer operator (cf. [15, Lemma 5], general transformations
and transfer operators can be treated by modifying this same idea). If the sum
converges then $h_{z}$ is an eigenfunction associated to eigenvalue $z$. We
choose $h$ to be the difference of the characteristic functions of two
intervals and then convergence follows from the bound on the $H_{p}^{t}$ norm
of characteristic functions. Consequently the essential spectral radius is at
least $k^{-1}$ in this case. (If $A$ is an interval, then $\mathbf{1}_{A}\circ
T^{\ell}=\mathbf{1}_{T^{-\ell}A}$ and so $T^{-\ell}A$ is composed of at most
$k^{\ell}$ intervals.) In particular, the essential spectrum on $H_{p}^{t}$ is
not smaller than if we were to use $\operatorname{BV}$.

######  Remark ($L^{\infty}$ bound).

Assumption (3) of Theorem 1 is unfortunate and we suspect that it could be
removed since, for all choices of Banach space which come to mind, the
conclusion of the theorem holds. When the Banach space is sufficiently large
the argument is to construct eigenfunctions or approximate eigenfunctions.
When the Banach space is smaller, as is the case we work with, the dual space
is bigger and the present work takes advantage of this to construct the
eigenvectors of the dual.

######  Remark (Zeta functions).

The study of resonances is intimately connected to zeta functions. Milnor &
Thurston [32, p.558] gave an example of an interval transformation for which
they showed that the zeta function cannot be extended meromorphically outside
a specified disc. The same is known for $\beta$-transformations when they
aren’t Markov (stated in [19, Theorem 2.4], using essentially [39]). In
general, if the transfer operator (acting on a relevant Banach space) has
essential spectral radius contained in a disc of radius $r$, then the zeta
function admits a meromorphic extension to a disc of radius $r^{-1}$ (see,
e.g., [23, 4] and [8, Part I]). If this connection between spectrum and zeta
function were true for general Banach spaces (as suggested by a weight of
evidence but currently an open problem as far as the authors are aware) then
the two results cited above would imply, in those settings, lower bounds on
the essential spectral radius for the transfer operator acting on any relevant
Banach space.

We would hope that results, similar to this work, can be obtained, both in the
case of piecewise $\mathcal{C}^{\infty}$ expanding transformations in higher
dimension and billiard maps (2D, hyperbolic with discontinuities). This seems
probable although in higher dimension there are additional complications
related to the geometry and propagation of discontinuities.

##  3\. Lower bound

Let $T:[0,1]\to[0,1]$, $\varphi:[0,1]\to\mathbb{C}$,
$\mathfrak{B}\supseteq\mathcal{C}^{\infty}(\mathcal{I})$ and
$\left\\{a_{k}\right\\}_{k=1}^{\infty}$ (orbit of the non-trivial
discontinuity, i.e., $a_{k}=T^{k}a$, where $a\in\partial\omega$,
$\omega\in\Omega$ and ${\\{T^{k}a\\}}_{k=1}^{\infty}$ is an infinite set) be
fixed according to the assumptions of Theorem 1 for the rest of this section.
Additionally, for notational simplicity, we drop the subscript and write
$\mathcal{L}$ for $\mathcal{L}_{\varphi}$.

Since the discontinuities of $T$ are key it is convenient to introduce some
additional notation for this. We may assume that $\Omega$ is the maximal
partition on which $T$ is continuous. Let $N=\\#\Omega$ and let
$0=c_{0}<c_{1}<\cdots<c_{N}=1$ be the points of discontinuity in the sense
that $\Omega=\left\\{(c_{\ell-1},c_{\ell})\right\\}_{\ell=1}^{N}$. Let
$\Gamma=\\{c_{0},\ldots,c_{N}\\}$. There are $2N$ sequences of points of the
form121212Here and subsequently, we use the superscript $+$/$-$ for the left
and right limits in the sense that $g(a_{k}^{+})=\lim_{\epsilon\rightarrow
0}g(a_{k}+\epsilon)$ and $g(a_{k}^{-})=\lim_{\epsilon\rightarrow
0}g(a_{k}-\epsilon)$. $\left\\{T^{k}(c_{\ell}^{-})\right\\}_{k=0}^{\infty}$ or
$\left\\{T^{k}(c_{\ell}^{+})\right\\}_{k=0}^{\infty}$. We say that these are
the discontinuity orbits and let $\Delta$ denote the set of all points in all
these orbits. Each one of these discontinuity orbits may be a finite or
infinite set of points. In the case that the transformation is Markov the set
would be finite. For convenience we use the convention that
$T^{-1}y=\\{x:y=T(x^{-})\\}\cup\\{x:y=T(x^{+})\\}$.

######  Lemma 3.1.

Without loss of generality we may suppose that ${\\{a_{k}\\}}_{k=1}^{\infty}$
is such that $T^{-n}a_{k}\cap\Gamma=\emptyset$ whenever $0\leq n<k$ and that
$T^{-1}a_{1}\cap\Gamma\neq\emptyset$.

###### Proof.

If this isn’t the case, then there would exist another discontinuity orbit
${\\{b_{k}\\}}_{k=1}^{\infty}$ such that $a_{m}=b_{n}$ for some
$m,n\in\mathbb{N}$, $m<n$. In this case we can choose ${\\{b_{k}\\}}_{k}$
instead of ${\\{a_{k}\\}}_{k}$. ∎

######  Lemma 3.2.

There exists $k_{0}\in\mathbb{N}$ such that
$T^{-1}a_{k}\cap\Delta=\\{a_{k-1}\\}$ for $k\geq k_{0}$.

###### Proof.

First observe that $a_{k}\notin\Gamma$ for every $k\in\mathbb{N}$ by Lemma
3.1. We must show that there exists $k_{0}\in\mathbb{N}$ such that
$T^{-1}a_{k}\cap\Delta=\\{a_{k-1}\\}$ whenever $k\geq k_{0}$. If this is
false, it means there is another discontinuity orbit
${\\{b_{k}\\}}_{k=1}^{\infty}\subset\Delta$ such that $Tb_{j}=a_{m}$ for some
$j\in\mathbb{N}$, $m\geq k_{0}$. If this happens we increase $k_{0}$ to be
equal to $m+1$. This can’t happen again with the same discontinuity orbit
since the orbit $\\{b_{k}\\}$ and $\\{a_{k}\\}$ are the same after this point.
Since there are only a finite number of discontinuity orbits we only need to
increase $k_{0}$ at most a finite number of times. ∎

The above two lemmas are sufficient for the purposes of this section but, in a
later section, we will prove Lemma 6.1, a more detailed yet similar result.
Let $k_{0}\in\mathbb{N}$ be fixed for the remainder of this section as
established in Lemma 3.2.

For each $n\in\mathbb{N}$, let $\Omega_{n}$ denote the maximal partition into
open intervals of a full measure subset of $[0,1]$ such that
$\left.T^{n}\right|_{\omega}$ is strictly monotone and continuous for each
$\omega\in\Omega$. Additionally, let
$\varphi_{n}=\prod_{k=0}^{n-1}\varphi\circ T^{k}$. Iterating the definition
(1) of the transfer operator we obtain, for all $n\in\mathbb{N}$,

(2) $\mathcal{L}^{n}h=\sum_{\omega\in\Omega_{n}}(\varphi_{n}\cdot
h)\circ{\left({\left.T^{n}\right|}_{\omega}\right)}^{-1}\cdot\mathbf{1}_{T^{n}\omega}.$

Since it was assumed that
$\mathcal{C}^{\infty}(\mathcal{I})\subseteq\mathfrak{B}$ and $\mathcal{L}$ is
continuous on $\mathfrak{B}$ we know that the set
$\bigcup_{n=0}^{\infty}\mathcal{L}^{n}(\mathcal{C}^{\infty}(\mathcal{I}))$ is
contained within $\mathfrak{B}$. Let $\mathfrak{A}_{0}\subseteq\mathfrak{B}$
denote the span of this set and let $\mathfrak{A}$ denote the completion of
$\mathfrak{A}_{0}$ in the $\left\|\cdot\right\|_{\mathfrak{B}}$-norm. Observe
that $\mathfrak{A}\subseteq\mathfrak{B}$ is a complete invariant subspace. In
particular $(\mathfrak{A},\left\|\cdot\right\|_{\mathfrak{B}})$ is a Banach
space. It may happen that $\mathfrak{A}=\mathfrak{B}$ but this can’t be
assumed in general.

######  Lemma 3.3.

Any $g\in\mathfrak{A}_{0}$ is continuous except for at a finite set of points
in $\Delta\cup\Gamma$. Furthermore, for any $k\in\mathbb{N}$, $g(a_{k}^{+})$
and $g(a_{k}^{-})$ are finite.

###### Proof.

We consider $g=\mathcal{L}^{n}h$ for some
$h\in\mathcal{C}^{\infty}(\mathcal{I})$, $n\in\mathbb{N}$. This means (2) that

$g=\sum_{\omega\in\Omega_{n}}(\varphi_{n}\cdot
h)\circ{\left({\left.T^{n}\right|}_{\omega}\right)}^{-1}\cdot\mathbf{1}_{T^{n}\omega}.$

Each summand is continuous on the interval $T^{n}\omega$ and zero elsewhere.
That it is only continuous is because $T$ is not required to be better. To
complete the proof we must show that the end points of each interval
$T^{n}\omega$ lie in $\Delta$. Observe that each $\omega\in\Omega_{n}$ is of
the form

$\omega=\omega_{i_{0}}\cap T^{-1}(\omega_{i_{1}})\cap...\cap
T^{-(n-1)}(\omega_{i_{n-1}}).$

where $\Omega=\\{\omega_{j}\\}_{j=1}^{N}$ and
$(i_{0},...,i_{n-1})\in{\\{1,\ldots,N\\}}^{n}$. From this we see that a point
can be an end point of the interval $T^{n}\omega$ only if it is an end point
of some $T^{k}\omega_{j}$ where $0\leq k\leq n$ and $j\in{\\{1,\ldots,N\\}}$.
This set of points is contained in the set $\Delta$ by definition. The
continuity and boundedness of $\varphi$ and $h$, means that the left and right
limits, $g(a_{k}^{+})$ and $g(a_{k}^{-})$, are finite. ∎

As a convenience, for any $h:[0,1]\to\mathbb{C}$ and $x\in[0,1]$, let

(3) $J\left(h,x\right)=h(x^{+})-h(x^{-}),$

i.e., this is the “jump” in $h$ at the point $x$. Lemma 3.3 implies that
$J\left(h,x\right)$ is defined for all $h\in\mathfrak{A}_{0}$, $x\in[0,1]$ and
is zero when $x\notin\Delta$. Since $T$ is not required to be orientation
preserving we need a way to keep track of this behaviour. For each $k$, let

(4) $\gamma_{k}=\begin{cases}1&\text{if $T$ is increasing at $a_{k}$},\\\
-1&\text{if $T$ is decreasing at $a_{k}$}.\end{cases}$

The following tells that the jumps behave, under the action of $\mathcal{L}$,
like a shift.

######  Lemma 3.4.

If $g\in\mathfrak{A}_{0}$ and $k\geq k_{0}$ then

$J\left(\mathcal{L}g,a_{k}\right)=\gamma_{k-1}\ \varphi(a_{k-1})\
J\left(g,a_{k-1}\right).$

###### Proof.

In order to calculate $J\left(\mathcal{L}g,a_{k}\right)$ we observe that
$\mathcal{L}g$ is given by a sum of terms of the form

$(\varphi\cdot
g)\circ{\left({\left.T\right|}_{\omega}\right)}^{-1}\cdot\mathbf{1}_{T\omega}.$

Because $k\geq k_{0}$ we know that $T^{-1}a_{k}\cap\Gamma=\emptyset$, in other
words, none of the end points of the $T\omega$ equal $a_{k}$ (Lemma 3.2). It
therefore remains to consider discontinuities in $(\varphi\cdot g)$ at points
in $T^{-1}a_{k}$. Since the discontinuities of $\varphi$ lie in $\Gamma$,
Lemma 3.1, $\varphi$ is continuous at points in $T^{-1}a_{k}$. By Lemma 3.3,
we know that $g$ can only have discontinuities in $\Delta$. Since $\Delta\cap
T^{-1}a_{k}=a_{k-1}$ this is the only term which contributes. The magnitude of
the jump is equal to $\varphi(a_{k-1})J\left(g,a_{k-1}\right)$. The sign
depends on whether ${\left.T\right|}_{\omega}$ is orientation preserving or
reversing for the $\omega$ which contains $a_{k-1}$ and so (4) is given by
$\gamma_{k-1}$. ∎

In the subsequent argument it will be convenient to guarantee the existence of
some element of $\mathfrak{A}$ which has a non-zero jump at $a_{k_{0}}$ but
has no jump at $a_{k}$ for all $k>k_{0}$. We can’t just take
$\mathbf{1}_{[a_{k_{0}},1]}$ because we can’t guarantee that such a function
is in $\mathfrak{A}_{0}$. We can however take advantage of the dynamics to
construct what we require.

######  Lemma 3.5.

There exists $h_{\mathcal{K}}\in\mathfrak{A}_{0}$ such that
$J\left(h_{\mathcal{K}},a_{k_{0}}\right)=1$ and, whenever $k>k_{0}$,
$J\left(h_{\mathcal{K}},a_{k}\right)=0$.

###### Proof.

If $h_{0}\in\mathcal{C}^{\infty}(\mathcal{I})$ (to be chosen later) then (2),

$\mathcal{L}^{k_{0}}h_{0}=\sum_{\omega\in\Omega_{k_{0}}}(\varphi_{k_{0}}\cdot
h_{0})\circ{\left({\left.\smash{T^{k_{0}}}\right|}_{\omega}\right)}^{-1}\cdot\mathbf{1}_{T^{k_{0}}\omega}\in\mathfrak{A}_{0}.$

Similar to the proof of Lemma 3.3, we see that a point is a discontinuity of
$\mathcal{L}^{k_{0}}h_{0}$ only if it is an end point of the interval
$T^{n}\omega$ where $0\leq n\leq{k_{0}}$, $\omega\in\Omega$. Lemma 3.1 tells
us that $T^{-n}a_{k}\cap\Gamma=\emptyset$ when $n<k$ and so
$J\left(\mathcal{L}^{k_{0}}h_{\mathcal{K}},a_{k}\right)=0$ whenever $k>k_{0}$.
On the other hand, $T^{-1}a_{1}\subseteq T^{-k_{0}}a_{k_{0}}$ and
$T^{-1}a_{1}\cap\Gamma\neq\emptyset$ according to Lemma 3.1. It is possible
that $T^{-k_{0}}a_{k_{0}}\cap\Gamma$ contains more than a single point. We
therefore choose $h_{0}\in\mathcal{C}^{\infty}(\mathcal{I})$ such that it is
non-zero at just one point in this set. It is even possible that
$T^{-k_{0}}a_{k_{0}}\cap\Gamma$ contains the same point twice since it might
exist as both the left and right limit. However this is also not a problem
since $h_{0}\in\mathcal{C}^{\infty}(\mathcal{I})$ can be chosen to be non-zero
at one side of a point in $\Gamma$ and zero the other side. This all means
that $J\left(\mathcal{L}^{k_{0}}h_{0},a_{k_{0}}\right)\neq 0$ and, scaling
appropriately, we insure that
$J\left(\mathcal{L}^{k_{0}}h_{0},a_{k_{0}}\right)=1$. ∎

Recall that
$\Lambda^{\inf}=\liminf_{n\to\infty}\left|\varphi_{n}(a_{1})\right|^{\frac{1}{n}}$.
We may assume that $\varphi(a_{k})>0$ for all $k\in\mathbb{N}$ because, if
this were not the case, then $\Lambda^{\inf}=0$ and Theorem 1 is trivially
true since the essential spectral radius is never less than zero. Hence, we
may also assume that $\Lambda^{\inf}>0$. Define the sequence
$\left\\{\alpha_{k}\right\\}$ by setting $\alpha_{k_{0}-1}=1$ and by requiring
that

(5)
$\alpha_{k}=\frac{\gamma_{k-1}}{\varphi(a_{k-1})}\alpha_{k-1}\quad\text{whenever
$k\geq k_{0}$}.$

This definition implies that
$\left|\alpha_{k}\right|=\left|\varphi_{k_{0}-1}(a_{1})\right|/\left|\varphi_{k}(a_{1})\right|$
and, moreover, that
$\limsup_{k\to\infty}\left|\alpha_{k}\right|^{1/k}={1}/{\Lambda^{\inf}}$. For
any $\left|\lambda\right|<\Lambda^{\inf}$, define the linear functional
$\ell_{\lambda}:\mathfrak{A}_{0}\to\mathbb{C}$,

(6)
$\ell_{\lambda}(h)=\sum_{k=k_{0}}^{\infty}\lambda^{k}\alpha_{k}\,J\left(h,a_{k}\right).$

######  Lemma 3.6.

If $\left|\lambda\right|<{\Lambda^{\inf}}$ then $\ell_{\lambda}$ extends to a
linear functional on $\mathfrak{A}$.

###### Proof.

For any $h\in\mathfrak{A}_{0}$ we know that
$\left|J\left(h,a_{k}\right)\right|\leq 2\left\|h\right\|_{L^{\infty}}\leq
2\left\|h\right\|_{\mathfrak{B}}$ (here we use the assumption that
$\left\|\cdot\right\|_{L^{\infty}}\leq\left\|\cdot\right\|_{\mathfrak{B}}$).
Additionally, since $\left|\lambda\right|<\Lambda^{\inf}$,
$C_{\lambda}=\sum_{k=k_{0}}^{\infty}\left|\lambda^{k}\alpha_{k}\right|$ is
finite. Consequently, $\left|\ell_{\lambda}(h)\right|\leq
2C_{\lambda}\left\|h\right\|_{\mathfrak{B}}$ for all $h\in\mathfrak{A}_{0}$
and so $\ell_{\lambda}$ extends to $\mathfrak{A}$. ∎

Let $h_{\mathcal{K}}\in\mathfrak{A}_{0}$, as constructed in Lemma 3.5, be
fixed for the remainder of this section.

######  Lemma 3.7.

If $\left|\lambda\right|<{\Lambda^{\inf}}$ then
$\ell_{\lambda}(h_{\mathcal{K}})=\lambda^{k_{0}}\alpha_{k_{0}}$. In particular
$\ell_{\lambda}$ is non-zero as an element of $\mathfrak{A}^{*}$, the dual of
$\mathfrak{A}$.

###### Proof.

Lemma 3.5 tells that $J\left(h_{\mathcal{K}},a_{k}\right)=0$ whenever
$k>k_{0}$, hence

$\ell_{\lambda}(h_{\mathcal{K}})=\sum_{k=k_{0}}^{\infty}\lambda^{k}\alpha_{k}\,J\left(h_{\mathcal{K}},a_{k}\right)=\lambda^{k_{0}}\alpha_{k_{0}}J\left(h_{\mathcal{K}},a_{k_{0}}\right).$

This means that
$\ell_{\lambda}(h_{\mathcal{K}})=\lambda^{k_{0}}\alpha_{k_{0}}$ since, by
Lemma 3.5, $J\left(h_{\mathcal{K}},a_{k_{0}}\right)=1$. ∎

Using $h_{\mathcal{K}}\in\mathfrak{A}_{0}$ we define the operator
$\mathcal{K}:\mathfrak{A}_{0}\to\mathfrak{A}_{0}$,

(7)
$\mathcal{K}h=\alpha_{k_{0}}^{-1}J\left(h,a_{k_{0}-1}\right)\,h_{\mathcal{K}}.$

Observe that, for all $h\in\mathfrak{A}_{0}$,
$\left\|\mathcal{K}h\right\|_{\mathfrak{B}}\leq
2\left|\smash{\alpha_{k_{0}}}\right|^{-1}\left\|h\right\|_{L^{\infty}}\left\|\smash{h_{\mathcal{K}}}\right\|_{\mathfrak{B}}\leq
C\left\|h\right\|_{\mathfrak{B}}$. Consequently $\mathcal{K}$ admits a
continuous extension to $\mathfrak{A}$ which, abusing notation we also denote
$\mathcal{K}$. Observe that this is a rank-one operator. Now let

(8)
$\mathcal{M}:\mathfrak{A}\to\mathfrak{A};\quad\mathcal{M}={\left.\mathcal{L}\right|}_{\mathfrak{A}}-\mathcal{K}.$

Since $\mathcal{M}$ and $\mathcal{L}$ differ by a compact operator they will
have the same essential spectral radius. We now show that the linear
functionals $\ell_{\lambda}$ are eigenvectors for the dual operator
$\mathcal{M}^{*}$.

######  Lemma 3.8.

If $\left|\lambda\right|<{\Lambda^{\inf}}$ then
$\mathcal{M}^{*}\ell_{\lambda}=\lambda\ell_{\lambda}$ as elements of
$\mathfrak{A}^{*}$.

###### Proof.

Let $h\in\mathfrak{A}_{0}$. We will calculate separately the two terms of
$\ell_{\lambda}(\mathcal{M}h)=\ell_{\lambda}(\mathcal{L}h)-\ell_{\lambda}(\mathcal{K}h)$.
For the $\mathcal{L}$ term,

$\ell_{\lambda}(\mathcal{L}h)=\sum_{k=k_{0}}^{\infty}\lambda^{k}\alpha_{k}\,J\left(\mathcal{L}h,a_{k}\right).$

By definition (5), $\alpha_{k}\gamma_{k-1}\varphi(a_{k-1})=\alpha_{k-1}$, by
Lemma 3.4, $J\left(\mathcal{L}g,a_{k}\right)=\gamma_{k-1}\ \varphi(a_{k-1})\
J\left(g,a_{k-1}\right)$ and so
$\lambda^{k}\alpha_{k}\,J\left(\mathcal{L}h,a_{k}\right)=\lambda\left[\lambda^{k-1}\alpha_{k-1}\,J\left(h,a_{k-1}\right)\right]$
for each $k\geq k_{0}$. Consequently,

(9) $\displaystyle\ell_{\lambda}(\mathcal{L}h)$
$\displaystyle=\lambda\sum_{k=k_{0}-1}^{\infty}\lambda^{k}\alpha_{k}\,J\left(h,a_{k}\right)$
$\displaystyle=\lambda^{k_{0}}J\left(h,a_{k_{0}-1}\right)+\lambda\ell_{\lambda}(h).$

For the $\mathcal{K}$ term, recall that
$\mathcal{K}h=\alpha_{k_{0}}^{-1}J\left(h,a_{k_{0}-1}\right)\,h_{\mathcal{K}}$
and so
$\ell_{\lambda}(\mathcal{K}h)=\alpha_{k_{0}}^{-1}J\left(h,a_{k_{0}-1}\right)\,\ell_{\lambda}(h_{\mathcal{K}})$.
Lemma 3.7 tells us that
$\ell_{\lambda}(h_{\mathcal{K}})=\lambda^{k_{0}}\alpha_{k_{0}}$ and so

(10)
$\ell_{\lambda}(\mathcal{K}h)=\lambda^{k_{0}}J\left(h,a_{k_{0}-1}\right).$

Summing the terms from the above calculations, (9) and (10), we have shown
that
$\ell_{\lambda}(\mathcal{M}h)=\ell_{\lambda}(\mathcal{L}h)-\ell_{\lambda}(\mathcal{K}h)=\lambda\ell_{\lambda}(h)$
for any $h\in\mathfrak{A}_{0}$. By continuity this holds also on
$\mathfrak{A}$. ∎

We denote the spectrum and the essential spectrum of an operator by
$\operatorname{Spec}(\cdot)$ and $\operatorname{EssSpec}(\cdot)$ respectively,
and their respective “filled” versions are denoted by
$\operatorname{\underline{EssSpec}}(\cdot)$ and
$\operatorname{\underline{Spec}}(\cdot)$ (see Section 4 for the precise
definitions). To complete the proof it is convenient to recall the different
spaces and operators we worked with. The Banach space
$(\mathfrak{A},\left\|\cdot\right\|_{\mathfrak{B}})$ is an
$\mathcal{L}$-invariant subspace of the Banach space
$(\mathfrak{B},\left\|\cdot\right\|_{\mathfrak{B}})$. We worked with the
operators

$\mathcal{L}:\mathfrak{B}\to\mathfrak{B},\quad\mathcal{M}:\mathfrak{A}\to\mathfrak{A},\quad\mathcal{K}:\mathfrak{A}\to\mathfrak{A},$

and we know that, by definition of $\mathcal{M}$,

$\left.\mathcal{L}\right|_{\mathfrak{A}}=\mathcal{M}+\mathcal{K}.$

Lemma 3.7 tells us that $\ell_{\lambda}\in\mathfrak{A}^{*}$ is non-zero and so
Lemma 3.8 implies that, whenever $\left|\lambda\right|<\Lambda^{\inf}$,
$\lambda$ is an eigenvalue of
$\mathcal{M}^{*}:\mathfrak{A}^{*}\to\mathfrak{A}^{*}$. Moreover the spectrum
is a closed set and so

$\operatorname{Spec}(\mathcal{M})=\operatorname{Spec}(\mathcal{M}^{*})\supseteq\\{\left|\lambda\right|\leq\Lambda^{\inf}\\}.$

Lemma 4.1 implies that
$\operatorname{\underline{EssSpec}}(\mathcal{M})\supseteq\\{\left|\lambda\right|\leq\Lambda^{\inf}\\}$.
Since $\mathcal{K}$ is compact,

$\operatorname{EssSpec}(\mathcal{M})=\operatorname{EssSpec}(\left.\mathcal{L}\right|_{\mathfrak{A}})$

and so
$\operatorname{\underline{EssSpec}}(\left.\mathcal{L}\right|_{\mathfrak{A}})\supseteq\\{\left|\lambda\right|\leq\Lambda^{\inf}\\}$
and, in particular,
$\operatorname{\underline{Spec}}(\left.\mathcal{L}\right|_{\mathfrak{A}})\supseteq\\{\left|\lambda\right|\leq\Lambda^{\inf}\\}$.
To finish the argument we need to compare the spectrum of the restriction
$\left.\mathcal{L}\right|_{\mathfrak{A}}$ with the spectrum of $\mathcal{L}$.
By Lemma 4.2,
$\operatorname{Spec}(\left.\mathcal{L}\right|_{\mathfrak{A}})\subseteq\operatorname{\underline{Spec}}(\mathcal{L})$,
and so we know that
$\operatorname{\underline{Spec}}(\mathcal{L})\supseteq\\{\left|\lambda\right|\leq\Lambda^{\inf}\\}$.
This implies (Lemma 4.1), that

$\operatorname{\underline{EssSpec}}(\mathcal{L})\supseteq\\{\left|\lambda\right|\leq\Lambda^{\inf}\\}$

and, in particular, the essential spectral radius of $\mathcal{L}$ is not
smaller than $\Lambda^{\inf}$. This concludes the proof of Theorem 1.

##  4\. Background on essential spectrum

In this section we isolate some relatively standard functional analytic
details which are useful for the present work. We say that $z$ lies in the
essential spectrum $\operatorname{EssSpec}(\mathcal{L})$ of a bounded operator
$\mathcal{L}$ if $(z\operatorname{id}-\mathcal{L})$ is not a Fredholm
operator. There are five different definitions of essential spectrum in common
usage (see, e.g., [17, I.4] where the one we use corresponds to the third).
The different definitions in general aren’t equivalent however the essential
spectral radius is equal for all five definitions [17, Corollary 4.11]. The
notion of essential spectrum we use here is invariant under compact
perturbations (this is true for four out or the five definitions) [17, Theorem
4.1].

Let $\operatorname{\underline{EssSpec}}(\mathcal{L})$ denote the complement of
the unbounded component of
$\mathbb{C}\setminus\operatorname{EssSpec}(\mathcal{L})$. I.e., this is the
union of $\operatorname{EssSpec}(\mathcal{L})$ and all the bounded components
of the complement. Similarly $\operatorname{\underline{Spec}}(\mathcal{L})$
denotes the complement of the unbounded component of
$\mathbb{C}\setminus\operatorname{Spec}(\mathcal{L})$. These are the “filled”
versions of the usual notions.

In our work we arrive at the point where we know that the spectrum contains a
complete disc. According to one of the definitions of the essential spectrum
this would imply that the entire disc is essential spectrum. However this
version of the essential spectrum might not be invariant under compact
perturbation and this is a property which we require in the present argument.
We know that [16, Theorem 4.3.18]:

  * If $\mathcal{L}$ is a bounded operator then $\operatorname{Spec}(\mathcal{L})\setminus\operatorname{\underline{EssSpec}}(\mathcal{L})$ consists of isolated eigenvalues with finite algebraic and geometric multiplicities.

This allows us to conclude the following observation.

######  Lemma 4.1.

Let $\mathcal{L}$ be a bounded operator and let $C>0$. If
$\operatorname{\underline{Spec}}(\mathcal{L})\supseteq\\{\left|\lambda\right|\leq
C\\}$ then
$\operatorname{\underline{EssSpec}}(\mathcal{L})\supseteq\\{\left|\lambda\right|\leq
C\\}$.

###### Proof.

Since
$\operatorname{\underline{Spec}}(\mathcal{L})\supseteq\\{\left|\lambda\right|\leq
C\\}$ there must exist a connected set $S\subset\mathbb{C}$ such that
$S\supseteq\\{\left|\lambda\right|\leq C\\}$ and131313We denote by $\partial
S$ the topological boundary of $S\subset\mathbb{C}$. $\partial
S\subset\operatorname{Spec}(\mathcal{L})$. Let $\lambda\in\partial S$ and, for
sake of contradiction, suppose that
$\lambda\notin\operatorname{\underline{EssSpec}}(\mathcal{L})$. This means
[16, Theorem 4.3.18] that $\lambda$ is an isolated eigenvalue which is in
contradiction to the fact that $\lambda$ is in the boundary of $S$.
Consequently $\partial
S\subset\operatorname{\underline{EssSpec}}(\mathcal{L})$ and so
$S\subseteq\operatorname{\underline{EssSpec}}(\mathcal{L})$. ∎

In general we couldn’t hope for better than the above inclusion (the result
requires $\operatorname{\underline{EssSpec}}(\mathcal{L})$ rather than
$\operatorname{EssSpec}(\mathcal{L})$) because there exist
examples141414Consider $\mathcal{L}:l^{2}(\mathbb{Z})\to l^{2}(\mathbb{Z})$
defined as $(a_{1},a_{2},a_{3},\ldots)\mapsto(0,a_{1},a_{2},\ldots)$. If
$\left|z\right|=1$ then the range of $(z\operatorname{id}-\mathcal{L})$ is
dense but not closed and so this set is included in every definition of
essential spectrum. If $\left|z\right|<1$ then
$(z\operatorname{id}-\mathcal{L})$ has a closed range, one-dimensional kernel,
and one-dimensional cokernel and so this set is in
$\operatorname{Spec}(\mathcal{L})$ but not in
$\operatorname{EssSpec}(\mathcal{L})$. where the entire unit disc is in the
spectrum but that only the unit circle is in the part of the spectrum which is
invariant under compact perturbation.

We also need to be able to connect the the essential spectrum of the
restriction of an operator to the spectrum of the operator itself. It would be
convenient if the spectrum of the restriction was always smaller but this
isn’t true in general. Nonetheless this type of relationship holds for the
part of the spectrum which we are interested in. The following lemma tells us
that the spectrum of $\left.\mathcal{L}\right|_{\mathfrak{A}}$ can be richer
than the spectrum of $\mathcal{L}$ only if it “fills” holes present in the
spectrum of $\mathcal{L}$.

######  Lemma 4.2.

Let $\mathcal{L}$ be a bounded operator on Banach space
$(\mathfrak{B},\left\|\cdot\right\|_{\mathfrak{B}})$. Suppose that
$\mathfrak{A}\subseteq\mathfrak{B}$ is a closed $\mathcal{L}$-invariant
subspace. Then

(11)
$\operatorname{Spec}(\left.\mathcal{L}\right|_{\mathfrak{A}})\subseteq\operatorname{\underline{Spec}}(\mathcal{L}).$

###### Proof.

We know151515Let $\lambda\in\partial\operatorname{Spec}(\mathcal{L})$. Choose
sequence $\lambda_{k}\to\lambda$ such that
$\lambda_{k}\notin\operatorname{Spec}(\mathcal{L})$ for all $k\in\mathbb{N}$.
Since $\left\|(\lambda_{k}-\mathcal{L})^{-1}\right\|\to\infty$ there exists
${\\{h_{k}\\}}_{k}$ such that $\left\|h_{k}\right\|\to 0$ but
$\left\|(\lambda_{k}-\mathcal{L})^{-1}h_{k}\right\|=1$ for each $k$. Let
$g_{k}=(\lambda_{k}-\mathcal{L})^{-1}h_{k}$, these are the approximate
eigenfunctions. that
$\partial\operatorname{Spec}(\left.\mathcal{L}\right|_{\mathfrak{A}})$ is
contained in the approximate point spectrum of
${\left.\mathcal{L}\right|_{\mathfrak{A}}}$. The approximate point spectrum of
$\left.\mathcal{L}\right|_{\mathfrak{A}}$ is contained in the approximate
point spectrum of $\mathcal{L}$, in particular in
$\operatorname{Spec}(\mathcal{L})$. In summary, we have
$\partial\operatorname{Spec}(\left.\mathcal{L}\right|_{\mathfrak{A}})\subseteq\operatorname{Spec}(\mathcal{L})$.
Consequently
$\operatorname{Spec}(\left.\mathcal{L}\right|_{\mathfrak{A}})\subseteq\operatorname{\underline{Spec}}(\mathcal{L})$.
∎

##  5\. Examples

This section is devoted to proving Theorem 2 by explicitly constructing a
family of examples.

For any $m\in\mathbb{N}$, $m\geq 4$ and $\rho\in(1,m)$ we define the piecewise
monotone (indeed affine) transformation $T_{m,\rho}:[0,1]\to[0,1]$,

$T_{m,\rho}(x)=\begin{cases}\frac{m}{m-3}x,&\mbox{if
}x\in[0,\frac{m-3}{m})=\omega_{1}\\\ m(x-\frac{m-3}{m}),&\mbox{if
}x\in[\frac{m-3}{m},\frac{m-2}{m})=\omega_{2}\\\ m(x-\frac{m-2}{m}),&\mbox{if
}x\in[\frac{m-2}{m},\frac{m-1}{m})=\omega_{3}\\\
\rho(x-\frac{m-1}{m}),&\mbox{if }x\in[\frac{m-1}{m},1]=\omega_{4}.\\\
\end{cases}$

Observe that $T_{m,\rho}$, restricted to $\omega_{1}$, $\omega_{2}$ or
$\omega_{3}$ is bijective onto $[0,1)$, while
$T_{m,\rho}(\omega_{4})=[0,\frac{\rho}{m}]$.

######  Lemma 5.1.

For any $m\geq 4$, there exists $\rho\in(0,m)$ such that $1$ is a non-trivial
discontinuity for the transformation $T_{m,\rho}$ and161616Recall that we
write $\Lambda$ when $\Lambda^{\inf}=\Lambda^{\sup}$, i.e. when the limit
exists.

$\Lambda(T_{m,\rho},\tfrac{1}{T_{m,\rho}^{\prime}})=\frac{1}{m}.$

###### Proof.

Notice that $T_{m,\rho}$ restricted to $\omega_{2}\cup\omega_{3}$ is
independent of $\rho$. Furthermore, $T_{m,\rho}$
$\omega_{2}\supset\omega_{1}\cup\omega_{2}$ and $T_{m,\rho}$
$\omega_{1}\supset\omega_{1}\cup\omega_{2}$. Therefore, there exists a point
$b\in\omega_{2}\cup\omega_{3}$ (actually a full Cantor set of points), which
remains forever in $\omega_{2}\cup\omega_{3}$ and enjoys a non-periodic
trajectory. Let us fix $\rho$ in such a way that

$T_{m,\rho}(1)=\frac{\rho}{m}=b.$

By the non-periodicity of the orbit of $b$, we have that $1$ is a non-trivial
discontinuity for the map $T_{m,\rho}$. Furthermore, since such an orbit will
visit only the sets $\omega_{2}$ and $\omega_{3}$, we have that

$\Lambda(T_{m,\rho},\tfrac{1}{T_{m,\rho}^{\prime}},1)=\lim_{n\to\infty}\biggl{(}\frac{1}{\rho}\cdot(1/m)^{n-1}\biggr{)}^{1/n}=\frac{1}{m}.$

To finish, we observe that $1$ is the only non-trivial discontinuity so that
$\Lambda(T_{m,\rho},\tfrac{1}{T_{m,\rho}^{\prime}})=\Lambda(T_{m,\rho},\tfrac{1}{T_{m,\rho}^{\prime}},1)$.
∎

On the other hand, the next lemma gives an estimate of the quantity which
corresponds to the essential spectral radius on BV.

######  Lemma 5.2.

For all $m\geq 4$ and $\rho\in(0,m)$,

$\lim_{n\to\infty}\left\|{\tfrac{1}{(T_{m,\rho}^{n})^{\prime}}}\right\|_{L^{\infty}}^{1/n}\geq\frac{m-3}{m}.$

###### Proof.

We take advantage of the fact that $0$ is a fixed point. For every
$n\in\mathbb{N}$, let $I_{n}=[0,\frac{1}{10}(\frac{m-3}{m})^{n}]$. We observe
that
$\left\|{1}/{(T^{n})^{\prime}}\right\|_{L^{\infty}([0,1])}\geq\left\|{1}/{(T^{n})^{\prime}}\right\|_{L^{\infty}(I_{n})}$.
On the other hand, for every $m\geq 4$,
$\left\|{1}/{(T^{n})^{\prime}}\right\|_{L^{\infty}(I_{n})}=(\frac{m-3}{m})^{n}$,
since $T^{k}(x)\in\omega_{1}$ for every $k\leq n$. ∎

Let $c\in(0,1)$ and choose $m$ sufficiently large such that

$\frac{m-3}{m}-\frac{1}{m}>c.$

Then, by Lemma 5.1, we can fix $\rho$ such that $1$ is a non-trivial
discontinuity for the transformation $T_{m,\rho}$ and
$\Lambda(T_{m,\rho},\tfrac{1}{T_{m,\rho}^{\prime}})=\frac{1}{m}$. On the other
hand, thanks to Lemma 5.2,
$\lim_{n\to\infty}\left\|{{1}/{(T_{m,\rho}^{n})^{\prime}}}\right\|_{L^{\infty}}^{1/n}\geq\frac{m-3}{m}$,
independent of $\rho$. This concludes the proof of Theorem 2.

##  6\. Custom Banach space

This section is devoted to the construction of custom Banach spaces, and hence
the proof of Theorem 3.

The argument of Keller [24] for the lower bound to the essential spectral
radius on $\operatorname{BV}$ suggests that we need to be more restrictive on
the possibility of jumps in the observables. A first choice could be
$\operatorname{SBV}$, the space of “special” bounded variation functions.
However this space is still too large in the sense that it still permits the
construction that Keller uses. Learning from this we permit discontinuities
only where we absolutely need to because of the discontinuities of the
interval transformation. This permits us to encode the asymptotic behaviour of
the discontinuities introducing norms adapted to their orbits.

Throughout this section we work under the assumptions of Theorem 3, in
particular we suppose that $T:[0,1]\to[0,1]$ is a $\mathcal{C}^{\infty}$
piecewise monotone transformation (with associated partition $\Omega$) and
that $\varphi:[0,1]\to\mathbb{C}$ has a $\mathcal{C}^{\infty}$ extension to
$\overline{\omega}$ for each $\omega\in\Omega$ and is uniformly bounded away
from zero. Moreover $T$ is uniformly expanding. Let
$\eta=\lim_{n\to\infty}\left\|\smash{\prod_{k=0}^{n-1}\varphi(T^{k})}\right\|_{L^{\infty}}^{1/n}$
and
$\lambda=\lim_{n\to\infty}\left\|1/(T^{n})^{\prime}\right\|_{L^{\infty}}^{1/n}<1$.
Fix, for the remainder of this section,
$\widetilde{\Lambda}>\Lambda^{\sup}(T,\varphi)$. Since $\lambda<1$ we may
choose $r\in\mathbb{N}$, $r\geq 1$ sufficiently large such that
$\eta\lambda^{r-1}<\widetilde{\Lambda}$.

######  Remark.

The assumptions of Theorem 3 can be relaxed to $T$ and $\varphi$ being merely
$\mathcal{C}^{r}$ as long as $r\in\mathbb{N}$ is sufficiently large such that
the above inequality is satisfied.

Recall that, as defined in Section 3, $\Gamma\subset\mathcal{I}$ denotes the
boundary of the partition elements $\omega\in\Omega$ and $\Delta$ denotes
$\cup_{k=0}^{\infty}T^{k}\Gamma$, the set of all images of all
discontinuities. The following is an extension and rephrasing of Lemma 3.1 and
Lemma 3.2.

######  Lemma 6.1.

There exists a finite set of points ${\\{b_{k}\\}}_{k=1}^{N_{2}}$ and finite
number (indexed by $1\leq j\leq N_{1}$) of countable sets of points
${\\{a_{j,k}\\}}_{k=0}^{\infty}$ such that, $a_{j,k}\notin\Gamma$,

$\Delta={\\{b_{k}\\}}_{k=1}^{N_{2}}\cup\bigcup_{j=1}^{N_{1}}{\\{a_{j,k}\\}}_{k=0}^{\infty},$

and,

(12) $T^{-n}a_{j,k}\cap\Delta=\\{a_{j,k-n}\\},\quad\quad\text{ for all $j$,
$k\geq n$}.$

###### Proof.

As mentioned at the beginning of Section 3, there are at most a finite number
of non-trivial discontinuity orbits and this is the origin of $N_{1}$. Lemma
3.2 implies the result stated here, with the minor difference that now we keep
track of all discontinuity orbits. For each of these infinite orbits there
exists some iterate after which the required property (12) is satisfied and
the orbit will not return to $\Gamma$ (Lemma 3.1 & Lemma 3.2). In this way we
define the $a_{j,k}$. The $b_{k}$ are defined to be the finite number of
points of $\Delta$ remaining. ∎

For the remainder of this section we suppose that the $b_{k}$ and $a_{j,k}$
are fixed as determined in Lemma 6.1. We write
$\partial\Omega_{n}=\cup_{\omega\in\Omega_{n}}\partial\omega$.

######  Lemma 6.2.

$T^{-n}(a_{j,k+n})\cap\partial\Omega_{n}=\emptyset$ for all $j,k,n$. In
particular $a_{j,k}\notin\partial\Omega_{n}$.

###### Proof.

Suppose, for sake of contradiction, there exists $y$ such that
$T^{n}y=a_{j,k+n}$ and $T^{p}y\in\Gamma$ for $p\leq n-1$. Then,
$T^{n-p}T^{p}y=a_{j,k+n}$ and so there exists $a\in\Gamma$ such that
$T^{n-p}a=a_{j,k+n}$. This is a contradiction by Lemma 6.1. The final claim
follows from the fact that $a_{j,k}\in T^{-n}(a_{j,k+n})$. ∎

We say that $h:[0,1]\to\mathbb{C}$ is piecewise $\mathcal{C}^{\infty}$ with
respect to a finite partition $0=p_{0}<p_{1}<...<p_{N}=1$, if
${\left.h\right|}_{(p_{l},p_{l+1})}$ admits a $\mathcal{C}^{\infty}$ extension
up to $[p_{l},p_{l+1}]$, for all $0\leq l\leq N-1$. We define
$\mathfrak{B}_{0}$ to be the set of all such $h$ for which there exists a
finite partition of $[0,1]$ with $p_{l}\in\Delta$, which makes $h$ piecewise
$\mathcal{C}^{\infty}$ with respect to it.

Recall that $J\left(\cdot,\cdot\right)$ was defined (3) to be the jump in a
piecewise $\mathcal{C}^{\infty}$ function at a given point.

######  Lemma 6.3.

Let $h\in\mathfrak{B}_{0}$, $k\in\mathbb{N}_{0}$, $0\leq j\leq N_{1}$. Then

$\left|J\left(\mathcal{L}_{\varphi}h,a_{j,k+1}\right)\right|=\left|\varphi(a_{j,k})\right|\
\left|J\left(h,a_{j,k}\right)\right|.$

###### Proof.

This holds, as proved in Lemma 3.4, since the crucial property is that $h$
only has discontinuities at points in $\Delta$. ∎

###  6.1. The Banach space

We use the notation $\operatorname{D}_{\operatorname{a}}{}$ and
$\operatorname{D}_{\operatorname{j}}$ to denote the absolutely continuous part
and jump part of the derivative respectively. In other words, for any
$h\in\operatorname{SBV}$ (see, e.g., [1]), the weak derivative has the form
$\operatorname{D}h=\operatorname{D}_{\operatorname{a}}{h}+\operatorname{D}_{\operatorname{j}}{h}$
where $\operatorname{D}_{\operatorname{a}}{h}\in L^{1}$ and
$\operatorname{D}_{\operatorname{j}}{h}$ can be written as a sum of delta
functions. Note that when $h$ is piecewise-$\mathcal{C}^{\infty}$
$\operatorname{D}_{\operatorname{a}}h=h^{\prime}$, wherever the latter is
well-defined.

We observe that

(13)
$\operatorname{D}_{\operatorname{a}}\mathfrak{B}_{0}\subseteq\mathfrak{B}_{0},\quad\quad\mathcal{L}_{\varphi}\mathfrak{B}_{0}\subseteq\mathfrak{B}_{0}.$

Recall that $r\in\mathbb{N}$ was fixed at the beginning of this section. Let
$\zeta={\\{\zeta_{a}\\}}_{a\in\Delta}$ be a set of positive numbers (to be
defined shortly). For any $h\in\mathfrak{B}_{0}$ we define

(14)
$\left\|h\right\|_{\mathfrak{B}_{\zeta,r}}=\sum_{t=0}^{r}\left\|\operatorname{D}_{\operatorname{a}}^{t}h\right\|_{L^{1}}+\sum_{t=0}^{r-1}\left\|\operatorname{D}_{\operatorname{a}}^{t}h\right\|_{\mathfrak{J}_{\zeta}}$

where

$\left\|h\right\|_{\mathfrak{J}_{\zeta}}=\sum_{a\in\Delta}\zeta_{a}\
\left|J\left(h,a\right)\right|.$

We refer to the first sum in this definition as the _continuous_ part of the
norm and the second sum as the _jump_ part of the norm.

This norm has some resemblance to $\operatorname{BV}$ or $\operatorname{SBV}$.
In the case when $r=0$, the norm corresponds to the $L^{1}$ norm. In the case
when $r=1$ and $\zeta_{a}$ is equal to unity for every $a\in\Delta$, then the
norm corresponds to the usual $\operatorname{BV}$ norm and the Banach space is
the subset of $\operatorname{SBV}$ (in turn a subset of $\operatorname{BV}$)
where discontinuities are only permitted at points in $\Delta$. As
anticipated, we now fix $\zeta$. For all $1\leq j\leq N_{1}$ and all
$k\in\mathbb{N}_{0}$, let

(15) $\zeta(j,k)=\frac{\widetilde{\Lambda}^{k}}{|\varphi_{k}(a_{j,0})|},$

where, as before, $\varphi_{k}=\prod_{j=0}^{k-1}\varphi\circ T^{j}$. We hence
define, for any $h\in\mathfrak{B}_{0}$,

(16)
$\left\|h\right\|_{\mathfrak{J}_{\zeta}}=\sum_{j=1}^{N_{1}}\sum_{k=0}^{\infty}\zeta(j,k)\left|J\left(h,a_{j,k}\right)\right|+\sum_{k=1}^{N_{2}}\left|J\left(h,b_{k}\right)\right|.$

This corresponds to setting $\zeta_{a_{j,k}}=\zeta(j,k)$, for all $j$, $k$ and
setting $\zeta_{a}=1$ otherwise.

The Banach space $\mathfrak{B}_{\zeta,r}$ is defined as the closure of
$\mathfrak{B}_{0}$ with respect to the
$\left\|\cdot\right\|_{\mathfrak{B}_{r,\zeta}}$ norm.

######  Lemma 6.4.

For all $h\in\mathfrak{B}_{0}$,
$\left\|h\right\|_{\mathfrak{B}_{\zeta,r}}<\infty$.

###### Proof.

We know that (13),
$\operatorname{D}_{\operatorname{a}}^{l}h\in\mathfrak{B}_{0}$, whenever $0\leq
l\leq r$, and the restriction of $\operatorname{D}_{\operatorname{a}}^{l}h$ to
any interval of continuity admits a $\mathcal{C}^{\infty}$ extension up to the
boundary of such interval. Notice that each $h\in\mathfrak{B}_{0}$ has only a
finite number of discontinuities and so the jump part of the norm is finite.
Moreover, $\operatorname{D}_{\operatorname{a}}^{l}h$ has finite $L^{1}$-norm.
∎

######  Lemma 6.5.

If $s+l\leq r$, then
$\left\|\smash{\operatorname{D}_{\operatorname{a}}^{l}h}\right\|_{\mathfrak{B}_{\zeta,s}}\leq\left\|h\right\|_{\mathfrak{B}_{\zeta,r}}$
for all $h\in\mathfrak{B}_{\zeta,r}$.

###### Proof.

By density it suffices to consider $h\in\mathfrak{B}_{0}$. According to the
definition of the norm (14),

$\displaystyle\left\|\smash{\operatorname{D}_{\operatorname{a}}^{l}h}\right\|_{\mathfrak{B}_{\zeta,s}}$
$\displaystyle=\sum_{t=0}^{s}\left\|\smash{\operatorname{D}_{\operatorname{a}}^{l+t}h}\right\|_{L^{1}}+\sum_{t=0}^{s-1}\left\|\smash{\operatorname{D}_{\operatorname{a}}^{l+t}h}\right\|_{\mathfrak{J}_{\zeta}}$
$\displaystyle\leq\sum_{t=0}^{s+l}\left\|\operatorname{D}_{\operatorname{a}}^{t}h\right\|_{L^{1}}+\sum_{t=0}^{s+l-1}\left\|\operatorname{D}_{\operatorname{a}}^{t}h\right\|_{\mathfrak{J}_{\zeta}}=\left\|\smash{h}\right\|_{\mathfrak{B}_{\zeta,s+l}}.$

Since it was assumed that $s+l\leq r$ the conclusion follows. ∎

######  Lemma 6.6.

There exists $C>0$ such that, for all $l+1\leq r$ and all
$h\in\mathfrak{B}_{\zeta,r}$,

$\left\|\smash{\operatorname{D}_{\operatorname{a}}^{l}h}\right\|_{\operatorname{BV}}\leq
C\left\|h\right\|_{\mathfrak{B}_{\zeta,r}}.$

###### Proof.

There exists $C>1$ such that, for all $k\in\mathbb{N}$,
$\left|\varphi_{k}(a_{j,0})\right|\leq C\widetilde{\Lambda}^{k}$ (recall that
$\widetilde{\Lambda}>\Lambda^{\sup}(T,\varphi)$, which is the maximum of
$\Lambda^{\sup}(T,\varphi,a)$ where $a$ ranges over the set of non-trivial
discontinuities) and so, using the definition of $\zeta$,

$\frac{1}{\zeta(j,k)}=\left|\varphi_{k}(a_{j,0})\right|\widetilde{\Lambda}^{-k}\leq
C.$

In particular, $1\leq C\zeta(j,k)$. Consequently, for all
$h\in\mathfrak{B}_{0}$,

$\begin{split}\sum_{a\in\Delta}|J(h,a)|&\leq{C}\sum_{j=1}^{N_{1}}\sum_{k=0}^{\infty}\zeta(j,k)\left|J\left(h,a_{j,k}\right)\right|+\sum_{k=1}^{N_{2}}\left|J\left(h,b_{k}\right)\right|\\\
&\leq C{\|h\|}_{\mathfrak{J}_{\zeta}}.\end{split}$

Observe that (since any discontinuities of $h$ lie in $\Delta$),

$\displaystyle{\|h\|}_{\operatorname{BV}}$
$\displaystyle={\|h\|}_{L^{1}}+{\|\operatorname{D}_{\operatorname{a}}{h}\|}_{L^{1}}+\sum_{a\in\Delta}\left|J(h,a)\right|,$
$\displaystyle{\|h\|}_{\mathfrak{B}_{\zeta,1}}$
$\displaystyle={\|h\|}_{L^{1}}+{\|\operatorname{D}_{\operatorname{a}}{h}\|}_{L^{1}}+{\|h\|}_{\mathfrak{J}_{\zeta}}.$

Hence, we have shown that, for all $h\in\mathfrak{B}_{0}$,
${\|h\|}_{\operatorname{BV}}\leq C{\|h\|}_{\mathfrak{B}_{\zeta,1}}$.

Using this together with Lemma 6.5 and the density of $\mathfrak{B}_{0}$
implies the statement of the lemma. ∎

###  6.2. Lasota-Yorke inequality

In the next lemmas we estimate the $\mathfrak{B}_{r,\zeta}$ norm of
$\mathcal{L}_{\varphi}^{n}$ and hence obtain a type of Lasota-Yorke
inequality. The following is the relevant distortion result in the present
setting.

######  Lemma 6.7.

There exists $C>0$ and a set ${\\{A_{l,p,n}\\}}_{l,p,n}$ of functions
$A_{l,p,n}:[0,1]\to\mathbb{C}$ such that, for all $h\in\mathfrak{B}_{0}$,
$0\leq p\leq r$, $n\in\mathbb{N}_{0}$,

(17)
$D_{a}^{p}\left(\mathcal{L}_{\varphi}^{n}h\right)=\sum_{l=0}^{p}\mathcal{L}_{\varphi}^{n}(A_{l,p,n}\cdot\operatorname{D}_{\operatorname{a}}^{l}h),$

the $A_{l,p,n}$ are $\mathcal{C}^{\infty}$ on each $\omega\in\Omega_{n}$,
$\left\|A_{l,p,n}\right\|_{L^{\infty}}\leq C$ and
$A_{p,p,n}={((T^{n})^{\prime})}^{-p}$.

The above lemma is proved in Section 7.

In two common cases the derivative of $\varphi$ is zero: (1) In the case of
the transfer operator associated to the measure of maximal entropy; (2) In the
case of the transfer operator associated to the SRB measure for a piecewise
affine transformation, here $\varphi=\frac{1}{\left|T^{\prime}\right|}$ and is
piecewise constant. In both these cases the terms that appear in Lemma 6.7
are:

$A_{j,k,n}=\begin{cases}((T^{n})^{\prime})^{-k}&\text{if $j=k$}\\\ 0&\text{if
$j<k$}.\end{cases}$

We don’t use the above in the present work but, if required, it can be
verified from the formulae given in Lemma 7.2.

######  Lemma 6.8.

There exists $C>0$ such that, for all $n\in\mathbb{N}_{0}$, $k\geq n$, $0\leq
t\leq r$ and $h\in\mathfrak{B}_{0}$,

$\left|J\left(\operatorname{D}_{\operatorname{a}}^{t}(\mathcal{L}^{n}_{\varphi}h)),a_{j,k}\right)\right|\leq
C\sum_{l=0}^{t}\left|J\left(\mathcal{L}^{n}_{\varphi}(\operatorname{D}_{\operatorname{a}}^{l}h),a_{j,k}\right)\right|.$

###### Proof.

By Lemma 6.7 there exist functions $A_{l,t,n}$ (with key properties) such that

$J\left(\operatorname{D}_{\operatorname{a}}^{t}(\mathcal{L}^{n}_{\varphi}h),a_{j,k}\right)=\sum_{l=0}^{t}J\left(\mathcal{L}_{\varphi}^{n}(A_{l,t,n}\cdot\operatorname{D}_{\operatorname{a}}^{l}h),a_{j,k}\right),$

and, by definition of $\mathcal{L}_{\varphi}$,

$J\left(\mathcal{L}_{\varphi}^{n}(A_{l,t,n}\cdot\operatorname{D}_{\operatorname{a}}^{l}h),a_{j,k}\right)=\sum_{\omega\in\Omega_{n}}J\left((A_{l,t,n}\cdot\varphi_{n}\cdot\operatorname{D}_{\operatorname{a}}^{l}h)\circ
T^{n}|_{\omega}^{-1},a_{j,k}\right).$

Lemma 6.1 implies that for all $a_{j,k}$, $k\geq n$, there exists a unique
$\omega^{\prime}\in\Omega_{n}$ such that
${T^{n}|_{\omega^{\prime}}}^{-1}(a_{j,k})=a_{j,k-n}\in\Delta$ and for
$\omega\neq\omega^{\prime}$, ${T^{n}|_{\omega}}^{-1}(a_{j,k})\notin\Delta$.
Furthermore,

$\sum_{\omega\in\Omega_{n}}J\left({(A_{l,t,n}\cdot\varphi_{n}\cdot\operatorname{D}_{\operatorname{a}}^{l}h)\circ{T^{n}|_{\omega}}^{-1}},a_{j,k}\right)\\\
=A_{l,t,n}(a_{j,k-n})J\left((\varphi_{n}\cdot\operatorname{D}_{\operatorname{a}}^{l}h)\circ{T^{n}|_{\omega^{\prime}}}^{-1},a_{j,k}\right).$

This is because $A_{l,t,n}\cdot\varphi_{n}$ is continuous at any point in
$T^{-n}(a_{j,k})$, $k\geq n$ (Lemma 6.2 & Lemma 6.7), and,
$\operatorname{D}_{\operatorname{a}}^{l}h$ is discontinuous only in $\Delta$
(13). Similarly,

$\displaystyle
J\left((\varphi_{n}\cdot\operatorname{D}_{\operatorname{a}}^{l}h)\circ{T^{n}|_{\omega^{\prime}}}^{-1},a_{j,k}\right)$
$\displaystyle=\sum_{\omega\in\Omega_{n}}J\left({(\varphi_{n}\cdot\operatorname{D}_{\operatorname{a}}^{l}h)\circ{T^{n}|_{\omega}}^{-1}},a_{j,k}\right)$
$\displaystyle=J\left(\mathcal{L}^{n}_{\varphi}(\operatorname{D}_{\operatorname{a}}^{l}h),a_{j,k}\right).$

Collecting together the above equalities we have shown that

$\left|J\left(\mathcal{L}_{\varphi}^{n}(A_{l,t,n}\cdot\operatorname{D}_{\operatorname{a}}^{l}h),a_{j,k}\right)\right|\leq\left\|A_{l,t,n}\right\|_{L^{\infty}}\left|J\left(\mathcal{L}^{n}_{\varphi}(\operatorname{D}_{\operatorname{a}}^{l}h),a_{j,k}\right)\right|.$

The statement of the lemma follows since Lemma 6.7 tells that
$\left\|A_{l,t,n}\right\|_{L^{\infty}}$ is bounded uniformly in $n$. ∎

For convenience we use the following notation for the continuous part and jump
part of the norm, respectively,

$\left\|h\right\|_{\mathfrak{C}^{r}}=\sum_{j=0}^{r}{\|\operatorname{D}_{\operatorname{a}}^{j}h\|}_{L^{1}},\quad\quad\left\|h\right\|_{\mathfrak{J}_{\zeta}^{r}}=\sum_{j=0}^{r-1}{\|\operatorname{D}_{\operatorname{a}}^{j}h\|}_{\mathfrak{J}_{\zeta}}.$

By change of variables, since $\left\|\varphi\cdot
T^{\prime}\right\|_{L^{\infty}}$ is finite, for all $h\in L^{1}$,
$n\in\mathbb{N}_{0}$,

(18)
$\left\|\mathcal{L}_{\varphi}h\right\|_{L^{1}}\leq\left\|\varphi_{n}\cdot\smash{\left(T^{n}\right)^{\prime}}\cdot
h\right\|_{L^{1}}.$

######  Lemma 6.9.

There exists some $C>0$ such that, for all $h\in\mathfrak{B}_{0}$,

${\|\mathcal{L}_{\varphi}h\|}_{\mathfrak{B}_{\zeta,r}}\leq
C{\|h\|}_{\mathfrak{B}_{\zeta,r}}.$

In particular, $\mathcal{L}_{\varphi}$ extends to a continuous operator on
$\mathfrak{B}_{\zeta,r}$.

###### Proof.

We estimate
${\|\mathcal{L}_{\varphi}h\|}_{\mathfrak{B}_{\zeta,r}}=\sum_{l=0}^{r}{\|\operatorname{D}_{\operatorname{a}}^{l}\mathcal{L}_{\varphi}h\|}_{L^{1}}+\sum_{l=0}^{r-1}{\|\operatorname{D}_{\operatorname{a}}^{l}\mathcal{L}_{\varphi}h\|}_{\mathfrak{J}_{\zeta}}$.
For any $0\leq l\leq r$, using Lemma 6.7 (for $n=1$) and (18),

$\displaystyle{\|\operatorname{D}_{\operatorname{a}}^{l}\mathcal{L}_{\varphi}h\|}_{L^{1}}$
$\displaystyle\leq\sum_{s=0}^{l}{\|\mathcal{L}_{\varphi}(A_{s,l,1}\operatorname{D}_{\operatorname{a}}^{s}h)\|}_{L^{1}}$
$\displaystyle\leq\sum_{s=0}^{l}{\|\varphi\cdot T^{\prime}\cdot
A_{s,l,1}\cdot\operatorname{D}_{\operatorname{a}}^{s}h\|}_{L^{1}}\leq
C\sum_{s=0}^{l}{\|\operatorname{D}_{\operatorname{a}}^{s}h\|}_{L^{1}},$

for some $C>0$. This takes care of the continuous part of the norm. As for the
jump part, we have, for any $0\leq l\leq r-1$

$\displaystyle{\|\operatorname{D}_{\operatorname{a}}^{l}\mathcal{L}_{\varphi}h\|}_{\mathfrak{J}_{\zeta}}$
$\displaystyle=\sum_{j=1}^{N_{1}}\sum_{k=1}^{\infty}\zeta(j,k)\left|J\left(\operatorname{D}_{\operatorname{a}}^{l}(\mathcal{L}_{\varphi}h),a_{j,k}\right)\right|$
$\displaystyle\ \
+\left(\sum_{k=1}^{N_{2}}\left|J\left(\operatorname{D}_{\operatorname{a}}^{l}\mathcal{L}_{\varphi}h,b_{k}\right)\right|+\sum_{j=1}^{N_{1}}\left|J\left(\operatorname{D}_{\operatorname{a}}^{l}(\mathcal{L}_{\varphi}h),a_{j,0}\right)\right|\right).$

Let us start with the final pair of sums. Crucially this is a finite number of
terms. By Lemma 6.6 and Lemma 6.7, we have that, for some $C>0$,

(19)
$\displaystyle\left|J\left(\operatorname{D}_{\operatorname{a}}^{l}\mathcal{L}_{\varphi}h,b_{k}\right)\right|$
$\displaystyle\leq\sum_{s=0}^{l}\left|J\left(\mathcal{L}_{\varphi}(A_{s,l,1}\cdot\operatorname{D}_{\operatorname{a}}^{s}h),b_{k}\right)\right|$
$\displaystyle\leq 2\ \\#\Omega\sum_{s=0}^{l}{\|\varphi\cdot
A_{s,l,1}\cdot\operatorname{D}_{\operatorname{a}}^{s}h\|}_{L^{\infty}}\leq
C{\|h\|}_{\mathfrak{B}_{\zeta,r}}.$

An identical estimate holds for the terms involving
$\left|J\left(\operatorname{D}_{\operatorname{a}}^{l}\mathcal{L}_{\varphi}h,a_{j,0}\right)\right|$.
This takes care of the finite sum.

For the first sum, the remaining term to be estimated, Lemma 6.8 implies,

$\left|J\left(\operatorname{D}_{\operatorname{a}}^{l}(\mathcal{L}_{\varphi}h),a_{j,k}\right)\right|\leq
C\sum_{s=0}^{l}\left|J\left(\mathcal{L}_{\varphi}(\operatorname{D}_{\operatorname{a}}^{s}h),a_{j,k}\right)\right|.$

Lemma 6.3 implies that this is bounded by
$C\sum_{s=0}^{l}\left|J\left(\operatorname{D}_{\operatorname{a}}^{s}h,a_{j,k-1}\right)\right|$.
Summing over $k$, this means that

$\sum_{j=1}^{N_{1}}\sum_{k=0}^{\infty}\zeta(j,k)\left|J\left(\operatorname{D}_{\operatorname{a}}^{l}(\mathcal{L}_{\varphi}h),a_{j,k}\right)\right|\leq
C\left\|h\right\|_{\mathfrak{J}_{\zeta}^{r}},$

as required. ∎

Subsequently we will prove inequalities concerning $\mathcal{L}_{\varphi}$ and
Lemma 6.9 means that it is sufficient to prove them on $\mathfrak{B}_{0}$.

######  Lemma 6.10.

For each $n\in\mathbb{N}$, there exists $C_{n}>0$ such that, if
$h\in\mathfrak{B}_{r,\zeta}$,

$\left\|\mathcal{L}_{\varphi}^{n}h\right\|_{\mathfrak{C}^{r}}\leq
C_{n}\left\|h\right\|_{\mathfrak{C}^{r-1}}+\left\|\varphi_{n}\right\|_{L^{\infty}}\left\|{1}/{(T^{n})^{\prime}}\right\|_{L^{\infty}}^{r-1}\left\|h\right\|_{\mathfrak{C}^{r}}.$

###### Proof.

By density it suffices to prove the statement for $h\in\mathfrak{B}_{0}$.
Using Lemma 6.7, and (18), one has that, for $0\leq q\leq r-1$,

$\displaystyle\left\|\operatorname{D}_{\operatorname{a}}^{q}{(\mathcal{L}^{n}_{\varphi}h)}\right\|_{L^{1}}$
$\displaystyle\leq\sum_{l=0}^{q}\left\|\mathcal{L}^{n}_{\varphi}\cdot(A_{l,q,n}\cdot\operatorname{D}_{\operatorname{a}}^{l}{h})\right\|_{L^{1}}\leq\sum_{l=0}^{q}\left\|\varphi_{n}(T^{n})^{\prime}\cdot
A_{l,q,n}\cdot\operatorname{D}_{\operatorname{a}}^{l}{h}\right\|_{L^{1}}$
$\displaystyle\leq\sum_{l=0}^{q}\left\|\varphi_{n}\cdot(T^{n})^{\prime}\cdot
A_{l,q,n}\right\|_{L^{\infty}}\left\|\smash{\operatorname{D}_{\operatorname{a}}^{l}{h}}\right\|_{L^{1}}.$

Lemma 6.7 and the assumption $\left\|\varphi\cdot
T^{\prime}\right\|_{L^{\infty}}<\infty$ implies that

$C_{n}=\max_{l\leq q\leq r}\left\|\varphi_{n}\cdot(T^{n})^{\prime}\cdot
A_{l,q,n}\right\|_{L^{\infty}}$

is finite. This means that,
$\left\|\operatorname{D}_{\operatorname{a}}^{q}{(\mathcal{L}^{n}_{\varphi}h})\right\|_{L^{1}}\leq
C_{n}\left\|h\right\|_{\mathfrak{C}^{q}}$ and, in particular,

$\left\|\mathcal{L}_{\varphi}h\right\|_{\mathfrak{C}^{r-1}}\leq
rC_{n}\left\|h\right\|_{\mathfrak{C}^{r-1}}.$

As for the top term, using $A_{r,r,n}={((T^{n})^{\prime})^{-r}}$ (Lemma 6.7),

$\begin{split}\left\|\operatorname{D}_{\operatorname{a}}^{r}{(\mathcal{L}_{\varphi}h)}\right\|_{L^{1}}&\leq\left\|\varphi_{n}\cdot(T^{n})^{\prime}\cdot
A_{r,r,n}\right\|_{L^{\infty}}\left\|\smash{\operatorname{D}_{\operatorname{a}}^{r}{h}}\right\|_{L^{1}}\\\
&\ \ +\sum_{l=0}^{r-1}\left\|\varphi_{n}\cdot(T^{n})^{\prime}\cdot
A_{l,r,n}\right\|_{L^{\infty}}\left\|\smash{\operatorname{D}_{\operatorname{a}}^{l}{h}}\right\|_{L^{1}}\\\
&\leq\left\|\smash{{\varphi_{n}}\cdot{((T^{n})^{\prime})^{-(r-1)}}}\right\|_{L^{\infty}}\left\|h\right\|_{\mathfrak{C}^{r}}+rC_{n}\left\|h\right\|_{\mathfrak{C}^{r-1}}.\end{split}$

The statement follows by summing the above estimates. ∎

######  Lemma 6.11.

Let $n\in\mathbb{N}$. There exists $C_{n}>0$ such that, for all
$h\in\mathfrak{B}_{0}$, $x\in[0,1]$ and $0\leq t\leq r-1$,

$|J(\operatorname{D}_{\operatorname{a}}^{t}(\mathcal{L}^{n}_{\varphi}h),x)|\leq
C_{n}{\|h\|}_{\mathfrak{B}_{\zeta,r}}.$

###### Proof.

By Lemma 6.6 and Lemma 6.9, for all $h\in\mathfrak{B}_{0}$ and $x\in[0,1]$,

$|J(\operatorname{D}_{\operatorname{a}}^{t}(\mathcal{L}^{n}_{\varphi}h),x)|\leq
2{\|\operatorname{D}_{\operatorname{a}}^{t}(\mathcal{L}^{n}_{\varphi}h)\|}_{L^{\infty}}\leq
2C{\|\mathcal{L}^{n}_{\varphi}h\|}_{\mathfrak{B}_{\zeta,r}}\leq
2{C}C^{n}{\|h\|}_{\mathfrak{B}_{\zeta,r}}.\qed$

######  Lemma 6.12.

There exists $C>0$ and, for each $n\in\mathbb{N}$, there exists a finite set
of linear functionals $\\{\rho_{l}\\}_{l\in Q_{n}}$ (on
$\mathfrak{B}_{r,\eta}$) such that, for all $h\in\mathfrak{B}_{r,\eta}$,

$\left\|\mathcal{L}_{\varphi}^{n}h\right\|_{\mathfrak{J}_{\zeta}^{r}}\leq\sum_{l\in
Q_{n}}|\rho_{l}(h)|+C\tilde{\Lambda}^{n}\left\|h\right\|_{\mathfrak{J}_{\zeta}^{r}}.$

###### Proof.

By density it suffices to consider $h\in\mathfrak{B}_{0}$. For all $0\leq
t\leq r-1$, $n\in\mathbb{N}_{0}$, using only the definition of
$\left\|\cdot\right\|_{\mathfrak{J}_{\zeta}}$,

(20)
$\begin{split}{\|\operatorname{D}_{\operatorname{a}}^{t}(\mathcal{L}^{n}_{\varphi}h)\|}_{\mathfrak{J}_{\zeta}}&=\sum_{k=1}^{N_{2}}|J(\operatorname{D}_{\operatorname{a}}^{t}(\mathcal{L}^{n}_{\varphi}h),b_{k})|\\\
&\ \
+\sum_{j=1}^{N_{1}}\sum_{k=0}^{n-1}\zeta(j,k)|J(\operatorname{D}_{\operatorname{a}}^{t}(\mathcal{L}^{n}_{\varphi}h),a_{j,k})|\\\
&\ \
+\sum_{j=1}^{N_{1}}\sum_{k=n}^{\infty}\zeta(j,k)\left|J(\operatorname{D}_{\operatorname{a}}^{t}(\mathcal{L}_{\varphi}^{n}h)),a_{j,k})\right|.\end{split}$

For each $n,t,j,k$, let

$\displaystyle\widetilde{\rho}_{n,t,k}(h)$
$\displaystyle=J(\operatorname{D}_{\operatorname{a}}^{t}(\mathcal{L}^{n}_{\varphi}h),b_{k}),$
$\displaystyle\rho_{n,t,j,k}(h)$
$\displaystyle=\zeta(j,k)(J(\operatorname{D}_{\operatorname{a}}^{t}(\mathcal{L}^{n}_{\varphi}h),a_{j,k})).$

Lemma 6.11 implies that these are all linear functionals on
$\mathfrak{B}_{\zeta,r}$. This accounts for the term $\sum_{l\in
Q_{n}}|\rho_{l}(h)|$ in the statement of the lemma. It remains to estimate the
final term of (20). We notice that, applying Lemma 6.3 to
$\operatorname{D}_{\operatorname{a}}^{t}h$, for $k\geq 1$, $0\leq j\leq
N_{1}$,

$\left|J\left(\mathcal{L}_{\varphi}(\operatorname{D}_{\operatorname{a}}^{t}h),a_{j,k}\right)\right|=\left|\varphi(a_{j,k-1})\right|\
\left|J\left(\operatorname{D}_{\operatorname{a}}^{t}h,a_{j,k-1}\right)\right|.$

Iterating this we obtain that, whenever $k\geq n$,

(21)
$\left|J\left(\mathcal{L}^{n}_{\varphi}(\operatorname{D}_{\operatorname{a}}^{t}h),a_{j,k}\right)\right|=\left|\varphi_{n}(a_{j,k-n})\right|\
\left|J\left(\operatorname{D}_{\operatorname{a}}^{t}h,a_{j,k-n}\right)\right|.$

Using also Lemma 6.8,

$\displaystyle\left|J\left(\operatorname{D}_{\operatorname{a}}^{t}(\mathcal{L}_{\varphi}^{n}h),a_{j,k}\right)\right|$
$\displaystyle\leq
C\sum_{i=0}^{t}\left|J\left(\mathcal{L}_{\varphi}^{n}(\operatorname{D}_{\operatorname{a}}^{i}h),a_{j,k}\right)\right|$
$\displaystyle=C\left|\varphi_{n}(a_{j,k-n})\right|\sum_{i=0}^{t}\left|J\left(\operatorname{D}_{\operatorname{a}}^{i}h,a_{j,k-n}\right)\right|.$

It follows by definition (15) that
$\zeta(j,k)\left|\varphi_{n}(a_{j,k-n})\right|=\tilde{\Lambda}^{n}\zeta(j,k-n)$.
Consequently,

$\zeta(j,k)\left|J\left(\operatorname{D}_{\operatorname{a}}^{t}(\mathcal{L}_{\varphi}^{n}h),a_{j,k}\right)\right|\leq
C\tilde{\Lambda}^{n}\zeta(j,k-n)\sum_{i=0}^{t}\left|J\left(\operatorname{D}_{\operatorname{a}}^{i}h,a_{j,k-n}\right)\right|.$

Summing over $j$ and $k\geq n$, we have shown that the final term of (20) is
bounded above by

$C\tilde{\Lambda}^{n}\sum_{j=1}^{N_{1}}\sum_{k=0}^{\infty}\zeta(j,k)\sum_{i=0}^{t}\left|J\left(\operatorname{D}_{\operatorname{a}}^{i}h,a_{j,k}\right)\right|=C\tilde{\Lambda}^{n}\left\|h\right\|_{\mathfrak{J}_{\zeta}^{t}}.$

The proof is completed by summing all the estimates over $t$. ∎

We are now in the position to combine the estimates obtained above. Since we
fixed $r$ such that $\eta\lambda^{r-1}<\widetilde{\Lambda}$, we can choose
$\widetilde{\eta}>\eta$ and $\widetilde{\lambda}>\lambda$ such that
$\widetilde{\eta}\widetilde{\lambda}^{r-1}\leq\widetilde{\Lambda}$. Let such
$\widetilde{\eta}$ and $\widetilde{\lambda}$ be fixed for the remainder of the
section.

######  Lemma 6.13.

There exists $C>0$ and, for all $n\in\mathbb{N}$, there exists $C_{n}>0$ and a
finite set of linear functionals ${\\{\rho_{j}\\}}_{j\in Q_{n}}$, such that,
for all $h\in\mathfrak{B}_{0}$,

$\begin{split}\left\|\mathcal{L}_{\varphi}^{n}h\right\|_{\mathfrak{B}_{\zeta,r}}&\leq
C_{n}\left\|h\right\|_{\mathfrak{C}^{r-1}}+C\tilde{\eta}^{n}\tilde{\lambda}^{n(r-1)}\left\|h\right\|_{\mathfrak{C}^{r}}\\\
&\ \ +\sum_{j\in
Q_{n}}|\rho_{j}(h)|+C\widetilde{\Lambda}^{n}\left\|h\right\|_{\mathfrak{J}_{\eta}^{r}}.\end{split}$

###### Proof.

Since $\tilde{\eta}>\eta$ and $\tilde{\lambda}>\lambda$, Lemma 6.10 implies
that, for some $C>0$,

$\left\|\mathcal{L}_{\varphi}^{n}h\right\|_{\mathfrak{C}^{r}}\leq
C_{n}\left\|h\right\|_{\mathfrak{C}^{r-1}}+C\tilde{\eta}^{n}\tilde{\lambda}^{n(r-1)}\left\|h\right\|_{\mathfrak{C}^{r}},$

as required for the first and second term in the statement of the lemma. The
third and fourth terms are given by Lemma 6.12. ∎

###  6.3. Essential spectral radius

Here we use the estimate of Lemma 6.13, together with a compactness result, in
order to estimate the essential spectral radius.

The following is a type of compact embedding between
$\left\|\cdot\right\|_{\mathfrak{B}_{\zeta,r}}$ and
$\left\|\cdot\right\|_{\mathfrak{C}^{s}}$.

######  Lemma 6.14.

Suppose that $s\in\mathbb{N}_{0}$, $p\in\mathbb{N}$, $s<p$. For every
$\epsilon>0$ there exists a finite set ${\\{S_{k}\\}}_{k}$ of subsets of
$\mathfrak{B}_{\zeta,p}$ such that
$\\{h:\left\|h\right\|_{\mathfrak{B}_{\zeta,p}}\leq 1\\}$ is covered by
$\bigcup_{k}S_{k}$ and, for each $k$,
$\left\|h-g\right\|_{\mathfrak{C}^{s}}\leq\epsilon$ for every $h,g\in S_{k}$.

###### Proof.

By Lemma 6.5 and Lemma 6.6 which implies that
$\\{h:\left\|\smash{\operatorname{D}_{\operatorname{a}}^{l}}h\right\|_{\mathfrak{B}_{\zeta,1}}\leq
1\\}$ is a subset of
$\\{h:\left\|\smash{\operatorname{D}_{\operatorname{a}}^{l}}h\right\|_{\operatorname{BV}}\leq
C\\}$,

$\left\\{h:\left\|h\right\|_{\mathfrak{B}_{\zeta,p}\leq
1}\right\\}\subseteq\bigcap_{l=0}^{p-1}\left\\{h:\left\|\smash{\operatorname{D}_{\operatorname{a}}^{l}}h\right\|_{\operatorname{BV}}\leq
C\right\\}.$

Fix, for the moment, $l$ and consider the set
$\left\\{h:\left\|\smash{\operatorname{D}_{\operatorname{a}}^{l}}h\right\|_{\operatorname{BV}}\leq
C\right\\}$. The compactness of $\operatorname{BV}$ in $L^{1}$ implies that,
for all $\epsilon>0$ there exists a finite cover
${\left\\{S_{k}\right\\}}_{k}$ of
$\left\\{h:\left\|h\right\|_{\operatorname{BV}}\leq C\right\\}$ such that,
$\left\|f-g\right\|_{L^{1}}\leq\epsilon$ whenever $f,g\in S_{k}$ for each $k$.
The sets $R_{k}=\\{h:\operatorname{D}_{\operatorname{a}}^{l}h\in S_{k}\\}$ are
a finite cover of
$\\{h:\left\|\smash{\operatorname{D}_{\operatorname{a}}^{l}}h\right\|_{\operatorname{BV}}\leq
C\\}$ such that
$\left\|\operatorname{D}_{\operatorname{a}}^{l}(f-g)\right\|_{L^{1}}\leq\epsilon$
whenever $f,g\in R_{k}$ for each $k$. This can be done for all $0\leq l\leq
p-1$ and so would imply that
$\left\|f-g\right\|_{\mathfrak{C}^{p-1}}\leq\epsilon$. This implies that the
statement holds true for all $s<p$. ∎

######  Lemma 6.15.

The essential spectral radius of $\mathcal{L}_{\varphi}$ on
$\mathfrak{B}_{\zeta,r}$ is at most $\widetilde{\Lambda}$.

###### Proof.

Let
$B_{0}=\left\\{\smash{h\in\mathfrak{B}_{\zeta,r}:\left\|h\right\|_{\mathfrak{B}_{\zeta,r}}\leq
1}\right\\}$ and, for $n\in\mathbb{N}$, let
$B_{n}=\mathcal{L}_{\varphi}^{n}B_{0}$. We will show that, for each
$n\in\mathbb{N}$ large, $B_{n}$ admits a finite cover such that the diameter
of each element of the cover, measured in the
$\left\|\cdot\right\|_{\mathfrak{B}_{\zeta,r}}$-norm, is not greater than
$C\widetilde{\Lambda}^{n}$. According to Nussbaum [33], this suffices to prove
that the essential spectral radius of
$\mathcal{L}_{\varphi}:\mathfrak{B}_{\zeta,r}\to\mathfrak{B}_{\zeta,r}$ is not
greater than $\widetilde{\Lambda}$.

Since $\widetilde{\eta}\widetilde{\lambda}^{r-1}\leq\widetilde{\Lambda}$,
Lemma 6.13 tells that, for all $h\in\mathfrak{B}_{\zeta,r}$, $n\in\mathbb{N}$,

$\displaystyle\left\|\mathcal{L}_{\varphi}^{n}h\right\|_{\mathfrak{B}_{\zeta,r}}\leq
C\widetilde{\Lambda}^{n}\left\|h\right\|_{\mathfrak{B}_{\zeta,r}}+C_{n}\left\|h\right\|_{\mathfrak{C}^{r-1}}+\sum_{j\in
Q_{n}}|\rho_{j}(h)|.$

Fix $n\in\mathbb{N}$. Using the compact embedding of Lemma 6.14 we can choose
a finite set ${\\{S_{k}\\}}_{k}$ of subsets of $B_{0}$ such that

$C_{n}\left\|h-g\right\|_{\mathfrak{C}^{r-1}}\leq\widetilde{\Lambda}^{n},\quad\quad\text{for
all $h,g\in S_{k}$, for each $k$}.$

Observe that the set $\\{\sum_{j\in Q_{n}}\rho_{j}(h):h\in
B_{0}\\}\subset\mathbb{C}$ is bounded because $B_{0}$ is bounded and the
$\rho_{j}$ are linear functionals. This means that we can further refine the
finite covering ${\\{S_{k}\\}}_{k}$ of $B_{0}$ such that it also satisfies

$\sum_{j\in
Q_{n}}|\rho_{j}(h-g)|\leq\widetilde{\Lambda}^{n},\quad\quad\text{for all
$h,g\in S_{k}$, for each $k$}.$

Observe that ${\\{\mathcal{L}^{n}_{\varphi}S_{k}\\}}_{k}$ is a finite cover of
$B_{n}$. For each $k$, the combination of the above estimates implies that,
for all $h,g\in S_{k}$,

$\left\|\mathcal{L}^{n}_{\varphi}h-\mathcal{L}^{n}_{\varphi}g\right\|_{\mathfrak{B}_{\zeta,r}}=\left\|\mathcal{L}^{n}_{\varphi}(h-g)\right\|_{\mathfrak{B}_{\zeta,r}}\leq(C+2)\widetilde{\Lambda}^{n}.$

I.e., as required, we have estimated the diameter of each of the sets
$\mathcal{L}^{n}_{\varphi}S_{k}$. ∎

This concludes the proof of Theorem 3 since the three statements of the
theorem are given by Lemma 6.4, Lemma 6.9 and Lemma 6.15, respectively. The
inclusion in $L^{\infty}$ is given by Lemma 6.6.

##  7\. Distortion

This section is devoted to the proof of Lemma 6.7. We work in the same setting
as Section 6, but none of this section relies on previous results.

######  Lemma 7.1.

For all $h\in\mathfrak{B}_{0}$,

$\operatorname{D}_{\operatorname{a}}{\left(\mathcal{L}_{\varphi}h\right)}=\mathcal{L}_{\varphi}\left(\tfrac{\varphi^{\prime}}{\varphi\cdot
T^{\prime}}\cdot
h\right)+\mathcal{L}_{\varphi}\left(\tfrac{1}{T^{\prime}}\cdot\operatorname{D}_{\operatorname{a}}{h}\right).$

###### Proof.

Since $\mathcal{L}_{\varphi}h=\sum_{k=1}^{n}\left(\varphi\cdot
h\right)\circ{\left.T\right|}_{\omega_{k}}^{-1}\cdot\mathbf{1}_{T\omega_{k}}$,
we calculate the derivative of each summand, restricted to $T\omega_{k}$,

$\displaystyle\operatorname{D}_{\operatorname{a}}\left(\left(\varphi\cdot
h\right)\circ{\left.T\right|}_{\omega_{k}}^{-1}\right)$
$\displaystyle=\frac{\operatorname{D}_{\operatorname{a}}{\left(\varphi\cdot
h\right)}}{T^{\prime}}\circ{\left.T\right|}_{\omega_{k}}^{-1}$
$\displaystyle=\left(\frac{\varphi^{\prime}}{T^{\prime}}\cdot
h\right)\circ{\left.T\right|}_{\omega_{k}}^{-1}+\left(\frac{\varphi}{T^{\prime}}\cdot\operatorname{D}_{\operatorname{a}}{h}\right)\circ{\left.T\right|}_{\omega_{k}}^{-1}.$

Summing over $k$ we obtain the result. ∎

######  Lemma 7.2.

There exist functions $A_{l,p,n}:[0,1]\to\mathbb{C}$, $0\leq p\leq r$ and
$0\leq l\leq p$ such that

$D_{a}^{p}\left(\mathcal{L}_{\varphi}^{n}h\right)=\sum_{l=0}^{p}\mathcal{L}_{\varphi}^{n}(A_{l,p,n}\cdot\operatorname{D}_{\operatorname{a}}^{l}h).$

Furthermore, these functions are $\mathcal{C}^{\infty}$ on $\omega$ for each
$\omega\in\Omega_{n}$ and satisfy,

(22) $\displaystyle A_{0,p+1,n}$
$\displaystyle=\frac{A_{0,p,n}^{\prime}}{(T^{n})^{\prime}}+\frac{A_{0,p,n}}{(T^{n})^{\prime}}\cdot\frac{\varphi_{n}^{\prime}}{\varphi_{n}},$
$\displaystyle A_{l,p+1,n}$
$\displaystyle=\frac{A_{l,p,n}^{\prime}}{(T^{n})^{\prime}}+\frac{A_{l,p,n}}{(T^{n})^{\prime}}\cdot\frac{\varphi_{n}^{\prime}}{\varphi_{n}}+\frac{A_{l-1,p,n}}{(T^{n})^{\prime}},\quad\text{when
$1\leq l\leq p$},$ $\displaystyle A_{p+1,p+1,n}$
$\displaystyle=\frac{A_{p,p,n}}{(T^{n})^{\prime}}.$

###### Proof.

Let $n\in\mathbb{N}$. First observe that $A_{0,0,n}=1$ and, in particular, is
$\mathcal{C}^{\infty}$ on $\omega$ for each $\omega$. In other words the lemma
holds for the case $p=0$. Now suppose that the lemma holds for some fixed $p$.
This means that,

$\displaystyle\operatorname{D}_{\operatorname{a}}^{p+1}\left(\mathcal{L}^{n}_{\varphi}h\right)$
$\displaystyle=\sum_{\ell=0}^{p}\operatorname{D}_{\operatorname{a}}\mathcal{L}^{n}_{\varphi}(A_{\ell,p,n}\cdot\operatorname{D}_{\operatorname{a}}^{\ell}h)$
$\displaystyle=\sum_{\ell=0}^{p}\sum_{\omega\in\Omega_{n}}\operatorname{D}_{\operatorname{a}}\left((\varphi_{n}\cdot
A_{\ell,p,n}\cdot\operatorname{D}_{\operatorname{a}}^{l}h)\circ{\left.T^{n}\right|}_{\omega}^{-1}\right)\cdot\mathbf{1}_{T^{n}\omega}.$

By Lemma 7.1 the term
$\operatorname{D}_{\operatorname{a}}\left((\varphi_{n}\cdot
A_{\ell,p,n}\cdot\operatorname{D}_{\operatorname{a}}^{l}h)\circ{\left.T^{n}\right|}_{\omega}^{-1}\right)$
is equal to

$\left(\frac{\varphi_{n}}{(T^{n})^{\prime}}\left[A_{\ell,p,n}^{{}^{\prime}}\cdot\operatorname{D}_{\operatorname{a}}^{\ell}h+A_{\ell,p,n}\cdot\operatorname{D}_{\operatorname{a}}^{\ell+1}h+\frac{\varphi_{n}^{\prime}}{\varphi_{n}}\cdot
A_{\ell,p,n}\cdot\operatorname{D}_{\operatorname{a}}^{\ell}h\right]\right)\circ{\left.T^{n}\right|}_{\omega}^{-1}.$

Comparing this with the expression for
$D_{a}^{p+1}\mathcal{L}_{\varphi}^{n}(h)$ (i.e., collecting same order
derivatives together), we obtain the claimed relationship between the terms
(22). That the $A_{l,p,n}$ are $\mathcal{C}^{\infty}$ in each
$\omega\in\Omega_{n}$ follows from the proven relationships (22). Indeed, both
$\varphi_{n}$ and $(T^{n})^{\prime}$ are so and bounded away from zero by
assumption. ∎

The following is [20, Lemma 6.1] transposed to the present setting.

######  Lemma 7.3.

Let ${\\{I_{k}\\}}_{k=0}^{\infty}$ be intervals and let $S_{k}:I_{k}\to
I_{k+1}$ be $\mathcal{C}^{r}$ contracting transformations with uniform
contraction171717I.e., there exists $\lambda<1$, $C>0$ such that
$(S^{k})^{\prime}(x)\leq C\lambda^{k}$ for all $k\in\mathbb{N}$, $x\in
I_{0}$.. Furthermore let $\theta_{k}:I_{k}\to(0,\infty)$ be $\mathcal{C}^{r}$
functions. Suppose that there exists $C>0$ such that, for any $x\in I_{k}$ for
any $k$, the $\mathcal{C}^{r}$ norm of $\theta_{k}$ is bounded on a
neighbourhood of $x$ by $C\theta_{k}(x)$. Let $S^{k}=S_{k-1}\circ\cdots\circ
S_{0}$.

There exists $L>0$ such that, for all $x\in I_{0}$,

$\left\|\textstyle\prod_{k=0}^{n-1}\theta_{k}\circ
S^{k}\right\|_{\mathcal{C}^{r}(I_{0})}\leq
L\textstyle\prod_{k=0}^{n-1}\theta_{k}\circ S^{k}(x).$

######  Lemma 7.4.

There exists $C>0$ such that the following holds. If $A$ is a positive
$\mathcal{C}^{r}$ function defined on an interval, such that for all $x$, the
$\mathcal{C}^{r}$ norm of $A$ in a neighbourhood of $x$ is bounded by $LA(x)$,
for some $L$, then,

$\left\|{A^{\prime}}/{A}\right\|_{\mathcal{C}^{r-1}}\leq CL^{r}.$

###### Proof.

The $q$th derivative of the quotient $\frac{A^{\prime}}{A}$ is equal to a sum
of terms of the form

$\frac{A^{(q_{1})}}{A}\cdot\frac{A^{(q_{2})}}{A}\cdots\frac{A^{(q_{j})}}{A}$

where the $q_{1}+q_{j}+\cdots+q_{j}=q+1$. That this is true for $q=0$ is
immediate. If we assume that the statement is true for $q$, then we can
observe that differentiating only produces terms of the same type.
Consequently $\left\|{A^{\prime}}/{A}\right\|_{\mathcal{C}^{r-1}}$ is bounded
by $L^{r}$ multiplied by a combinatorial constant $C$ which depends only on
$r$. ∎

Lemma 7.2 proved the existence of the $A_{l,p,n}$ and, from the formula (22)
for $A_{p+1,p+1,n}$, the fact that $A_{p,p,n}={((T^{n})^{\prime})}^{-p}$. To
complete the proof of Lemma 6.7, it remains to show that there exists $C>0$
such that, for $\left\|A_{l,p,n}\right\|_{L^{\infty}}\leq C$, in particular
that the bound is independent of $n$.

Let $n\in\mathbb{N}_{0}$, $\omega\in\Omega_{n}$, $x\in\omega$. Since
${\left.T^{n}\right|}_{\omega}$ is invertible,

(23) $B_{l,p,n,\omega}=A_{l,p,n}\circ{\left.T^{n}\right|}_{\omega}^{-1}$

is well defined on $T^{n}\omega$.

######  Lemma 7.5.

Let $0\leq p\leq r$. There exists $C_{p}>0$ such that, for all
$n\in\mathbb{N}_{0}$, $\omega\in\Omega_{n}$, $0\leq l\leq p$,

$\left\|B_{l,p,n,\omega}\right\|_{\mathcal{C}^{r-p}(T^{n}\omega)}\leq C_{p}.$

###### Proof.

First note that $B_{0,0,n}=1$ for all $n\in\mathbb{N}_{0}$ and so the
statement of the lemma holds for $p=0$.

Proceeding by induction on $p$, let us suppose that the statement of the lemma
is true for $p$. By Lemma 7.2,

$A_{l,p+1,n}=\frac{A_{l,p,n}^{\prime}}{(T^{n})^{\prime}}+\frac{A_{l,p,n}}{(T^{n})^{\prime}}\cdot\frac{\varphi_{n}^{\prime}}{\varphi_{n}}+\frac{A_{l-1,p,n}}{(T^{n})^{\prime}},$

and consequently,

(24)
$B_{l,p+1,n,\omega}=B_{l,p,n,\omega}^{\prime}+B_{l,p,n,\omega}\cdot\left({\Phi_{n}^{\prime}}/{\Phi_{n}}\right)+B_{l-1,p,n,\omega}\cdot\Theta_{n},$

where, for convenience we write
$\Phi_{n}=\varphi_{n}\circ{\left.T^{n}\right|}_{\omega}^{-1}$ and
$\Theta_{n}=\frac{1}{(T^{n})^{\prime}}\circ{\left.T^{n}\right|}_{\omega}^{-1}$.
(In the case $l=0$ or $l=p+1$ not all of the three terms are present but this
only makes the following estimate easier.) Taking the $\mathcal{C}^{r-(p+1)}$
norm181818$\mathcal{C}^{r}$ is the closure of $\mathcal{C}^{\infty}$ with
respect to the norm ${\|h\|}_{\mathcal{C}^{r}}=\sup_{0\leq k\leq
r}2^{r-k}{\|h^{(k)}\|}_{L^{\infty}}$and so
$(\mathcal{C}^{r},{\|\cdot\|}_{\mathcal{C}^{r}})$ is a Banach algebra. and
using the inductive hypothesis,

$\left\|B_{l,p+1,n}\right\|_{\mathcal{C}^{r-(p+1)}}\leq
C_{p}\left(2+\left\|{\Phi_{n}^{\prime}}/{\Phi_{n}}\right\|_{\mathcal{C}^{r}}+\left\|\Theta_{n}\right\|_{\mathcal{C}^{r}}\right).$

It was assumed that $\varphi$ is $\mathcal{C}^{\infty}(\mathcal{I})$ and
bounded away from zero, consequently there exists $C>0$ such that, for all
$x$, the $\mathcal{C}^{r}$ norm of $\varphi$ in a neighbourhood of $x$ is
bounded by $C\varphi(x)$. The same property holds also for
$\frac{1}{T^{\prime}}$. This means that Lemma 7.3 applies in both cases. This
yields directly $\left\|\Theta_{n}\right\|_{\mathcal{C}^{r}}\leq L$. As for
the quotient ${\Phi_{n}^{\prime}}/{\Phi_{n}}$, we also apply Lemma 7.4 for the
family of functions $\Phi_{n}$, which yields
$\left\|{\Phi_{n}^{\prime}}/{\Phi_{n}}\right\|_{\mathcal{C}^{r}}\leq CL^{r}$,
independently on $n$. ∎

Let $n\in\mathbb{N}_{0}$, $\omega\in\Omega_{n}$, $x\in\omega$. By definition
(23), $A_{l,p,n}(x)=B_{l,p,n,\omega}(T^{n}x)$. Lemma 7.5 provides, in
particular, a uniform $L^{\infty}$ bound on $B_{l,p,n,\omega}$ and this
concludes the proof of Lemma 6.7.

## Acknowledgements

We thank Carlangelo Liverani for motivating us to consider this problem (by
encouraging us to prove what we now know to be impossible) and for many
helpful discussions. We are grateful to Viviane Baladi, Roberto Castorrini,
Marco Lenci, Stefano Luzzatto, David Seifert and Daniel Smania for helpful
comments, corrections and suggestions. We also would like to thank the
anonymous referees for correcting some mistakes and pointing out relevant
missing references. Partially supported by PRIN Grant “Regular and stochastic
behaviour in dynamical systems” (PRIN 2017S35EHN) and MIUR Excellence
Department Project awarded to the Department of Mathematics, University of
Rome Tor Vergata, CUP E83C18000100006.

## Statements and Declarations

The authors have no relevant financial or non-financial interests to disclose.
Data sharing not applicable to this article as no datasets were generated or
analysed during the current study. Also, no llamas were involved in the
preparation of this manuscript.

## References

  * [1] L. Ambrosio, N. Fusco & D. Pallara. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, (2000).
  * [2] V. Araújo & I. Melbourne. Exponential decay of correlations for nonuniformly hyperbolic flows with a $\mathcal{C}^{1+\alpha}$ stable foliation, including the classical Lorenz attractor. _Annales Henri Poincaré_ 17, 2975–3004 (2016).
  * [3] A. Arbieto & D. Smania. Transfer operators and atomic decomposition. _Preprint_ arXiv:1903.06943v3 (2020).
  * [4] V. Baladi & G. Keller. Zeta functions and transfer operators for piecewise monotone transformations. _Commun. Math. Phys._ 127, 459–477 (1990).
  * [5] V. Baladi. Positive transfer operators and decay of correlations. _Advanced Series in Nonlinear Dynamics, 16._ World Scientific Publishing Co., Inc., River Edge, NJ, (2000).
  * [6] V. Baladi & M. Tsujii. Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. _Ann. Inst. Fourier (Grenoble)_ 57, 127–154 (2007).
  * [7] V. Baladi & M. Tsujii. Dynamical determinants and spectrum for hyperbolic diffeomorphisms. In: K. Burns, D. Dolgopyat, Ya. Pesin (eds.) Probabilistic and Geometric Structures in Dynamics, 29–68, _Contemp. Math., 469_ , Amer. Math. Soc., Providence, RI (2008).
  * [8] V. Baladi. Dynamical zeta functions and dynamical determinants for hyperbolic maps – A functional approach. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge. A series of Modern Surveys in Mathematics 68. Springer (2018).
  * [9] V. Baladi, M. Demers & C. Liverani. Exponential decay of correlations for finite horizon Sinai billiard flows. _Inventiones_ 211, 39–177 (2018).
  * [10] O. Butterley. An alternative approach to generalised $\operatorname{BV}$ and the application to expanding interval maps. _Discrete Contin. Dyn. Syst._ 33, 3355–3363 (2013).
  * [11] O. Butterley. Area expanding $\mathcal{C}^{1+\alpha}$ suspension semiflows. _Commun. Math. Phys.,_ 325, 803–820 (2014).
  * [12] O. Butterley, N. Kiamari & C. Liverani. Locating Ruelle-Pollicott resonances. _Nonlinearity_ 35, 513 (2022).
  * [13] N. Chernov & R. Markarian. Chaotic billiards. Mathematical surveys and monographs 127. _American Mathematical Soc._ , ISBN: 0-8218-4096-7 (2006).
  * [14] V. Climenhaga & D.J. Thompson. Intrinsic ergodicity beyond specification: $\beta$-shifts, S-gap shifts, and their factors. _Isr. J. Math.,_ 192, 785–817 (2012).
  * [15] P. Collet & S. Isola. On the essential spectrum of the transfer operator for expanding Markov maps. _Commun. Math. Phys.,_ 139, 551–557 (1991).
  * [16] E.B. Davies. Linear operators and their spectra. Cambridge studies in Advanced Mathematics, 106. Cambridge University Press, Cambridge (2007).
  * [17] D.E. Edmunds & W.D. Evans. Spectral theory and differential operators. Oxford Clarendon Press (1987).
  * [18] F. Faure, S. Gouëzel & E. Lanneau. Ruelle spectrum of linear pseudo-Anosov maps. _J. l’École polytechnique – Mathématiques_ 6, 811–877 (2019).
  * [19] L. Flatto, J.C. Lagarias & B. Poonen. The zeta function of the beta transformation. _Ergodic Theory Dynam. Systems_ 14, 237–266 (1994).
  * [20] S. Gouëzel & C. Liverani. Banach spaces adapted to Anosov systems. _Ergodic Theory Dynam. Systems_ 26, 1, 189–217 (2006).
  * [21] S. Gouëzel & C. Liverani. Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties. _J. Diff. Geometry_ 79, 433–477 (2008).
  * [22] V.M. Gundlach & Y. Latushkin. A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Hölder spaces. _Ergodic Theory Dynam. Systems_ 23, 175–191 (2003).
  * [23] F. Hofbauer & G. Keller. Zeta-functions and transfer-operators for piecewise linear transformations. _J. reine und angew. Math._ 352, 100–113 (1984).
  * [24] G. Keller. On the rate of convergence to equilibrium in one-dimensional systems. _Commun. Math. Phys._ 96, 181–193 (1984).
  * [25] G. Keller. Generalized bounded variation and applications to piecewise monotonic transformations. _Z. für Wahrscheinlichkeitstheorie und verw. Gebiete._ 69, 461–478 (1985).
  * [26] G. Keller. Markov extensions, zeta functions, and Fredholm theory for piecewise invertible dynamical systems. _Trans. Am. Math. Soc._ 314, 433–497 (1989).
  * [27] G. Keller & C. Liverani. Stability of the spectrum for transfer operators. _Ann. Scuola Norm. Sup. Pisa Cl. Sci._ 28, 141–152 (1999).
  * [28] G. Keller & H.H. Rugh. Eigenfunctions for smooth expanding circle maps. _Nonlinearity_ 17, 1723–1730 (2004).
  * [29] C. Liverani. Rigorous numerical investigation of the statistical properties of piecewise expanding maps – a feasibility study. _Nonlinearity_ , 14, 463–490 (2001).
  * [30] C. Liverani. Invariant measures and their properties. A functional analytic point of view. _Dynamical systems. Part II._ Scuola Normale Superiore, Pisa. Centro di Ricerca Matematica “Ennio De Giorgi” 185–237 (2004).
  * [31] A. Lasota & J. Yorke. On the existence of invariant measures for piecewise monotonic transformations. _Trans. Amer. Math. Soc._ 186, 481–488 (1974).
  * [32] J. Milnor & W. Thurston. On iterated maps of the interval. _Alexander, J.C. (eds) Dynamical Systems. Lecture Notes in Mathematics_ 1342, 465–563 (1988).
  * [33] R.D. Nussbaum. The radius of the essential spectrum. _Duke Math. J._ 37, 473–478 (1970).
  * [34] W. Parry. On the $\beta$-expansion of real numbers. _Acta Math. Acad. Sci. Hungary_ 11, 401–416 (1960).
  * [35] M. Rychlik. Bounded variation and invariant measures. _Studia Mathematica_ 76, 69–80 (1983).
  * [36] D. Ruelle. An extension of the theory of Fredholm determinants. _Publ. Math. Inst. Hautes Études Sci._ 72, 175–193 (1990).
  * [37] D. Smania. Transfer operators, atomic decomposition and the bestiary. _Preprint_ arXiv:1903.06976v2 (2021).
  * [38] R.S. Strichartz. Multipliers on fractional Sobolev spaces. _J. of Math. and Mechanics_ , 16, 1031–1060 (1967).
  * [39] Y. Takahashi. $\beta$-transformations and symbolic dynamics. Proceedings of the Second Japan-USSR Symposium on Probability Theory, 455–464 (1973). Lecture Notes in Mathematics 330. Springer.
  * [40] D. Thomine. A spectral gap for transfer operators of piecewise expanding maps. _Discrete Contin. Dyn. Syst._ , 30, 917–944 (2011).
  * [41] M. Viana. What’s new on Lorenz strange attractors? _The Mathematical Intelligencer_ 22, 6–19 (2000).