# Persistent Non-Statistical Dynamics
in One-Dimensional Maps

Douglas Coates  University of Exeter, Exeter, UK<EMAIL_ADDRESS>and
Stefano Luzzatto  Abdus Salam International Centre for Theoretical Physics
(ICTP), Trieste, Italy<EMAIL_ADDRESS>www.stefanoluzzatto.net

(Date: 2 September 2023)

###### Abstract.

We study a class $\widehat{\mathfrak{F}}$ of one-dimensional full branch maps
introduced in [CoaLuzMub22], admitting two indifferent fixed points as well as
critical points and/or singularities with unbounded derivative. We show that
$\widehat{\mathfrak{F}}$ can be partitioned into 3 pairwise disjoint
subfamilies
$\widehat{\mathfrak{F}}=\mathfrak{F}\cup\mathfrak{F}_{\pm}\cup\mathfrak{F}_{*}$
such that all $g\in\mathfrak{F}$ have a unique physical measure equivalent to
Lebesgue, all $g\in\mathfrak{F}_{\pm}$ have a physical measure which is a
Dirac-$\delta$ measure on one of the (repelling) fixed points, and all
$g\in\mathfrak{F}_{*}$ are non-statistical and in particular have no physical
measure. Moreover we show that these subfamilies are _intermingled_ : they can
all be approximated by maps in the other subfamilies in natural topologies.

###### 2020 Mathematics Subject Classification:

37A10, 37C40, 37C83, 37E05

D.C. was partially supported by the ERC project 692925 _NUHGD_ and by the
Abdus Salam ICTP visitors program.

##  1\. Introduction

We study the existence and, especially, the _non-existence_ of physical
measures in a large family of interval maps $\widehat{\mathfrak{F}}$ which was
introduced in [CoaLuzMub22]. We start with heuristic and conceptual overview
of results. Then in Section 2 we give the formal definition of the family
$\widehat{\mathfrak{F}}$ and the precise technical statements of our results.
In Section 3 we recall the main construction and key estimates from
[CoaLuzMub22] which will be required, and in Sections 4-6 we prove our
results.

###  1.1. Overview of Results

The family $\widehat{\mathfrak{F}}$ consists of full branch maps with two
orientation preserving branches, which are all in the same topological
conjugacy class of uniformly expanding maps such as $f(x)=2x$ mod 1, in
particular they are all topologically conjugate to each other. Depending on a
number of parameters, they may however exhibit quite different features: the
two fixed points are always topologically repelling but may be either
hyperbolic or neutral and the branches may have critical points and/or
singularities with infinite derivative. $\widehat{\mathfrak{F}}$ contains
uniformly expanding maps and well-known intermittent maps as well as many
other maps which, as far as we know have not been studied before, see
[CoaLuzMub22] for an extensive discussion and review of the literature. Figure
1 shows some possible graphs of the maps in $\widehat{\mathfrak{F}}$, and see
Section 2.1 for the formal definition.

Figure 1. Graph of $g$ for various possible values of parameters.

The first main result of this paper is a complete classification of maps in
$\widehat{\mathfrak{F}}$ from the point of view of the kind of physical
measure they admit (or not). In particular we will prove the following.

######  Theorem A.

The family of maps $\widehat{\mathfrak{F}}$ is the union of three non-empty
pairwise disjoint subfamilies

(1)
$\widehat{\mathfrak{F}}=\mathfrak{F}\cup\mathfrak{F}_{\pm}\cup\mathfrak{F}_{*}$

satisfying the following properties:

1) all $g\in\mathfrak{F}$ have a physical measure equivalent to Lebesgue;

2) all $g\in\mathfrak{F}_{\pm}$ have a physical measure supported on a
repelling fixed point;

3) all $g\in\mathfrak{F}_{*}$ are non-statistical and in particular have no
physical measure.

The definitions of physical measure and and non-statistical maps are given in
Section 1.2, and the families
$\mathfrak{F},\mathfrak{F}_{\pm},\mathfrak{F}_{*}$ are defined explicitly in
Section 2.2 in terms of various parameters of the maps in
$\widehat{\mathfrak{F}}$ . Item 1) in Theorem A, i.e. the existence of
physical measures equivalent to Lebesgue for maps in $\mathfrak{F}$, was
proved in [CoaLuzMub22], where it was also shown that such physical measure
satisfy a number of statistical properties such as decay of correlations and
various limit theorems Our focus in this paper is therefore on the
complementary families $\mathfrak{F}_{\pm}$ and $\mathfrak{F}_{*}$. We note
that the three families $\mathfrak{F},\mathfrak{F}_{\pm},\mathfrak{F}_{*}$
form a partition of $\widehat{\mathfrak{F}}$, _there are no cases in which we
are not able to obtain a conclusion_ (although, as we shall see in the proofs,
there are some boundary cases in which more sophisticated arguments are
required).

A natural question concerns the “ _size_ ” and “ _structure_ ” of the families
$\mathfrak{F},\mathfrak{F}_{\pm},\mathfrak{F}_{*}$ inside
$\widehat{\mathfrak{F}}$, and perhaps one of the most surprising and
unexpected results of this paper is that they are _intermingled_ and _dense_
in some natural subsets of $\widehat{\mathfrak{F}}$ in some appropriate
topologies. Due to the presence of the discontinuity we cannot use the
standard $C^{r}$, or even $C^{0}$, metric on $\widehat{\mathfrak{F}}$, since
the maps are not even continuous. However there are pairs of maps
$f,g\in\widehat{\mathfrak{F}}$ whose _difference_ $f-g\in C^{r}$, and which
therefore may be considered as $C^{r}$ _perturbations_ one of the other. This
observation motivates the definition of a natural extended metric on
$\widehat{\mathfrak{F}}$ defined as follows: for any
$f,g\in\widehat{\mathfrak{F}}$ and $r\geq 0$ we let

$d_{r}(f,g)\coloneqq\begin{cases}\|f-g\|_{C^{r}}&\text{ if }f-g\in C^{r}\\\
\infty&\text{ otherwise}.\end{cases}$

For simplicity we will just refer to it as $C^{r}$ _metric_ on
$\widehat{\mathfrak{F}}$111Notice that we can “normalize” this extended metric
to give a standard bounded metric on $\widehat{\mathfrak{F}}$ by defining
$\tilde{d}_{r}(f,g)\coloneqq{\tilde{d}_{r}(f,g)}/{(1+\tilde{d}_{r}(f,g))}$
when $d_{r}(f,g)<\infty$ and $\tilde{d}_{r}(f,g)\coloneqq 1$ otherwise. The
metrics $d_{r}$ and $\tilde{d}_{r}$ lead to equivalent topologies and so for
our purposes it does not really matter which one we use.. There are of course
many maps which are at infinite distance from each other but, as we shall see,
there is nevertheless a very rich structure in every neighbourhood of maps
$f\in\widehat{\mathfrak{F}}$ as we have the following surprising and
remarkable result.

######  Theorem B.

Each of the families $\mathfrak{F}$, $\mathfrak{F}_{\pm}$, $\mathfrak{F}_{*}$
are $C^{0}$-dense in $\widehat{\mathfrak{F}}$.

Theorem B says, more precisely, than any $f\in\widehat{\mathfrak{F}}$,
belonging to any of the classes $\mathfrak{F}$, $\mathfrak{F}_{\pm}$,
$\mathfrak{F}_{*}$, can be approximated arbitrarily closely in the $C^{0}$
metric by maps in the other two classes. For example, maps with a physical
measure equivalent to Lebesgue, including uniformly expanding maps, can be
$C^{0}$ approximated both by maps with physical measures given by Dirac-delta
measures on the fixed points and also by maps without physical measures.
Similarly, maps without physical measures can be $C^{0}$ approximated both by
maps with a physical measure equivalent to Lebesgue and by maps with Dirac-
delta physical measures on the fixed points.

Theorem B will be proved as a special case of a more technical but much more
general result, see Theorem 4 below, which also implies that maps in the
families $\mathfrak{F}$, $\mathfrak{F}_{\pm}$, $\mathfrak{F}_{*}$ can be
approximated by maps in the other families in arbitrarily regular metrics,
depending on the maps. In particular, for the case $r=1$ we define the set

$\widetilde{\mathfrak{F}}\coloneqq\\{f\in\widehat{\mathfrak{F}}:\text{both
fixed points are neutral fixed points}\\}$

and we have the following perhaps even more surprising and remarkable result.

######  Theorem C.

Each of the families $\mathfrak{F}$, $\mathfrak{F}_{\pm}$, $\mathfrak{F}_{*}$
are $C^{1}$-dense in $\widehat{\mathfrak{F}}$.

Theorem C says that every $C^{1}$ neighbourhood of every map
$f\in\widetilde{\mathfrak{F}}$ contains maps belonging to _all three families_
$\mathfrak{F}$, $\mathfrak{F}_{\pm}$, $\mathfrak{F}_{*}$. In particular every
map with a physical measure equivalent to Lebesgue can be $C^{1}$ approximated
by maps without physical measures, and vice-versa.

Finally, we address the question of how _persistent_ is the dynamics
corresponding to the families $\mathfrak{F}$, $\mathfrak{F}_{\pm}$,
$\mathfrak{F}_{*}$, i.e. how _“large”_ each family is and how _“robust”_ with
respect to perturbations. Given Theorems B and C above, none of these families
can be _open_ in either the $C^{0}$ or the $C^{1}$ metrics as defined above,
however we will see that there are several ways in which we can argue that
each family is _large_ in some sense. In particular, letting
$\operatorname{supp}f\coloneqq\\{x:f(x)\neq 0\\}$ denote the support of
function, we will prove the following result.

######  Theorem D.

Let $f\in\mathfrak{F}_{*}$. If $g\in\widetilde{\mathfrak{F}}$ is such that
$\overline{\operatorname{supp}(f-g)}\subset(-1,0)\cup(0,1),$ then
$g\in\mathfrak{F}_{*}$.
The same statement is true if we replace $\mathfrak{F}_{*}$ with either of the
other two subclasses $\mathfrak{F},\mathfrak{F}_{\pm}$.

Theorem D shows that the families $\mathfrak{F}$, $\mathfrak{F}_{\pm}$,
$\mathfrak{F}_{*}$ are open under a particular class of perturbations which is
slightly more restrictive than those allowed under the general $C^{r}$ metric
defined above but still quite substantial. This is of course particularly
remarkable when applied to maps in $\mathfrak{F}_{\pm}$ whose physical
measures are Dirac-delta measures on fixed points, and to maps in
$\mathfrak{F}_{*}$ without physical measures. As far as we know there are no
other previously known examples of systems with non-statistical behaviour
which is as robust as this. There are also other ways in which the families
$\mathfrak{F}$, $\mathfrak{F}_{\pm}$, $\mathfrak{F}_{*}$ are persistent but
these need to be formulated in terms of the parameters which define the maps
and we therefore postpone the statements to Section 2.4 below.

###  1.2. Physical Measures

In this section we give the precise definition of physical measure and non-
statistical dynamics and give a brief discussion of relevant previously
existing results.

####  1.2.1. Topological and Statistical Limits

For completeness we start with some general background notions which help to
motivate the definitions and the results. Given a set $X$, a map $f:X\to X$
determines a Dynamical System by defining the $n$’th iterate
$f^{n}=f\circ\cdots\circ f$ by the $n$-fold composition of $f$ with itself,
and for any $x_{0}\in X$ we define the $n$’th iterate of $x_{0}$ under $f$ as
the image $x_{n}=f^{n}(x)$. We can think of $X$ as a _state space_ , or the
collection of all possible configurations of some system, the map $f$ as a
_force_ or _mechanism_ which acts on the system, and $x_{0}$ as an _initial
condition_. Then the sequence

$\mathcal{O}(x):=\\{x_{n}\\}_{n=0}^{\infty},$

which we call the (forward) _orbit_ of $x_{0}$, denotes the _evolution in
time_ of the system starting from the initial condition $x_{0}$. The main
objective of the Theory of Dynamical Systems is essentially that of describing
the structure of orbits and how they depend on the initial condition $x_{0}$
and on the map $f$. If $X$ is a _topological space_ we can define the _omega-
limit_ set $\omega(x):=\\{y\in X:y\text{ is an accumulation point of the
sequence }\mathcal{O}(x)\\},$ which gives information about the asymptotic
nature of the orbit from a _topological_ point of view. If $X$ is a _measure
space_ we can define the Dirac-delta point mass measures $\delta_{x_{k}}$ at
each point of the orbit and use this to describe the orbit by equidistributing
the mass on the first $n$ terms of the orbit, giving a sequence

$\mu_{n}(x_{0})\coloneqq\frac{1}{n}\sum_{k=0}^{n-1}\delta_{x_{k}}$

of probability measures associated to the initial condition $x_{0}$. If this
sequence _converges_ , for example in the weak star-topology, i.e. if there
exists a probability measure $\mu$ such that

(2) $\mu_{n}(x_{0})\to\mu,$

then $\mu_{n}$ approximates $\mu$ but, most importantly, for all sufficiently
large $n\geq 0$, $\mu$ _approximates_ $\mu_{n}$, and therefore $\mu$ gives an
asymptotic description of the orbit from a _statistical_ point of view.

Notice that if $X$ is a metric space we can describe each orbit _from both a
topological and a statistical point of view_. In many cases these two
descriptions are intuitively consistent one with the other, for example if $X$
is a complete metric space and $f$ is a contraction and $p\in X$ is the unique
fixed point of $f$, then is easy to check that for any initial condition
$x_{0}\in X$ the points of the orbit of $x_{0}$ converge to $p$ and therefore
$\omega(x_{0})=\\{p\\}$ and $\mu_{n}(x_{0})\to\delta_{p}$. Similarly, for
irrational circle rotations it is relatively easy to check that the orbit of
every point $x_{0}\in\mathbb{S}^{1}$ is dense in $\mathbb{S}^{1}$, and
therefore $\omega(x_{0})=\mathbb{S}^{1}$, and it is a classical (but non-
trivial) result that every orbit is uniformly distributed and therefore
$\mu_{n}(x)\to$ _Lebesgue_ on $\mathbb{S}^{1}$. However this is not always the
case and sometimes, such as in several examples which will be discussed below,
and in some of the cases mentioned in our results, the topological and
statistical description depend on the initial condition and yield quite
different pictures of what we consider as the “typical” dynamics of the
system.

####  1.2.2. Definition of Physical Measures

Given a probability measure $\mu$ we define its _basin_

$\mathscr{B}_{\mu}\coloneqq\left\\{x:\mu_{n}(x)\to\mu\right\\}.$

The set $\mathscr{B}_{\mu}$ may very well be empty but, if
$\mathscr{B}_{\mu}\neq\emptyset$ then a natural question is to study it’s
size. Suppose $X$ is a measure space with a normalized reference (Lebesgue)
measure denoted by $Leb$.

######  Definition 1.1.

A probability measure $\mu$ is a _physical measure_ (with _full measure
basin_) for $f$ if

$Leb(\mathscr{B}_{\mu})=1$

More generally, we say that $\mu$ is a physical measure if
$Leb(\mathscr{B}_{\mu})>0$ but the examples we consider in this paper will
always have full measure basin so for simplicity, unless otherwise specified,
we will always implicitly assume that physical measures have full measure
basins.

####  1.2.3. Physical measures on attractors

There are plenty of examples of dynamical systems with physical measures, for
example: the Dirac-delta $\delta_{p}$ in the fixed point for a contraction (in
which case every point actually belongs to the basin); Lebesgue measure for
irrational rotations (in which case also, every point belongs to the basin)
and also for piecewise affine expanding circle maps such as $f(x)=2x$ mod 1
(in which case the basin has full Lebesgue measure but its complement is non-
empty and indeed consist of an uncountable set of points). It follows by a
classical theorem of Birkhoff [Bir31] that if a probability measure $\mu$ is
invariant ($\mu(f^{-1}(A))=\mu(A)$ for every measurable set $A$), ergodic
($f^{-1}(A)=A\Rightarrow\mu(A)=0$ or $\mu(A)=1$), and _absolutely continuous
with respect to Lebesgue_ ($Leb(A)=0\Rightarrow\mu(A)=0$) then $\mu$ is a
physical measure and if $\mu$ is _equivalent to Lebesgue_
($Leb(A)=0\Leftrightarrow\mu(A)=0$), then $\mu$ is a physical measure with
full measure basin.

Such ergodic invariant measures equivalent to, or absolutely continuous with
respect to, Lebesgue have been proved to exist in many classes of _uniformly
and non-uniformly expanding maps_. There is a huge literature so we just
mention a few significant papers such as [AlvLuzPin05, AraLuzVia09, Buz00,
DiaHolLuz06, LasYor73, Pin06, Tsu00c, Tsu01a, Tsu05] but highlight in
particular those related to one-dimensional maps with neutral fixed points
[Dol04, BahGalNis18, BahSau16, BalTod16, BruLep13, BruTerTod19, CamIso95,
CoaHolTer19, CriHayMar10, Cui21, Dua12, FisLop01, FreFreTod13, FreFreTod16,
FroMurSta11, Ino00, Kor16, LivSauVai99, Mel08, MelTer12, NicTorVai16, Pia80,
PolSha09, PolWei99, PomMan80, Ruz15, Ruz18, Sar01, SheStr13, Ter13, Ter15,
Tha00, Tha05, Tha80, Tha83, Tha95, Tha95a, You99, Zwe00, Zwe03, Zwe98].

In higher dimensions, where we may have a non-trivial attractor $\Lambda$ of
zero Lebesgue measure, if $\Lambda$ satisfies some hyperbolicity conditions
then pioneering work of Anosov, Sinai, Ruelle, and Bowen in the 1960s and
1970s showed that the absolute continuity can be replaced by a weaker but more
technical condition of _absolute continuity of conditional measures_ and
_absolute continuity of the stable foliation_ , and measures satisfying such
properties are often referred to as _Sinai-Ruelle-Bowen_ , or _SRB_ , measures
[AnoSin67, Bow75, Rue76, Sin72]. Over the following decades then there has
been a tremendous amount of research which has extended their results to
increasingly general classes of systems, we mention here just some of the more
recent papers [AlvDiaLuz17, Bur21, BuzCroSar22, CliLuzPes17, CliLuzPes23,
Vec22] and refer the reader to those for a more comprehensive list of
references, and the formulation of a far-reaching conjecture of Palis to the
effect that “typical” dynamical systems have (a finite number of) physical
measures [Pal08, Pal15].

####  1.2.4. Physical measures on repellors

A rarer, but in many ways more interesting and intriguing, class of examples
of physical measures consists of systems in which the physical measure is
supported on an invariant set $\Lambda$ which has zero Lebesgue measure and is
topologically _repelling_ , in the sense that points close to $\Lambda$ are
mapped _away_ from $\Lambda$ rather than _towards_ $\Lambda$. The simplest
example of this phenomenon, which is also very relevant for the class of maps
which we consider in this paper, is given by well known Manneville-Pomeau
_intermittency map_ $f(x)=x+x^{2}$ mod 1 [PomMan80]. It is easy to see that
$f(0)=0$, and so the origin is a fixed point, and that $f^{\prime}(x)=1+2x$,
so that $f^{\prime}(0)=1$ and $f^{\prime}(x)>1$ for all $x>0$. In particular
every point in any small neighbourhood of the origin, except the origin
itself, eventually leaves such a neighbourhood, and in this sense the origin
is topologically repelling. However it can be shown that asymptotically most
points of the orbits belong to arbitrarily small neighbourhoods of the origin,
and in fact _the empirical measures $\mu_{n}(x)$ converge to the Dirac-delta
measure $\delta_{0}$ at the origin_, which is therefore a physical measure
with full basin, see [Alv20] and some of the paper mentioned above. Notice
that the Manneville-Pomeau intermittency map is included in our family
$\widehat{\mathfrak{F}}$, see discussion in [CoaLuzMub22], and the result just
mentioned is a special case of our Theorem 2 below. In the 1990s, Hofbauer and
Keller proved the even more surprising result that there are many examples in
smooth quadratic unimodal maps for which a similar phenomena occurs [HofKel90,
HofKel95].

####  1.2.5. Non-Statistical Maps

Equally, if not even more, interesting are dynamical systems which _do not
admit any physical measure._ The simplest, and somewhat trivial, way in which
this can occur is when the basin of every probability measure $\mu$ has zero
Lebesgue measure, such as in the identity map for which
$\mu_{n}(x)\to\delta_{x}$, for every $x$ or for rational circle rotations for
which all orbits are periodic. A much more sophisticated and interesting way
in which a map can fail to have physical measures is when there exists a full
measure set of points for which the sequence of measures $\mu_{n}(x)$ _does
not converge_ , in this case we say that the orbit of $x$ is _non-
statistical_. More formally, letting

$\mathscr{N}:=\\{x\in X:\mu_{n}(x)\text{ does not converge}\\}$

we can make the following definition.

######  Definition 1.2.

A map $f:X\to X$ is _non-statistical_ if

$Leb(\mathscr{N})=1$

Notice that non-convergence of $\mu_{n}(x)$ means that there must exist at
least two measures, $\hat{\mu},\tilde{\mu}$, and two subsequences
$n_{i}\to\infty,n_{j}\to\infty$ such that $\mu_{n_{i}}(x)\to\hat{\mu}$ and
$\mu_{n_{j}}(x)\to\tilde{\mu}$. This means that there is an infinite sequence
of times for which the statistics of the orbit is extremely well described by
the measure $\hat{\mu}$ and another sequence of times for which the statistics
of the orbit is extremely well described by the measure $\tilde{\mu}$. We can
think of these sequences as defining a series of _timescales_ at which we see
completely different statistical behaviour and therefore the observed
frequency of visits to any particular region does not stabilize as
$n\to\infty$.

There is quite a large bibliography of research exploring the notion of the
non-existence of physical measures from different points of view and giving a
number, albeit quite limited, of examples [AarThaZwe05, AraPin21, Bac99,
BarKirNak20, BerBie22, ColVar01, CroYanZha20, Her18, HofKel90, HofKel95,
HuYou95, Ino00, JarTol04, KanKirLi16, KarAsh11, Kel04, KirLiNak22, KirLiSom10,
KirNakSom19, KirNakSom21, KirNakSom22, KirSom17, Kle06, LabRod17, Pal15,
Tak08, Tak94, Tal20, Tal22, Zwe02]. We give here only a short and non
comprehensive review of some of these and refer the reader to the original
papers for additional information.

Arguably the first example of a non-statistical system is the _Bowen eye_ ,
attributed by Takens [Tak94] to Bowen in the early 70s. The Bowen eye is a two
dimensional vector field with an eye-like region whose boundary is formed by
two saddle connections between two fixed points and under carefully chosen
conditions orbits tend to oscillate between the two in a non-statistical way.

This example is somewhat “mild” because the dynamics is very simple but,
around the same time Hofbauer and Keller [HofKel90, HofKel95, Kel04] showed
that there are (uncountably) many parameters in the logistic family
$f_{\lambda}(x)\coloneqq\lambda x(1-x)$ for which $f_{\lambda}$ is
_topologically mixing_ but _non-statistical_. Very recently Talebi [Tal22]
generalized and extended this result to the setting of complex rational maps.
Another approach to the construction of non-statistical examples, or at least
examples with positive Lebesgue measure of non-statistical points, is by
constructing _wandering domains_ with non-statistical dynamics [ColVar01,
KirSom17, BerBie22] and a further example of non-statistical behaviour appears
in [CroYanZha20] where the authors construct a skew product
$F:\mathbb{T}^{2}\times\mathbb{R}\to\mathbb{T}^{2}\times\mathbb{R}$ which
gives rise to non-statistical behaviour using the fact that skew translations
over Anosov diffeomorphisms share properties with Brownian motion. Some
results related to ours, for interval maps with two neutral fixed points, were
also obtained in [Zwe00, AarThaZwe05] in a somewhat more abstract setting and
with a particular focus on the existence and properties of a sigma-finite
invariant measure.

As far as we know, none of the existing results considers a class of maps
anywhere near as large as the family $\widehat{\mathfrak{F}}$ considered here,
nor gives such a complete and a systematic characterization of the various
kinds of physical measures as given in Theorem A. Most importantly, none of
the existing results comes anywhere close to constructing examples of
toplogically mixing maps without physical measures which are so _prevalent_
and _persistent_ , as described in Theorems B, C, D.

##  2\. Statement of Results

We now give the precise definition of the family of maps
$\widehat{\mathfrak{F}}$ and the subfamilies $\mathfrak{F}$,
$\mathfrak{F}_{\pm}$, $\mathfrak{F}_{*}$ as well as some more general
technical theorems which imply the main theorems in Section 1.1 above.

###  2.1. Doubly Intermittent Full Branch Maps

We consider the class of maps introduced in [CoaLuzMub22]. For completeness we
recall the precise definitions. Let $I,I_{-},I_{+}$ be compact intervals, let
$\mathring{I},\mathring{I}_{-},\mathring{I}_{+}$ denote their interiors, and
suppose that $I=I_{-}\cup I_{+}$ and
$\mathring{I}_{-}\cap\mathring{I}_{+}=\emptyset$.

(A0) $g:I\to I$ is _full branch_ : the restrictions
$g_{-}:\mathring{I}_{-}\to\mathring{I}$ and
$g_{+}:\mathring{I}_{+}\to\mathring{I}$ are orientation preserving $C^{2}$
diffeomorphisms and the only fixed points are the endpoints of $I$.

To simplify the notation we assume that $I=[-1,1],I_{-}=[-1,0],I_{+}=[0,1]$
but our results will be easily seen to hold in the general setting. For
$\iota>0$, we let
$U_{0-}:=(-\iota,0],U_{0+}:=[0,\iota),U_{-1}:=g(U_{0+}),U_{+1}:=g(U_{0-})$ be
one-sided neighbourhoods of the endpoint of the intervals $I=I_{-},I_{+}$.

(A1) There exists constants $\ell_{1},\ell_{2}\geq 0$,
$\iota,k_{1},k_{2},a_{1},a_{2},b_{1},b_{2}>0$ such that, if
$\ell_{1},\ell_{2}>0$, $k_{1},k_{2}\neq 1$,

(3) $g(x)=\begin{cases}x+b_{1}{(1+x)}^{1+\ell_{1}}&\text{in }U_{-1},\\\
1-a_{1}{|x|}^{k_{1}}&\text{in }U_{0-},\\\ -1+a_{2}{x}^{k_{2}}&\text{in
}U_{0+},\\\ x-b_{2}{(1-x)}^{1+\ell_{2}}&\text{in }U_{+1},\end{cases}$

If $\ell_{1}=0$ and/or $\ell_{2}=0$ we replace the corresponding lines in (3)
with $g|_{U_{\pm 1}}(x)\coloneqq\pm 1+(1+b_{1})(x+1)\mp\eta(x),$ where $\eta$
is $C^{2}$, $\eta(\pm 1)=0,\eta^{\prime}(\pm 1)=0$, and
$\eta^{\prime\prime}(x)>0$ on $U_{-1}$ and $\eta^{\prime\prime}(x)<0$ on
$U_{+1}$. If $k_{1}=1$ and/or $k_{2}=1$, then we replace the corresponding
lines in (3) with the assumption that $g^{\prime}(0_{-})=a_{1}>1$ and/or
$g^{\prime}(0_{+})=a_{2}>1$ respectively, and that $g$ is monotone in the
corresponding neighbourhood, which makes the definition much less restrictive.

It is easy to see that the definition in (3) yields maps with dramatically
different derivative behaviour depending on the values of
$\ell_{1},\ell_{2},k_{1},k_{2}$, including having neutral or expanding fixed
points and points with zero or infinite derivative.

Our final assumption can be _intuitively thought of as saying that $g$ is
uniformly expanding outside the neighbourhoods $U_{0\pm}$ and $U_{\pm 1}$_.
This is however much stronger than what is needed and therefore we formulate a
weaker and more general assumption for which we need to describe some aspects
of the topological structure of maps satisfying condition . First of all we
define

(4) $\Delta^{-}_{0}:=g^{-1}(0,1)\cap I_{-}\quad\text{ and
}\quad\Delta^{+}_{0}:=g^{-1}(-1,0)\cap I_{+}.$

Then we define iteratively, for every $n\geq 1$, the sets

(5) $\Delta_{n}^{-}:=g^{-1}(\Delta_{n-1}^{-})\cap I_{-}\quad\text{ and
}\quad\Delta_{n}^{+}:=g^{-1}(\Delta_{n-1}^{+})\cap I_{+}$

as the $n$’th preimages of $\Delta_{0}^{-},\Delta_{0}^{+}$ inside the
intervals $I_{-},I_{+}$. It follows from  that $\\{\Delta_{n}^{-}\\}_{n\geq
0}$ and $\\{\Delta_{n}^{+}\\}_{n\geq 0}$ are $\bmod\;0$ partitions of $I_{-}$
and $I_{+}$ respectively, and that the partition elements depend
_monotonically_ on the index in the sense that $n>m$ implies that
$\Delta_{n}^{\pm}$ is closer to $\pm 1$ than $\Delta_{m}^{\pm}$, in particular
the only accumulation points of these partitions are $-1$ and $1$
respectively. Then, for every $n\geq 1$, we let

(6) $\delta_{n}^{-}:=g^{-1}(\Delta_{n-1}^{+})\cap\Delta_{0}^{-}\quad\text{ and
}\quad\delta_{n}^{+}:=g^{-1}(\Delta_{n-1}^{-})\cap\Delta_{0}^{+}.$

Notice that $\\{\delta_{n}^{-}\\}_{n\geq 1}$ and $\\{\delta_{n}^{+}\\}_{n\geq
1}$ are $\bmod\;0$ partitions of $\Delta_{0}^{-}$ and $\Delta_{0}^{+}$
respectively and also in these cases the partition elements depend
monotonically on the index in the sense that $n>m$ implies that
$\delta_{n}^{\pm}$ is closer to $0$ than $\delta_{m}^{\pm}$, (and in
particular the only accumulation point of these partitions is 0). Notice
moreover, that $g^{n}(\delta_{n}^{-})=\Delta_{0}^{+}$ and
$g^{n}(\delta_{n}^{+})=\Delta_{0}^{-}.$ We now define two non-negative
integers $n_{\pm}$ which depend on the positions of the partition elements
$\delta_{n}^{\pm}$ and on the sizes of the neighbourhoods $U_{0\pm}$ on which
the map $g$ is explicitly defined. If $\Delta_{0}^{-}\subseteq U_{0-}$ and/or
$\Delta_{0}^{+}\subseteq U_{0+}$, we define $n_{-}=0$ and/or $n_{+}=0$
respectively, otherwise we let

(7) $n_{+}:=\min\\{n:\delta_{n}^{+}\subset U_{0+}\\}\quad\text{ and }\quad
n_{-}:=\min\\{n:\delta_{n}^{-}\subset U_{0-}\\}.$

We can now formulate our final assumption as follows.

(A2)

There exists a $\lambda>1$ such that for all $1\leq n\leq n_{\pm}$ and for all
$x\in\delta_{n}^{\pm}$ we have $(g^{n})^{\prime}(x)>\lambda$.

Following [CoaLuzMub22], we let

$\widehat{\mathfrak{F}}\coloneqq\\{g:I\to I\text{ which satisfy
\ref{itm:A0}-\ref{itm:A2}}\\}$

The class $\widehat{\mathfrak{F}}$ contains many maps which have been studied
in the literature, including uniformly expanding maps and various well known
intermittency maps with a single neutral fixed point, we refer the reader to
[CoaLuzMub22] for a detailed literature review.

###  2.2. Physical measures on repelling fixed points and non-statistical
dynamics

It is proved in [CoaLuzMub22] that every $g\in\widehat{\mathfrak{F}}$ admits a
unique (up to scaling) $\sigma$-_finite ergodic invariant measure_ $\hat{\mu}$
_equivalent to Lebesgue_ and that many properties depend on the constants

$\beta^{-}:=\ell_{1}k_{2},\quad\beta^{+}:=\ell_{2}k_{1},\quad\text{ and
}\quad\beta:=\max\\{\beta^{+},\beta^{-}\\}.$

Notice that $\beta^{-},\beta^{+}\in[0,\infty)$ and can take any value in the
allowed range, depending on the values of $\ell_{1},\ell_{2},k_{1},k_{2}$.
They determine the level of _“stickiness”_ of the fixed points $-1$ and $+1$
respectively, given by the combination of the constants $\ell_{1},\ell_{2}$,
which determine the order of tangency of the graph of $g$ with the diagonal,
and the the constants $k_{1},k_{2}$, which give the order of the singular or
critical points. The larger the value of $\beta^{-},\beta^{+}$ the
_“stickier”_ are the corresponding fixed points. We can now define explicitly
the subfamilies in (1) by letting

(8)
$\mathfrak{F}\coloneqq\\{\beta\in[0,1)\\},\qquad\mathfrak{F}_{\pm}\coloneqq\\{\beta\geq
1,\ \beta^{-}\neq\beta^{+}\\}\qquad\mathfrak{F}_{*}\coloneqq\\{\beta\geq 1,\
\beta^{-}=\beta^{+}\\}.$

It is clear that these families are pairwise disjoint and that their union is
exactly $\widehat{\mathfrak{F}}$. Notice that $\beta\in[0,1)$ implies that
_both_ $\beta^{-},\beta^{+}\in[0,1)$, whereas $\beta\geq 1$ only implies that
_at least one_ of $\beta^{-},\beta^{+}\geq 1$. It is proved in [CoaLuzMub22]
that the $\sigma$-finite invariant measure $\hat{\mu}$ is finite, and can
therefore be rescaled to a probability measure $\mu$, _if and only if_
$\beta\in[0,1)$.

######  Theorem 1 ([CoaLuzMub22]).

If $g\in\mathfrak{F}$ then $g$ admits a physical measure equivalent to
Lebesgue.

As mentioned above, this proves 1) in Theorem A. We are therefore interested
in the families $\mathfrak{F}_{\pm}$ and $\mathfrak{F}_{*}$, neither of which
can contain any map with a physical measure equivalent to Lebesgue. The maps
in $\mathfrak{F}_{\pm}$ are those where one fixed point is _stickier_ than the
other, whereas the maps in $\mathfrak{F}_{*}$ are those for which the
stickiness is the same, at least as far as it can be measured by the constants
$\beta^{-},\beta^{+}$. It turns out that the typical statistical behaviour is
completely different in these two cases.

######  Theorem 2.

If $g\in\mathfrak{F}_{\pm}$ then $g$ admits a physical measure with full
basin. Moreover:
1) if $\beta^{-}>\beta^{+}$, the physical measure is the Dirac-delta measure
$\delta_{-1}$ on the fixed point -1;
2) if $\beta^{+}>\beta^{-}$, the physical measure is the Dirac-delta measure
$\delta_{1}$ on the fixed point +1 .

When $\beta^{+}=\beta^{-}$, the two neutral fixed points are, in some sense,
“equally sticky” and it is natural to conjecture that typical orbits would
oscillate between the two, spending positive proportion of time near each one.
There are, however, several quite different ways in which this can happen. Let

(9) $\Omega\coloneqq\\{\nu_{p}\coloneqq
p\delta_{1}+(1-p)\delta_{-1}:p\in[{0,1}]\\}$

be the space of all convex combinations of the Dirac-delta measures
$\delta_{-1}$ and $\delta_{1}$. We then have the following quite remarkable
statement for maps in $\mathfrak{F}_{*}$.

######  Theorem 3.

If $g\in\mathfrak{F}_{*}$, then for Lebesgue almost every $x\in I$, the
sequence $\mu_{n}(x)$ does not converge and so $g$ is non-statistical.
Moreover, the set of accumulation points of the sequence $\mu_{n}(x)$ is
$\Omega$.

Theorems 1, 2, and 3, clearly imply Theorem A. For $g\in\mathfrak{F}_{*}$,
Theorem 3 also gives additional and non-trivial information about the limit
points of the sequence $\mu_{n}(x)$.

######  Remark 2.1.

It is interesting to note that, contrary to what might be expected, there are
no cases in which there are _physical_ measures which are non-trivial convex
combinations $\nu_{p}$ with $p\in(0,1)$ of the Dirac-delta measures. One may
expect that this may be achieved at least for some carefully chosen values of
the multiplicative paramaters $a_{1},a_{2},b_{1}.b_{2}$ in (3) but in fact our
results show that these play no significant role, at least at this level of
the description of the dynamics.

###  2.3. Density of $\mathfrak{F},\mathfrak{F}_{\pm},\mathfrak{F}_{*}$ in
$\widehat{\mathfrak{F}}$

We now address the issue of the density of the families
$\mathfrak{F},\mathfrak{F}_{\pm},\mathfrak{F}_{*}$, as stated in Theorems B
and C. We will actually state a much more general result which says that each
map $f\in\widehat{\mathfrak{F}}$ can be approximated arbitrarily closely in
the $C^{r}$ topology by maps in _any_ of the families
$\mathfrak{F},\mathfrak{F}_{\pm},\mathfrak{F}_{*}$, for some $r$ which depends
both on $f$ and on the family in which we want to approximate $f$. We recall
first of all the _ceiling function_ $\lceil
x\rceil:=min\\{\kappa\in\mathbb{N}:x\leq\kappa\\}.$ Then, for every
$f\in\widehat{\mathfrak{F}}$, we define

$r_{\pm}\coloneqq
r_{\pm}(f)\coloneqq\max\\{\lceil\ell_{1}\rceil,\lceil\ell_{2}\rceil\\},\qquad\tilde{r}\coloneqq\tilde{r}(f)\coloneqq\begin{cases}\lceil
1/k_{2}\rceil&\text{if }0\leq\beta^{+}<1\leq\beta^{-}\\\ \lceil
1/k_{1}\rceil&\text{if }0\leq\beta^{-}<1\leq\beta^{+}\\\ \min\\{\lceil
1/k_{2}\rceil\lceil 1/k_{1}\rceil\\}&\text{otherwise, }\end{cases}$

and

$r_{*}\coloneqq
r_{*}(f)\coloneqq\begin{cases}\min\\{\lceil\ell_{1}\rceil,\lceil\ell_{2}\rceil\\}&\text{if
}\beta\in[0,1)\\\ \lceil\ell_{2}\rceil&\text{if }\beta^{+}<1\leq\beta^{-}\\\
\lceil\ell_{1}\rceil&\text{if }\beta^{-}<1\leq\beta^{+}\\\
\min\left\\{\lceil\ell_{2}k_{1}/k_{2}\rceil,\lceil\ell_{1}\rceil\right\\}&\text{if
}\beta^{+},\beta^{-}\geq 1\text{ and }k_{2}\geq k_{1}\\\
\min\left\\{\lceil\ell_{1}k_{2}/k_{1}\rceil,\lceil\ell_{2}\rceil\right\\}&\text{if
}\beta^{+},\beta^{-}>1\text{ and }k_{1}\geq k_{2}.\end{cases}$

Notice that $r_{\pm},r_{*},r$ are all well defined non-negative _integers_
because of the way there are defined using the ceiling function. Moreover,
$r_{*}=0$ if and only if at least one of the fixed points is hyperbolic, and
$r_{\pm}=0$ if and only if both fixed points are hyperbolic (e.g. if $f$ is
uniformly expanding). If both fixed points are neutral then we necessarily
have $r_{\pm},r_{*},r>0$ and therefore $r_{\pm},r_{*},r\geq 1$, since they are
all integers and defined in terms of ceiling functions.

######  Theorem 4.

For every $f\in\widehat{\mathfrak{F}}$ and every $\varepsilon>0$ there exists
$\tilde{f}\in\mathfrak{F}$, $f_{\pm}\in\mathfrak{F}_{\pm}$ and
$f_{*}\in\mathfrak{F}_{*}$ such that

$d_{\tilde{r}}(f,\tilde{f})<\varepsilon,\qquad
d_{r_{\pm}}(f,f_{\pm})<\varepsilon,\qquad d_{r_{*}}(f,f_{*})<\varepsilon.$

Theorem 4 immediately implies Theorems B and C since we always have
$r_{\pm},r_{*},\tilde{r}\geq 0$ and, as mentioned in the previous paragraph,
if $f\in\widetilde{\mathfrak{F}}$ (i.e. if both fixed points are neutral) then
we always have $r_{\pm},r_{*},\tilde{r}\geq 1$. However, it also shows that in
some cases we can have approximations in much higher topologies. For example,
consider a map $f\in\mathfrak{F}_{*}$, which therefore has no physical
measure. By definition we have $\ell_{1}k_{2}=\ell_{2}k_{1}=\beta$, where
$\beta\geq 1$ is arbitrary. For definiteness let us suppose that $\beta=1$ and
then, given any _arbitrarily large_ positive integer $R$, there exists a map
$f\in\mathfrak{F}_{*}$ such that $\ell_{1}=\ell_{2}=R$ and $k_{1}=k_{2}=1/R$.
This implies $\tilde{r}=r_{\pm}=R$ and therefore, from Theorem 4, we get that
the map $f$, which does not have any physical measure, can be approximated
arbitrarily closely in the $C^{R}$ topology by maps in $\mathfrak{F}$, which
have a physical measure equivalent to Lebesgue, _and_ by maps in
$\mathfrak{F}_{\pm}$, which have physical measures which are Dirac-delta
measures on a fixed point. Notice that we do not need to consider $r_{*}$
since taking $g=f$ the last approximation is trivial.

###  2.4. “Openness” of $\mathfrak{F},\mathfrak{F}_{\pm},\mathfrak{F}_{*}$ in
$\widehat{\mathfrak{F}}$

Finally, it just remains to discuss the “openness” of the families
$\mathfrak{F},\mathfrak{F}_{\pm},\mathfrak{F}_{*}$ as described in Theorem D.
Now that we have the formal definitions of the maps in
$\widehat{\mathfrak{F}}$ the statement in Theorem D is actually almost
immediate and therefore we just give the proof.

###### Proof of Theorem D.

By assumption the map $g$ is in the class $\widehat{\mathfrak{F}}$ and since
$\overline{\operatorname{supp}f-g}\subset(-1,0)\cup(0,1)$ we necesarrily have
that $g$ satisfies  with the same parameters as $f$. Thus,
$\beta^{+}(g)=\beta^{+}(f)$, $\beta^{-}(g)=\beta^{-}(f)$ and so $g$ must lies
in the same subclass $\mathfrak{F},\mathfrak{F}_{\pm},\mathfrak{F}_{*}$ as
$f$. ∎

We also mention, without giving formal statements, a couple of other natural
ways in which maps in $\mathfrak{F},\mathfrak{F}_{\pm},\mathfrak{F}_{*}$ can
be perturbed without falling outside of their original family, essentially by
perturbing some of the parameters through which they are defined. Notice first
of all that the conditions which determine whether $g$ belongs to
$\mathfrak{F}$, $\mathfrak{F}_{\pm}$, or $\mathfrak{F}_{*}$, do not depend on
the constants $a_{1},a_{2},b_{1},b_{2}$ and therefore we can choose these
arbitrarily without changing the values of $\beta_{1},\beta_{2}$. Sufficiently
small perturbations of these parameters do not invalidate condition (A2) and
therefore each subfamily $\mathfrak{F}$, $\mathfrak{F}_{\pm}$,
$\mathfrak{F}_{*}$ is also “ _open_ ” in the sense that there exists an open
neighbourhood of the parameters $a_{1},a_{2},b_{1},b_{2}$ for which the
corresponding maps still belong to the same subfamily.

In addition to the perturbations mentioned above, which do not change the
values of the parameters $\beta^{-},\beta^{+}$, we can also perturb the
parameters $\ell_{1},\ell_{2},k_{1},k_{2}$ which make up
$\beta^{-},\beta^{+}$. This may of course affect which subfamily the perturbed
map belongs to as the subfamilies are precisely defined in terms of the values
of $\beta^{-},\beta^{+}$, and indeed for these kinds of perturbations the
situation is slightly different depending on which subfamily we consider. The
maps in $\mathfrak{F}$ are characterized by the property that
$\beta^{-},\beta^{+}\in[0,1)$ and therefore there is an open set of
sufficiently small perturbations of $\ell_{1},\ell_{2},k_{1},k_{2}$ such that
this still holds as well as condition (A2), thus guaranteeing the the
perturbed map is still in $\mathfrak{F}$. Similarly, maps in
$\mathfrak{F}_{\pm}$ are characterized by the property that at least one of
$\beta^{-},\beta^{+}$ is $\geq 1$ and so again there is a large set of
perturbations of $\ell_{1},\ell_{2},k_{1},k_{2}$ which preserve that
condition. Notice however that this may not always contain an open set of
parameters, for example if $\beta^{-}<1$ and $\beta^{+}=1$, in which case we
can only perturb $\ell_{2}$ and $k_{1}$ in such a way that $\beta^{+}$ does
not decrease. Finally, maps in $\mathfrak{F}_{*}$ are defined in the most
restrictive way since they require $\beta^{-}=\beta^{+}$ and this condition is
not preserved for an open set of choices of parameters
$\ell_{1},\ell_{2},k_{1},k_{2}$. Nevertheless it is still a relatively large
and persistent family since we can define a

> _three-parameter family completely contained in $\mathfrak{F}^{*}$_:

for _any_ $\beta\geq 1,s,t>0$ there is a map $g$ with
$\ell_{1}=s,\ell_{2}=t,k_{1}=\beta/t,k_{2}=\beta/s,$ which implies that
$\beta^{-}=\ell_{1}k_{2}=\ell_{2}k_{1}=\beta^{+}$, and thus
$g\in\mathfrak{F}^{*}$.

##  3\. The Induced Map

In this section we recall some details of the construction of the induced map
carried out in [CoaLuzMub22] and a key estimate from [CoaLuzMub22], see
Proposition 3.1 below, which will play a crucial role in our proofs.

###  3.1. Topological construction

We recall first of all from [CoaLuzMub22] the topological structure of the
_first return maps_ on the intervals $\Delta_{0}^{-},\Delta_{0}^{+}$ defined
in (4). From the definitions of the sets $\Delta_{n}^{\pm}$ and
$\delta_{n}^{\pm}$ in (5) and (6), and from the fact that each branch of $g$
is a $C^{2}$ diffeomorphism, it follows that for every $n\geq 1$, the maps
$g:\delta_{n}^{-}\to\Delta_{n-1}^{+}$ and
$g:\delta_{n}^{+}\to\Delta_{n-1}^{-}$ are $C^{2}$ diffeomorphisms, and, for
$n\geq 2$, the same is true for the maps
$g^{n-1}:\Delta_{n-1}^{-}\to\Delta_{0}^{-},$ and
$g^{n-1}:\Delta_{n-1}^{+}\to\Delta_{0}^{+},$ and therefore for every $n\geq
1$, the maps $g^{n}:\delta_{n}^{-}\to\Delta_{0}^{+}$ and
$g^{n}:\delta_{n}^{+}\to\Delta_{0}^{-}$ are $C^{2}$ diffeomorphisms. We can
therefore define two _full branch_ maps
$\widetilde{G}^{-}:\Delta_{0}^{-}\to\Delta_{0}^{+}$ and
$\widetilde{G}^{+}:\Delta_{0}^{+}\to\Delta_{0}^{-}$ by
$\widetilde{G}^{\pm}|_{\delta_{n}^{\pm}}:=g^{n}.$ Then for every $i,j\geq 1$
we let

(10) $\delta_{i,j}^{-}:=g^{-i}(\delta^{+}_{j})\cap\delta^{-}_{i}\quad\text{
and }\quad\delta^{+}_{i,j}:=g^{-i}(\delta^{-}_{j})\cap\delta^{+}_{i}$

Then, for $i\geq 1$, the sets $\\{\delta_{i,j}^{-}\\}_{j\geq 1}$ and
$\\{\delta_{i,j}^{+}\\}_{j\geq 1}$ are partitions of $\delta_{i}^{-}$ and
$\delta_{i}^{+}$ respectively and so
$\mathscr{P}^{-}:=\\{\delta_{i,j}^{-}\\}_{i,j\geq 1}$ and
$\mathscr{P}^{+}:=\\{\delta_{i,j}^{+}\\}_{i,j\geq 1}$ are partitions of
$\Delta_{0}^{-},\Delta_{0}^{+}$ respectively, with the property that for every
$i,j\geq 1$, the maps $g^{i+j}:\delta_{i,j}^{-}\to\Delta_{0}^{-}$ and
$g^{i+j}:\delta_{i,j}^{+}\to\Delta_{0}^{+}$ are $C^{2}$ diffeomorphisms.
Notice that $i+j$ is the _first return time_ of points in $\delta_{i,j}^{-}$
and $\delta_{i,j}^{+}$ to $\Delta_{0}^{-}$ and $\Delta_{0}^{+}$ respectively,
and we have thus constructed _two full branch first return induced maps_
$G^{-}:=\widetilde{G}^{+}\circ\widetilde{G}^{-}:\Delta_{0}^{-}\to\Delta_{0}^{-}$
and
$G^{+}:=\widetilde{G}^{-}\circ\widetilde{G}^{+}:\Delta_{0}^{+}\to\Delta_{0}^{+}.$
for which we have $G^{-}|_{\delta_{i,j}^{-}}=g^{i+j}$ and
$G^{+}|_{\delta_{i,j}^{+}}=g^{i+j}.$

We now focus on one of these two full branch first return maps, for
definiteness let’s say $G^{-}$ (but we could just as well take $G^{+}$) and
for simplicity omit the superscript from the notation and write

(11) $\Delta_{0}:=\Delta_{0}^{-},\quad\delta_{i,j}:=\delta_{i,j}^{-},\quad
G:=G^{-}.$

It is proved in [CoaLuzMub22] that $G:\Delta_{0}\to\Delta_{0}$ is a full
branch _Gibbs-Markov_ map with respect to the partition $\\{\delta_{i,j}\\}$,
and therefore admits an _invariant ergodic probability measure_ $\hat{\mu}$
which is _equivalent to Lebesgue_ and with Lipschitz continuous density. Then,
by standard results, the _induced measure_

(12) $\mu\coloneqq\sum_{n=0}^{\infty}g^{n}_{*}(\hat{\mu}|\\{\tau\geq n\\})$

is a _sigma-finite, ergodic, $g$-invariant_ measure which, since $\bigcup
g^{n}(\\{\tau\geq n\\})=I\ (\text{mod}\ 0)$ by construction, is _equivalent to
Lebesgue_. It is easy to check that $\mu(I)<\infty$ if and only if $\tau\in
L^{1}(\hat{\mu})$, and it follows from Proposition 2.6 of [CoaLuzMub22] (which
we recall in Proposition 3.1 below) that $\tau\in L^{1}(\hat{\mu})$ if and
only if $\beta\in[0,1)$, i.e. if and only if $g\in\mathfrak{F}$, as already
mentioned above. Moreover, since $G$ is a _first return_ induced map, the
measure $\mu$ does not add any measure to the inducing domain and therefore
$\mu|\Delta_{0}=\hat{\mu}.$

###  3.2. Inducing Times

Our arguments revolve around the _distribution_ of various _observables_ on
$\Delta_{0}$ with respect to the probability $\hat{\mu}$. First we define
$\tau^{\pm}(x),\tau:\Delta_{0}\to\mathbb{N}$ by

(13) $\tau^{+}(x):=\\#\\{1\leq i\leq\tau:g^{i}(x)\in
I_{+}\\}\quad\tau^{-}(x):=\\#\\{1\leq j\leq\tau:g^{j}(x)\in
I_{-}\\},\quad\tau\coloneqq\tau^{+}+\tau^{-}$

where $\tau^{\pm}$ _count the number of iterates of $x$ in $I_{-},I_{+}$
respectively before returning to $\Delta_{0}$_. Notice that
$\tau^{+}|_{\delta_{i,j}}\equiv i$ and $\tau^{-}|_{\delta_{i,j}}\equiv j$,
that $\tau|_{\delta_{i,j}}\equiv i+j$ is exactly the _first return time_ to
$\Delta_{0}$. The following key technical result from [CoaLuzMub22] gives the
distribution of these functions. We say that $f\sim g$ if $f(t)/g(t)\to 1$ as
$t\to\infty$.

######  Proposition 3.1 ([CoaLuzMub22]*Proposition 2.6).

There exist constants $C_{+},C_{-}>0$ such that

(14) $\hat{\mu}(\tau^{+}>t)\sim
C_{+}t^{-1/\beta^{+}},\qquad\hat{\mu}(\tau^{-}>t)\sim
C_{-}t^{-1/\beta^{-}},\qquad\hat{\mu}(\tau>t)\sim(C_{+}+C_{-})t^{-1/\beta}.$

We will also be interested in the associated Birkhoff sums
$\tau_{k}^{\pm},\tau_{k}:\Delta_{0}\to\mathbb{N}$, under the _induced map_
$G$, defined by

(15) $\tau_{k}^{-}\coloneqq\sum_{\ell=0}^{k-1}\tau^{-}\circ
G^{\ell},\qquad\tau_{k}^{+}\coloneqq\sum_{\ell=0}^{k-1}\tau^{+}\circ
G^{\ell},\qquad\tau_{k}\coloneqq\tau_{k}^{+}+\tau_{k}^{-}.$

These give us the total time which points spend in the left and right
intervals after $k$ iterations of the induced map. For future reference, note
that $\tau_{k}^{-},\tau_{k}^{+},\tau_{k}$ are Birkhoff sums of
$\tau^{-},\tau^{+},\tau$ respectively, along the orbit of a point under the
induced map $G:\Delta_{0}\to\Delta_{0}$ and therefore, by ergodicity and
invariance of the probability measure $\hat{\mu}$ under $G$, since they are
all non-negative observables,

(16)
$\frac{\tau^{-}_{k}}{k}\to\int\tau^{-}d\hat{\mu},\qquad\qquad\frac{\tau^{+}_{k}}{k}\to\int\tau^{+}d\hat{\mu},\qquad\qquad\frac{\tau_{k}}{k}\to\int\tau
d\hat{\mu},$

as $k\to\infty$, irrespective of whether the integrals are finite or not, for
$\hat{\mu}$ almost every $x\in\Delta_{0}$. Therefore, since $\hat{\mu}$ is
equivalent to Lebesgue, for Lebesgue almost every $x\in\Delta_{0}$.
Proposition 3.1 implies that $\tau^{-},\tau^{+},\tau\in L^{1}$ if and only if
$\beta^{-},\beta^{+},\beta\in[0,1)$ respectively, and therefore, from (16) we
have that if $\beta^{-},\beta^{+},\beta\in[0,1)$, then we have respectively
that

(17)
$\frac{\tau^{-}_{k}}{k}\to\smallint\tau^{-}d\hat{\mu}<\infty,\qquad\qquad\frac{\tau^{+}_{k}}{k}\to\smallint\tau^{+}d\hat{\mu}<\infty,\qquad\qquad\frac{\tau_{k}}{k}\to\smallint\tau
d\hat{\mu}<\infty,$

and if $\beta^{-},\beta^{+},\beta\geq 1$, then we have respectively that

(18)
$\frac{\tau^{-}_{k}}{k}\to\infty,\qquad\qquad\frac{\tau^{+}_{k}}{k}\to\infty,\qquad\qquad\frac{\tau_{k}}{k}\to\infty.$

##  4\. Proof of Theorem 2

We split the proof of Theorem 2 into 3 parts. First we show that Lebesgue
almost every point spends asymptotically all its time in either $I^{-}$ or
$I^{+}$. Then we show that in fact such orbits spend most of the time in
arbitrarily small neighbourhoods of the corresponding fixed points $-1$ and
$+1$ when measured along the subsequence $\tau_{k}$. Finally we show that this
implies that the same holds for the full sequence of iterates, thus proving
the Theorem.

###  4.1. Statistics of orbits in $I^{-}$ and $I^{+}$

We now go into a bit more detail on the behaviour of the induced observables
$\tau^{-}_{k},\tau^{+}_{k},\tau_{k}$. Recall that by definition
$\tau_{k}=\tau^{+}_{k}+\tau^{-}_{k}$ and therefore

(19)
$\frac{\tau^{-}_{k}}{\tau_{k}}+\frac{\tau^{+}_{k}}{\tau_{k}}=\frac{\tau_{k}^{-}+\tau_{k}^{+}}{\tau_{k}}=\frac{\tau_{k}}{\tau_{k}}=1$

where ${\tau^{-}_{k}}/{\tau_{k}}$ and ${\tau^{+}_{k}}/{\tau_{k}}$ are simply
the proportion of time that the orbit of $x$ spends on the left and right
intervals respectively in its first $\tau_{k}$ iterates corresponding to $k$
iterations of the induced map. The main result of this section shows that when
$\beta\geq 1$, the largest of $\beta^{-}$ and $\beta^{+}$ “gets everything”.

######  Proposition 4.1.

Suppose $\beta\geq 1$. Then

(20) $\beta^{-}>\beta^{+}\implies\frac{\tau^{-}_{k}}{\tau_{k}}\to
1\quad\quad\text{ and
}\quad\quad\beta^{+}>\beta^{-}\implies\frac{\tau^{+}_{k}}{\tau_{k}}\to 1$

for Lebesgue almost every point $x\in\Delta_{0}$.

It then follows of course from (19) and (20) that when $\beta\geq 1$ we have
also $\beta^{-}>\beta^{+}\implies{\tau^{+}_{k}}/{\tau_{k}}\to 0$ and
$\beta^{+}>\beta^{-}\implies{\tau^{-}_{k}}/{\tau_{k}}\to 0.$ so Proposition
4.1 implies that whenever at least one of $\beta_{1},\beta_{2}$ is $\geq 1$
and $\beta_{1}\neq\beta_{2}$, Lebesgue almost every point spends
asymptotically all its time either in the left or right interval.

###### Proof of Proposition 4.1.

We will prove the result when $\beta^{-}>\beta^{+}$, the case
$\beta^{+}>\beta^{-}$ follows by exactly the same arguments. Notice first of
all that from (19) we have

$1=\frac{\tau_{k}^{-}+\tau_{k}^{+}}{\tau_{k}}=\frac{\tau_{k}^{-}}{\tau_{k}}+\frac{\tau_{k}^{+}}{\tau_{k}}=\frac{\tau_{k}^{-}}{\tau_{k}}\left(1+\frac{\tau_{k}^{+}}{\tau_{k}^{-}}\right).$

and it is therefore sufficient to show that

(21) $\frac{\tau^{+}_{k}}{\tau_{k}^{-}}\to 0$

as this implies ${\tau_{k}^{-}}/{\tau_{k}}\to 1$ as required. By assumption
$\beta\geq 1$ and therefore $\beta^{-}\geq 1$, and it is therefore sufficient
to consider the following 3 subcases.

1)

$\beta^{-}\geq 1>\beta^{+}>0$.

In this case we can write

$\frac{\tau^{+}_{k}}{\tau_{k}^{-}}=\frac{\tau^{+}_{k}}{k}\frac{k}{\tau_{k}^{-}}$

By (17) we have that ${\tau^{+}_{k}}/{k}$ is bounded and by (18) we have that
${k}/{\tau_{k}^{-}}\to 0$, implying that ${\tau^{+}_{k}}/{\tau_{k}^{-}}\to 0$
and thus giving (21).

2) $\beta^{-}>\beta^{+}>1$ .

In this case, the statements in (17) and (18) do not allow us to immediately
draw any definite conclusions and we need to refer to a non-trivial result of
[GalHolPer21] which applies precisely to our case. Indeed, since the map $G$
is Gibbs-Markov, the functions $\tau^{\pm}$ are constant on each partition
element $\delta_{i,j}$, and the distribution of $\tau^{\pm}$ is given by (14),
Proposition 2.8 of [GalHolPer21] applied to our setting satisfies gives the
following result.

######  Lemma 4.2 ([Proposition 2.8).

[GalHolPer21] ] For all $\epsilon>0$,

$\beta^{-}>1\implies k^{\beta^{-}-\epsilon}\lesssim\tau^{-}_{k}\lesssim
k^{\beta^{-}+\epsilon}\quad\text{ and }\quad\beta^{+}>1\implies
k^{\beta^{+}-\epsilon}\lesssim\tau^{+}_{k}\lesssim k^{\beta^{+}+\epsilon}.$

This implies
${\tau^{+}_{k}}/{\tau_{k}^{-}}\lesssim{k^{\beta^{+}+\epsilon}}/{k^{\beta^{-}-\epsilon}}$
almost surely, and thus implies (21) if $\epsilon$ is sufficiently small.

3) $\beta^{-}>\beta^{+}=1$.

Lemma 4.2 requires $\beta^{-}$ and $\beta^{+}$ to be _strictly_ greater than 1
and therefore we cannot apply the argument above completely to this setting
but we can conclude that

(22)
$\frac{\tau^{+}_{k}}{\tau_{k}^{-}}\lesssim\frac{\tau_{k}^{+}}{k^{\beta^{-}-\epsilon}}$

Letting $q:={1}/{(\beta^{-}-\epsilon)}$ and using the definition of
$\tau_{k}^{+}$ in (15) it will be convenient to write this as

(23)
$\frac{\tau^{+}_{k}}{\tau_{k}^{-}}\lesssim\frac{\tau_{k}^{+}}{k^{1/q}}=\frac{\tau^{+}+(\tau^{+}\circ
G)+\cdots+(\tau^{+}\circ G^{k-1})}{k^{1/q}}$

Since $\hat{\mu}$ is $G$ invariant, the summands $\tau^{+}\circ G^{\ell}$ are
identically distributed and Proposition 3.1 gives

$\mu((\tau^{+})^{q}>t)=\mu(\tau^{+}>t^{1/q})\sim
C_{+}t^{-1/q\beta^{+}}=C_{+}t^{-(\beta^{-}-\epsilon)/\beta^{+}}$

which implies that $(\beta^{-}-\epsilon)/\beta^{+}>1$ and therefore
$\tau^{+}\in L^{q}(\hat{\mu})$ if $\epsilon$ is small enough. Then we can
apply the following classical and remarkable result.

######  Lemma 4.3.

[Saw66]*Corollary to Lemma 3 Let $(X,\mu)$ be a probability space and suppose
$\varphi_{n}$ are identically distributed random variables with
$\varphi_{n}\in L^{q}$ for some $q\in(0,1)$. Then. $\mu$-almost surely,

$\frac{\varphi_{1}+\cdots+\varphi_{n}}{n^{1/q}}\to 0.$

Applying 4.3 to (23) implies (21) and thus completes the proof. ∎

###  4.2. Statistics of orbits near the fixed points for the subsequence
$\tau_{k}$

Proposition 4.1 tells us that depending on the relative values of
$\beta^{-},\beta^{+}$ orbits spend asymptotically all their time inside either
the left or right subintervals $I^{-},I^{+}$. We are however especially
interested in how much time orbits spend close to the two fixed points and in
this section we will show that actually most of the time spent in these
intervals is spent in arbitrarily small neighbourhoods of the corresponding
fixed points. To formalize this statement, for $\varepsilon>0$ we define the
intervals

(24) $U^{+}_{\varepsilon}\coloneqq(1-\varepsilon,1),\quad\text{ and }\quad
U^{-}_{\varepsilon}\coloneqq(-1,-1+\varepsilon),$

and then define the functions $S_{n,\varepsilon}^{\pm}:[-1,1]\to\mathbb{N}$
and $S_{n,\varepsilon}:[-1,1]\to\mathbb{N}$ by

(25)
$S_{n,\varepsilon}^{-}\coloneqq\sum_{k=0}^{n-1}\mathbb{1}_{U_{\varepsilon}^{-}}\circ
g^{k},\qquad
S_{n,\varepsilon}^{+}\coloneqq\sum_{k=0}^{n-1}{\mathbb{1}}_{U_{\varepsilon}^{+}}\circ
g^{k},\quad\text{ and }\quad
S_{n,\varepsilon}=S_{n,\varepsilon}^{-}+S_{n,\varepsilon}^{+}.$

The functions $S_{n,\varepsilon}^{-}$ and $S_{n,\varepsilon}^{+}$ simply count
the number of iterates of a point which belong to the neighbourhoods
$U^{-}_{\varepsilon}$ or $U^{+}_{\varepsilon}$ respectively, in the first $n$
iterates.

######  Proposition 4.4.

For every $\varepsilon>0$ and Lebesgue almost-every $x\in\Delta_{0}$,

(26) $\beta^{-}\geq
1\implies\frac{S_{\tau_{k},\varepsilon}^{-}}{\tau_{k}^{-}}\to
1,\qquad\beta^{+}\geq
1\implies\frac{S_{\tau_{k},\varepsilon}^{+}}{\tau_{k}^{+}}\to
1,\qquad\beta\geq 1\implies\frac{S_{\tau_{k},\varepsilon}}{\tau_{k}}\to 1.$

###### Proof.

Recall the definition of the partitions $\\{\Delta_{n}^{\pm}\\}$ in (5) and,
for $\varepsilon>0$, let

$N_{\varepsilon}^{\pm}\coloneqq\max\left\\{N:U_{\varepsilon}^{\pm}\subseteq\bigcup_{k=N}^{\infty}\Delta_{k}^{\pm}\right\\}.$

Then it is easy to see from the definition of the partition
$\\{\delta_{i,j}\\}$ in (10) and (11) and the properties of the induced map
that all points in $\delta_{i,j}$ with $i,j\geq N_{\varepsilon}$ will spend
all but $N_{\varepsilon}^{+}$ and $N_{\varepsilon}^{-}$ iterates inside
$U_{\varepsilon}^{-}$ and $U_{\varepsilon}^{+}$ respectively before they
return at time $\tau=i+j$. More formally, if $x\in\delta_{i,j}$ and $i>N^{+}$
then $g^{k}(x)\in U^{+}_{\varepsilon}$ for $k<i-N^{+}$ and $g^{k}\not\in
U^{+}_{\varepsilon}$ for $i-N^{+}<k\leq i$, similarly if $x\in\delta_{i,j}$
and $j>N^{-}$ then $g^{i+k}(x)\in U^{+}_{\varepsilon}$ for $1\leq k<j-N^{+}$
and $g^{i+k}\not\in U^{+}_{\varepsilon}$ for $j-N^{+}<k\leq j$. Therefore,
from the definitions in (13), (25) it follows that for every $x\in\Delta_{0}$
we have

(27) $\tau^{+}(x)-N_{\varepsilon}^{+}-1\leq
S^{+}_{\tau(x),\varepsilon}(x)\leq\tau^{+}(x),\quad\text{ and
}\quad\tau^{-}(x)-N_{\varepsilon}^{-}-1\leq
S^{-}_{\tau(x),\varepsilon}(x)\leq\tau^{-}(x).$

Notice that this holds even if $\tau^{\pm}(x)\leq N_{\varepsilon}^{\pm}$ since
then the left hand side of the corresponding inequality is negative. From the
definition of $\tau_{k}$ we can write

$S_{\tau_{k}(x),\varepsilon}^{\pm}(x)=S_{\tau(x),\varepsilon}^{\pm}(x)+S_{\tau(G(x)),\varepsilon}^{\pm}(G(x))+\cdots+S_{\tau(G^{k-1}(x)),\varepsilon}^{\pm}(G^{k-1}(x)).$

Applying the inequalities in (27) to each term in the sum above we get

(28)
$\tau^{\pm}_{k}(x)-k(N_{\varepsilon}^{\pm}+1)=\sum_{m=0}^{k-1}\tau^{\pm}\circ
G^{m}(x)-k(N_{\varepsilon}^{\pm}+1)\leq
S_{\tau_{k}(x),\varepsilon}^{\pm}(x)\leq\sum_{m=0}^{k-1}\tau^{\pm}\circ
G^{m}(x)=\tau^{\pm}_{k}(x)$

and so, dividing through by $\tau^{\pm}_{k}(x)$, gives

$1-\frac{k}{\tau^{\pm}_{k}(x)}(N^{\pm}_{\varepsilon}+1)\leq\frac{S_{\tau_{k}(x),\varepsilon}^{\pm}(x)}{\tau_{k}^{\pm}(x)}\leq
1.$

From (18) we have $k/{\tau_{k}^{\pm}(x)}\to 0$ for $\hat{\mu}$ almost every
$x\in\Delta_{0}$, yielding (26). From (28) we also get

$\tau_{k}(x)-k(N_{\varepsilon}^{+}+N_{\varepsilon}^{-}+2)\leq
S_{\tau_{k}(x),\varepsilon}^{+}(x)+S_{\tau_{k}(x),\varepsilon}^{-}(x)\leq\tau_{k}(x).$

Dividing through by $\tau_{k}(x)$ and applying (18) as above, we get (26) and
complete the proof. ∎

###  4.3. Statistics of orbits near the fixed points

We are now ready to prove Theorem 2.

###### Proof of Theorem 2.

We will show that for every $\varepsilon>0$ and Lebesgue almost every point
$x\in\Delta_{0}$,

(29) $\beta^{-}>\beta^{+}\implies\frac{S_{n,\varepsilon}^{-}}{n}\to
1,\qquad\beta^{+}>\beta^{-}\implies\frac{S_{n,\varepsilon}^{+}}{n}\to 1.$

This clearly implies the statement of Theorem 2. Notice first of all that from
Propositions 4.1 and 4.4 we immediately get that for every $\varepsilon>0$ and
Lebesgue almost every point $x\in\Delta_{0}$,

(30)
$\beta^{-}>\beta^{+}\implies\frac{S_{\tau_{k},\varepsilon}^{-}}{\tau_{k}}\to
1\quad\quad\text{ and
}\quad\quad\beta^{+}>\beta^{-}\implies\frac{S_{\tau_{k},\varepsilon}^{+}}{\tau_{k}}\to
1.$

We therefore just need to replace the convergence along the subsequence
$\tau_{k}$ with convergence along the full sequence. Suppose first that
$\beta^{+}>\beta^{-}$. Let $x\in\Delta_{0}^{-}$ and consider the sequence of
iterates $g^{i}(x)$ for $1\leq i\leq\tau(x)$. Recall from the construction of
the induced map that the iterates for which $g^{i}(x)\in U^{-}_{\varepsilon}$
all lie at the “beginning” of the sequence, i.e. once the orbit leaves the
neighbourhood $U^{-}_{\varepsilon}$ it cannot return to it before the next
return to $\Delta^{-}_{0}$. More formally, either $g^{i}(x)\notin
U^{-}_{\varepsilon}$ for all $0\leq i\leq\tau(x)$ (i.e. the finite piece of
orbit never enters $U^{-}_{\varepsilon}$, or there exists an integer $1\leq
m_{\varepsilon}<\tau(x)$ such that $g^{i}(x)\in U^{-}_{\varepsilon}$ for all
$0<i\leq m_{\varepsilon}$ and $g^{i}(x)\notin U^{-}_{\varepsilon}$ for all
$m_{\varepsilon}\leq i\leq\tau(x)$. This means that the ratio
${S_{n,\varepsilon}^{+}}/{n}$ is always larger _in-between_ returns that at
the _following_ return or, more formally, letting $k\geq 1$ be the smallest
integer such that $\tau_{k}\geq n$, we have

$1\geq\frac{S_{n,\varepsilon}^{+}}{n}\geq\frac{S_{\tau_{k},\varepsilon}^{+}}{\tau_{k}}\
$

By (30) this implies (29) in the case $\beta^{+}>\beta^{-}$. The case
$\beta^{-}>\beta^{+}$ follows replacing $-$ by $+$ in (11) and carrying out
exactly the same argument. ∎

##  5\. Proof of Theorem 3

In this section we assume throughout that $g\in\mathfrak{F}_{*}$ and therefore
$\beta^{-}=\beta^{+}=\beta$. The main technical result in the proof of Theorem
3 is the following.

######  Proposition 5.1.

For Lebesgue almost every $x\in\Delta_{0}$ we have

(31)
$\limsup_{k\to\infty}\frac{S_{n,\varepsilon}^{+}(x)}{n}=\limsup_{k\to\infty}\frac{S_{n,\varepsilon}^{-}(x)}{n}=1.$

This immediately implies that the sequence $\mu_{n}(x)$ does not converge and
that therefore $g$ does not admit any physical measures, and thus implies the
first part of Theorem 3. In Section 5.1 we reduce the proof of Proposition 5.1
to a more technical statement formulated in Proposition 5.4. In Section 5.2 we
prove Proposition 5.4 and finally, in Section 5.3 we will use (31) to show
that the set of limit points of the sequence $\mu_{n}(x)$ consists of the set
$\Omega$, completing the proof of Theorem 3.

###  5.1. Using the Gibbs-Markov structure

in this section we reduce the proof of Proposition 5.1 to a statement about
the positive measure of certain sets which are defined in terms of the Gibbs-
Markov structure of the map $G$ and of the observable
$\tau^{-},\tau^{+},\tau$. First of all let

$\mathcal{A}^{+}\coloneqq\left\\{x:\limsup_{k\to\infty}\frac{\tau^{+}\circ
G^{k}(x)}{\tau_{k}(x)}=+\infty\right\\}\quad\text{ and
}\quad\mathcal{A}^{-}\coloneqq\left\\{x:\limsup_{k\to\infty}\frac{\tau^{-}\circ
G^{k}(x)}{\tau_{k}(x)}=+\infty\right\\}.$

Notice that the ratio ${\tau^{+}\circ G^{k}(x)}/{\tau_{k}(x)}$ compares the
value of $\tau^{+}$ at the $k$’th return time, i.e. essentially the number of
iterates the orbit spends near the endpoint $1$ between the $k$’th and $k+1$
return time, with the total accumulated length $\tau_{k}$ of the orbit up to
it’s $k$’th return. If this ratio is big it means that time spent near 1
dominates whatever behaviour the orbit may have exhibited up to that time, and
for points in $\mathcal{A}^{+}$, this ratio is _arbitrarily large_ infinitely
often. Similar observations hold for $\mathcal{A}^{-}$ and therefore points
$x\in\mathcal{A}^{+}\cap\mathcal{A}^{-}$ satisfy (31), as formalized in the
following Lemma.

######  Lemma 5.2.

$x\in\mathcal{A}^{+}\Longrightarrow\limsup_{n\to\infty}\frac{S_{n,\varepsilon}^{+}}{n}=1\quad\text{
and }\quad
x\in\mathcal{A}^{-}\Longrightarrow\limsup_{n\to\infty}\frac{S_{n,\varepsilon}^{-}}{n}=1$

In particular, if $x\in\mathcal{A}^{+}\cap\mathcal{A}^{-}$ then $x$ satisfies
(31)

###### Proof.

Fixing $\varepsilon>0$ and choosing the subsequence
$n_{k}\coloneqq\tau_{k}+\tau^{+}\circ G^{k}$ we have

$\frac{S_{n_{k},\varepsilon}^{+}}{n_{k}}=\frac{\tau^{+}_{k}+\tau^{+}\circ
G^{k}}{\tau_{k}+\tau^{+}\circ
G^{k}}=\frac{\tau^{+}_{k}}{\tau_{k}^{+}}\frac{1}{1+\tau^{+}\circ
G^{k}/\tau_{k}}+\frac{1}{1+\tau_{k}/\tau^{+}\circ G^{k}}$

and so $x\in\mathcal{A}^{+}$ implies
$\limsup_{n\to\infty}S_{n,\varepsilon}^{+}/n=1$. Similarly, we get the
statement for points in $\mathcal{A}^{-}$. ∎

It follows from Lemma 5.2 that Proposition 5.1 follows if we prove that

(32) $Leb(\mathcal{A}^{+})=Leb(\mathcal{A}^{-})=1.$

Notice moreover that since the invariant measure $\hat{\mu}$ is equivalent to
Lebesgue on $\Delta_{0}$ it is enough to show that
$\hat{\mu}(\mathcal{A}^{+})=\hat{\mu}(\mathcal{A}^{-})=1$. Moreover, both sets
$\mathcal{A}^{+}$ and $\mathcal{A}^{-}$ are _invariant_ for the map $G$, i.e.
$G^{-1}(\mathcal{A}^{\pm})=\mathcal{A}^{\pm}$, and therefore, by ergodicity of
$\hat{\mu}$ it is sufficient to show that

(33) $\hat{\mu}(\mathcal{A}^{+})>0\quad\text{ and
}\quad\hat{\mu}(\mathcal{A}^{-})>0$

since this implies (32). To prove (33) it is sufficient to show that there
exist subsets $\widetilde{\mathcal{A}}^{\pm}\subseteq\mathcal{A}^{\pm}$ with
$\hat{\mu}(\widetilde{\mathcal{A}}^{\pm})>0$. For simplicity we will just
define the prove the result for ${\mathcal{A}}^{+}$ and
$\widetilde{\mathcal{A}}^{+}$, which therefore for convenience will will just
denote by $\widetilde{\mathcal{A}}$, as the result for ${\mathcal{A}}^{-}$ and
$\widetilde{\mathcal{A}}^{-}$ follows by identical arguments. To define
$\widetilde{\mathcal{A}}$ we first need to make use of the fact that return
time $\tau$ satisfies a stable law.

######  Lemma 5.3.

There is a non-degenerate stable random variable $Y$ such that for almost
every $t\in\mathbb{R}$

$\lim_{t\to\infty}\mu\left(\frac{\tau_{k}}{k^{\beta}}-d_{k}\leq
t\right)=\mu(Y\leq t),$

where $d_{k}=0$ if $\beta>1$ and $d_{k}\approx\log k$ if $\beta=1$. In
particular, there is $T>0$ such that $\mu(Y<T)>0$.

We note that to simplify the notation we use the symbol $\approx$ to denote
the fact that the ratio between quantities is uniformly bounded above and
below by a constant independent of $k$.

###### Proof.

The fact that $\tau$ satisfies stable law can be see from [AarDen2001, Theorem
6.1], however the formulation given there it is not immediate that the
sequence $d_{k}$ should grow like $\log k$ when $\beta=1$. So, instead we will
argue by [Gou2010, Theorem 1.5]. Let $X_{0},X_{1},\ldots$ be a sequence of
positive, independent and identically distributed random variables with the
same distribution as $\tau$: $\mathbb{P}(X_{0}\leq t)=\mu(\tau\leq t)$. From
Proposition 3.1 we know that $t^{1/\beta}\mathbb{P}(X_{0}\leq t)\to C>0$ and
so the classical probability literature (see for example [Nol20]) informs us
that there exists a sequence $d_{k}$ such that for almost every
$t\in\mathbb{R}$ we have
$\mathbb{P}({\sum_{j=0}^{k-1}X_{j}}/{k^{\beta}}-d_{k}\leq t)=\mathbb{P}(Y\leq
t),$ where $Y$ is a non-degenerate stable random variable. Moreover, we know
from [Nol20] that $d_{k}=0$ if $\beta>1$ and $d_{k}\approx\log k$ if
$\beta=1$. As $\tau$ is constant on the partition elements $\delta_{i,j}$, and
as $F$ is a topologically mixing Gibbs-Markov map, we know from [Gou2010,
Theorem 1.5] that $\tau_{k}$ will satisfy the same distributional convergence
as $\sum_{j=0}^{k-1}X_{j}$. Finally, as $Y$ is non-degenerate we know that
there must exists some $T>0$ such that $\mu(Y<T)>0$, which yields the Lemma. ∎

Now let $d_{k}$ and $T>0$ be as in Lemma 5.3 and let $a_{k}$ be a sequence
such that

(34) $a_{k}\to+\infty\ \quad\text{ and
}\quad\sum_{k=1}^{\infty}\frac{1}{a_{k}^{1/\beta}k(T+d_{k})}=+\infty$

(as $d_{k}\lesssim\log(k)$, regardless the value of $\beta$, we could take for
example $a_{k}^{1/\beta}=\log\log k$), and let

$A_{k}\coloneqq\\{\tau_{k}<k^{\beta}(T+d_{k})\text{ and }\tau^{+}\circ
G^{k}>a_{k}k^{\beta}(T+d_{k})\\}\quad\text{ and let
}\quad\widetilde{\mathcal{A}}\coloneqq\bigcap_{n=1}^{\infty}\bigcup_{k\geq
n}A_{k}$

be the set of points which belong to infinitely many $A_{k}$. We will prove
the following

######  Proposition 5.4.

$\widetilde{\mathcal{A}}\subseteq\mathcal{A}^{+}$ and
$\hat{\mu}(\widetilde{\mathcal{A}})>0$.

Proposition 5.4 clearly implies $\hat{\mu}(\mathcal{A}^{+})>0$, and an
identical argument gives $\hat{\mu}(\mathcal{A}^{-})>0$, and therefore, via
(33) and (32), implies Proposition 5.1. In the next section we prove
Proposition 5.4.

###  5.2. Proof of Proposition 5.4

We will first show the first part of the statement, that
$\widetilde{\mathcal{A}}\subseteq\mathcal{A}^{+}$, which is essentially a
direct consequence of the definitions.

######  Lemma 5.5.

$\widetilde{\mathcal{A}}\subseteq\mathcal{A}^{+}$

###### Proof.

If $x\in A_{k}$ then, by definition, as $k\to\infty$ we have

$\frac{\tau_{k}}{\tau^{+}\circ
G^{k}}<\frac{k^{\beta}(T+d_{k})}{a_{k}k^{\beta}(T+d_{k})}=\frac{1}{a_{k}}\to
0.$

So, when $k$ is sufficiently large, the ratio $\tau_{k}/\tau^{+}\circ G^{k}$
is arbitrarily small and $\tau^{+}\circ G^{k}/\tau_{k}$ is arbitrarily large.
It follows that for $x\in\widetilde{\mathcal{A}}$ we have
$\limsup_{k\to\infty}\tau^{+}\circ G^{k}(x)/\tau_{k}(x)=\infty$ and so
$\widetilde{\mathcal{A}}\subseteq\mathcal{A}$. ∎

To show that $\hat{\mu}(\widetilde{\mathcal{A}})>0$, and thus complete the
proof of Proposition 5.4, we will use a version of the second Borel-Cantelli
lemma, originally due to Rényi, which says that

(35) $\hat{\mu}(A_{k}\cap
A_{n})\lesssim\hat{\mu}(A_{k})\hat{\mu}(A_{n})\quad\text{ and
}\quad\sum_{k=1}^{\infty}\hat{\mu}(A_{k})=+\infty\quad\Longrightarrow\quad\hat{\mu}(\widetilde{\mathcal{A}})=\hat{\mu}\left(\bigcap_{n=1}^{\infty}\bigcup_{k\geq
n}A_{k}\right)>0.$

A more precise statement, see [Pet2002, Theorem 2.1], includes a an explicit
lower bound on the measure of $\widetilde{\mathcal{A}}$ but we will not need
that here. To verify the assumptions of (35), we observe first of all that the
sets $A_{k}$ can be written as the intersection

$A_{k}=B_{k}\cap G^{-k}C_{k}$

where

$B_{k}\coloneqq\left\\{\tau_{k}<k^{\beta}(T+d_{k})\right\\}\quad\text{ and
}\quad C_{k}\coloneqq\left\\{\tau^{+}>a_{k}k^{\beta}(T+d_{k})\right\\}.$

Notice that $B_{k}$ is a union of elements of the partition
$\mathcal{P}^{(k)}$ (on which $\tau_{k}$ is constant) whereas
$\mathcal{C}_{k}$ is a union of elements of the partition $\mathcal{P}$ (on
which $\tau^{+}$ is constant), and therefore $G^{-k}\mathcal{C}_{k}$ is a
union of elements of the partition $\mathcal{P}^{(k+1)}$. In particular,
$A_{k}$ is a union of elements of $\mathcal{P}^{(k+1)}$.

Our proof will use in a crucial way the standard bounded distortion property
of Gibbs-Markov maps which implies that for any measurable set
$\tilde{\omega}\subset\Delta_{0}$, any $k\geq 1$, and any
$\omega^{(k)}\in\mathcal{P}^{(k)}$ we have

(36)
$\hat{\mu}(\omega^{(k)}\cap\tilde{\omega})\approx\hat{\mu}(\omega^{(k)})\hat{\mu}(G^{k}(\tilde{\omega})).$

We recall that we use the symbol $\approx$ to denote that the ratio between
two quantities is uniformly bounded above and below by a constant independent
of $k$, $\omega^{(k)}$ or $\tilde{\omega}$. The standard formulation of
bounded distortion is
${\hat{\mu}(\omega^{(k)}\cap\tilde{\omega})}/{\hat{\mu}(\omega^{(k)})}\approx{\hat{\mu}(G^{k}(\tilde{\omega}))}/{\hat{\mu}(G^{k}(\omega^{(k)}))},$
see Corollary 3.11 in [CoaLuzMub22], but since
$\omega^{(k)}\in\mathcal{P}^{(k)}$, we have $G^{k}(\omega^{(k)})=\Delta_{0}$,
and since $\hat{\mu}(\Delta_{0})=1$ this implies (36). We will also need the
following result which is where we use the convergence to a stable law, as
stated in Lemma 5.3.

######  Lemma 5.6.

The measure $\hat{\mu}(B_{k})$ is uniformly bounded below.

###### Proof.

Notice first of all that can re-write the condition defining the sets $B_{k}$
as

$B_{k}=\left\\{\frac{\tau_{k}}{k^{\beta}}-d_{k}<T\right\\}.$

By Lemma 5.3, we have $\hat{\mu}(B_{k})\to\hat{\mu}(Y<T)$ and so our choice of
$T$ implies the statement. ∎

We can now verify the conditions of the Borel-Cantelli lemma in (35) in the
following two lemmas, thus completing the proof of Proposition 5.4, and
therefore of (32) and (31).

######  Lemma 5.7.

For any $k\geq 1$,

(37) $\hat{\mu}(A_{k})\approx\hat{\mu}(C_{k})\quad\text{ and therefore
}\quad\sum_{k=1}^{\infty}\hat{\mu}(A_{k})=+\infty.$

###### Proof.

It follows from the definition of the set $A_{k}$, (36), and Lemma 5.6, that

(38)
$\hat{\mu}(A_{k})=\sum_{\begin{subarray}{c}\omega^{(k)}\in\mathcal{P}^{(k)}\\\
\omega^{(k)}\subset B_{k}\end{subarray}}\hat{\mu}(\omega^{(k)}\cap
G^{-k}C_{k})\approx\sum_{\begin{subarray}{c}\omega^{(k)}\in\mathcal{P}^{(k)}\\\
\omega^{(k)}\subset
B_{k}\end{subarray}}\hat{\mu}(\omega^{(k)})\hat{\mu}(C_{k})\approx\hat{\mu}(B_{k})\hat{\mu}(C_{k})\approx\hat{\mu}(C_{k}),$

which is the first statement in (37). Now, from (14), keeping in mind that
$\beta^{+}=\beta^{-}=\beta\geq 1$, we have $\hat{\mu}(\tau^{+}>t)\approx
1/t^{1/\beta}$ and therefore, as $d_{k}$ is only non-zero when $\beta=1$,
$\hat{\mu}(A_{k})\approx\hat{\mu}(C_{k})\approx
1/(a_{k}^{1/\beta}k(T+d_{k})^{1/\beta})\approx 1/(a_{k}^{1/\beta}k(T+d_{k}))$
which, by (34), implies (37), completing the proof. ∎

######  Lemma 5.8.

For every $n,k\geq 1$ with $n\neq k$ we have $\hat{\mu}(A_{n}\cap
A_{k})\lesssim\hat{\mu}(A_{k})\hat{\mu}(A_{n})$

###### Proof.

For every $n\geq 1$ we have $\hat{\mu}(A_{n})=\hat{\mu}(B_{n}\cap
G^{-n}C_{n})\leq\hat{\mu}(G^{-n}C_{n})$ and so, supposing without loss of
generality that $k>n$ and letting $k=n+m$ and using also that $\hat{\mu}$ is
$G$-invariant, we get

$\hat{\mu}(A_{n}\cap A_{n+m})\leq\hat{\mu}(G^{-n}C_{n}\cap
G^{-k}C_{k})=\hat{\mu}(G^{-n}(C_{n}\cap G^{-m}C_{k}))=\hat{\mu}(C_{n}\cap
G^{-m}C_{k}).$

Since $C_{k}$ is a union of elements of $\mathcal{P}$ and $G^{-m}C_{k}$ is a
union of elements of $\mathcal{P}^{(m+1)}$, by (36) we get

$\hat{\mu}(C_{n}\cap
G^{-m}C_{k})=\sum_{\begin{subarray}{c}\omega^{(m)}\in\mathcal{P}^{(m)}\\\
\omega^{(m)}\subset C_{n}\end{subarray}}\hat{\mu}(\omega^{(m)}\cap
G^{-m}C_{k})\approx\sum_{\begin{subarray}{c}\omega^{(m)}\in\mathcal{P}^{(m)}\\\
\omega^{(m)}\subset
C_{n}\end{subarray}}\hat{\mu}(\omega^{(m)})\hat{\mu}(C_{k})\leq\hat{\mu}(C_{n})\hat{\mu}(C_{k}).$

Combining these two expressions and using again the first part of (37) gives
the statement. ∎

###  5.3. The space of accumulation points of $\mu_{n}(x)$

In the previous subsection we have shown that (31) holds for Lebesgue almost
every $x$ and that therefore in particular the sequence $\mu_{n}(x)$ does not
converge. To complete the proof of Theorem 3 it therefore just remains to show
that every measure $\nu_{p}$, for every $p\in[0,1]$, actually occurs as a
limit point of the sequence and that these are the only measures which can
occur as such limit points. This is a consequence of the following two lemmas.
The first is a general result which applies whenever $\beta\geq 1$ and which
essentially almost every orbit eventually spends most of it’s time in
arbitrarily small neighbourhoods of 1 and/or -1.

######  Lemma 5.9.

For almost every $x$, and for every $0<\varepsilon<1$,

(39) $\frac{S_{n,\varepsilon}^{+}(x)+S_{n,\varepsilon}^{-}(x)}{n}\to 1.$

###### Proof.

Let $K_{\varepsilon}=[-1,1]\setminus\left(U_{\varepsilon}^{+}\cup
U_{\varepsilon}^{-}\right)$ and let

(40)
$K_{n,\varepsilon}(x)\coloneqq\sum_{k=0}^{n-1}\mathbb{1}_{K_{\varepsilon}}\circ
g^{k}(x)=n-(S_{n,\varepsilon}^{+}(x)+S_{n,\varepsilon}^{-}(x))$

be the number of times the orbit of $x$ belongs to $K_{\varepsilon}$ before
time $n$. Notice that there exists $N_{\varepsilon}$, depending only on
$\varepsilon$, such that
$K_{\varepsilon}\subset\bigcup_{j=0}^{N_{\varepsilon}}\Delta_{j}^{\pm}$ and so
the orbit of $x$ enters $K_{\varepsilon}$ at most $2N_{\varepsilon}$ times
between one return and the next and so for every $k\geq 1$ we have
${K_{\tau_{k,\varepsilon}}}\leq 2N_{\varepsilon}k$. Thus, letting $k_{n}$ be
the unique integer such that $\tau_{k_{n}}(x)\leq n<\tau_{k_{n}+1}(x)$ we have

(41)
$\frac{K_{n,\varepsilon}}{n}\leq\frac{{K_{\tau_{k_{n}+1},\varepsilon}}}{\tau_{k_{n}}}\leq
2N_{\varepsilon}\frac{k_{n}+1}{\tau_{k_{n}}}$

Since $\beta\geq 1$ we have that $\tau_{k}/k\to\infty$, recall (18), and
therefore $k/\tau_{k}\to 0$ and so $({k_{n}}+1)/{\tau_{k_{n}}}\to 0$. By (41)
this implies ${K_{n,\varepsilon}}/{n}\to 0$ and therefore, substituting into
(40) proves the Lemma. ∎

The next lemma is an intuitive but non-trivial link between the limit points
of $\mu_{n}(x)$ and $S_{n,\varepsilon}^{+}(x)/n$.

######  Lemma 5.10.

For all $p\in[0,1]$, $\mu_{n_{k}}(x)\to\nu_{p}$ if and only if
$S_{n_{k},\varepsilon}^{+}(x)/n_{k}\to p$ for all small $\varepsilon>0$.

###### Proof.

First suppose that $\mu_{n_{k}}(x)\to\nu_{p}$ and fix some small
$\varepsilon>0$. Then, from the definition of the function
$S_{n,\varepsilon}^{+}$ and the measure $\mu_{n}(x)$ we have

$\frac{S_{n_{k},\varepsilon}^{+}}{n_{k}}=\frac{1}{n_{k}}\sum_{k=0}^{n-1}\mathbb{1}_{U_{\varepsilon}^{+}}\circ
g^{k}=\frac{1}{n_{k}}\sum_{k=0}^{n-1}\int\mathbb{1}_{U_{\varepsilon}^{+}}d\delta_{g^{i}(x)}=\int\mathbb{1}_{U_{\varepsilon}^{+}}d\mu_{n_{k}}(x).$

Now let $\tilde{\varphi}_{\varepsilon},\hat{\varphi}_{\varepsilon}$ be two
continuous functions such that
$\tilde{\varphi}_{\varepsilon}\leq\mathbb{1}_{U_{\varepsilon}^{+}}\leq\hat{\varphi}_{\varepsilon}$
and which are equal to 1 in some neighbourhood of $1$, equal to 0 in some
neighbourhood of $-1$. Then

$\int\tilde{\varphi}_{\varepsilon}d\mu_{n_{k}}(x)\leq\int\mathbb{1}_{U_{\varepsilon}^{+}}d\mu_{n_{k}}(x)\leq\int\hat{\varphi}_{\varepsilon}d\mu_{n_{k}}(x)$

and, since $\mu_{n_{k}}(x)\to\nu_{p}$ by assumption,

$\int\tilde{\varphi}_{\varepsilon}d\mu_{n_{k}}(x)\to\int\tilde{\varphi}_{\varepsilon}d\nu_{p}=p\quad\text{
and
}\quad\int\hat{\varphi}_{\varepsilon}d\mu_{n_{k}}(x)\to\int\hat{\varphi}_{\varepsilon}d\nu_{p}=p$

which, substituting into the expressions above, implies
$S_{n_{k},\varepsilon}^{+}(x)/n_{k}\to p$. Now suppose that

(42) $\frac{S_{n_{k},\varepsilon}^{+}(x)}{n_{k}}\to p\quad\text{ and
}\quad\frac{S_{n_{k},\varepsilon}^{-}(x)}{n_{k}}\to 1-p$

for all $\varepsilon>0$ (where the second statement follows by the first and
the second part of (31)). To show that $\mu_{n_{k}}\to\nu_{p}$ we will show
that for any continuous and bounded function $\varphi$, we have $\int\varphi
d\mu_{n_{k}}\to\varphi(-1)(1-p)+\varphi(1)p=\int\varphi d\nu_{p}$. Note first
that by definition of the measures $\mu_{n_{k}}(x)$,

(43) $\int\varphi d\mu_{n_{k}}=\frac{1}{n_{k}}\sum_{i=0}^{n_{k}-1}\varphi\circ
g^{i}(x).$

Now, for any $\tilde{\varepsilon}>0$, let $\varepsilon>0$ be sufficiently
small so that

(44) $\sup_{y\in U^{\pm}_{\varepsilon}}|\varphi(\pm
1)-\varphi(y)|<\tilde{\varepsilon},$

and split up the time averages (43) of $\varphi$ as follows:

(45) $\frac{1}{n_{k}}\sum_{i=0}^{n_{k}-1}\varphi\circ
g^{i}(x)=\frac{1}{n_{k}}\sum_{\begin{subarray}{c}0\leq i<n_{k}:\\\ g^{i}(x)\in
U_{\varepsilon}^{-}\end{subarray}}\varphi\circ
g^{i}(x)+\frac{1}{n_{k}}\sum_{\begin{subarray}{c}0\leq i<n_{k}:\\\ g^{i}(x)\in
U_{\varepsilon}^{+}\end{subarray}}\varphi\circ
g^{i}(x)+\frac{1}{n_{k}}\sum_{\begin{subarray}{c}0\leq i<n_{k}:\\\
g^{i}(x)\notin U_{\varepsilon}^{\pm}\end{subarray}}\varphi\circ g^{i}(x).$

For the first two terms of (45), by (44), we have

$(\varphi(\pm
1)-\tilde{\varepsilon})\frac{S_{n_{k},\varepsilon}^{\pm}(x)}{n_{k}}\leq\frac{1}{n_{k}}\sum_{\begin{subarray}{c}0\leq
i<n_{k}:\\\ g^{i}(x)\in U_{\varepsilon}^{\pm}\end{subarray}}\varphi\circ
g^{i}(x)\leq(\varphi(\pm
1)+\tilde{\varepsilon})\frac{S_{n_{k},\varepsilon}^{\pm}(x)}{n_{k}}$

which, by (44), for $n_{k}$ sufficiently large, gives

(46)
$(\varphi(1)-2\tilde{\varepsilon})p\leq\frac{1}{n_{k}}\sum_{\begin{subarray}{c}0\leq
i<n_{k}:\\\ g^{i}(x)\in U_{\varepsilon}^{+}\end{subarray}}\varphi\circ
g^{i}(x)\leq(\varphi(1)+2\tilde{\varepsilon})p$

and

(47)
$(\varphi(-1)-2\tilde{\varepsilon})(1-p)\leq\frac{1}{n_{k}}\sum_{\begin{subarray}{c}0\leq
i<n_{k}:\\\ g^{i}(x)\in U_{\varepsilon}^{-}\end{subarray}}\varphi\circ
g^{i}(x)\leq(\varphi(-1)+2\tilde{\varepsilon})(1-p)$

For the third term of (45) we have

$-\|\varphi\|_{\infty}\frac{n_{k}-S_{n_{k},\varepsilon}^{+}(x)-S_{n_{k},\varepsilon}^{-}(x)}{n_{k}}\leq\frac{1}{n_{k}}\sum_{\begin{subarray}{c}0\leq
i<n_{k}:\\\ g^{i}(x)\notin U_{\varepsilon}^{\pm}\end{subarray}}\varphi\circ
g^{i}(x)\leq\|\varphi\|_{\infty}\frac{n_{k}-S_{n_{k},\varepsilon}^{+}(x)-S_{n_{k},\varepsilon}^{-}(x)}{n_{k}}$

and therefore, for $n_{k}$ sufficiently large,

(48)
$-\tilde{\varepsilon}\|\varphi\|_{\infty}\leq\frac{1}{n_{k}}\sum_{\begin{subarray}{c}0\leq
i<n_{k}:\\\ g^{i}(x)\notin U_{\varepsilon}^{\pm}\end{subarray}}\varphi\circ
g^{i}(x)\leq\tilde{\varepsilon}\|\varphi\|_{\infty}.$

Substituting (46)-(48) into (45), and using (43), we then get that for $n_{k}$
sufficiently large we have

$\int\varphi
d\mu_{n_{k}}\geq(\varphi(1)-2\tilde{\varepsilon})p+(\varphi(-1)-2\tilde{\varepsilon})(1-p)-\tilde{\varepsilon}\|\varphi\|_{\infty}$

and

$\int\varphi
d\mu_{n_{k}}\leq(\varphi(1)+2\tilde{\varepsilon})p+(\varphi(-1)+2\tilde{\varepsilon})(1-p)+\tilde{\varepsilon}\|\varphi\|_{\infty}.$

Since $\tilde{\varepsilon}$ can be chosen arbitrarily small by choosing
$n_{k}$ sufficiently large, it follows that

$\int\varphi d\mu_{n_{k}}\to\varphi(1)p+\varphi(-1)(1-p)=\int\varphi d\nu_{p}$

and since $\varphi$ was also arbitrary, this implies that
$\mu_{n_{k}}\to\nu_{p}$ as $k\to\infty$. ∎

###### Proof of Theorem 3.

Equation (31) immediately implies that the sequence $\mu_{n}(x)$ cannot
converge and therefore $g$ has no physical measures. Moreover, since (39)
holds for every $\varepsilon>0$, the limit points of $\mu_{n}(x)$ must have
support in $\\{+1,-1\\}$ and therefore must be convex combinations of the form
$\nu_{p}\coloneqq p\delta_{1}+(1-p)\delta_{-1}$ for $p\in[0,1]$, and thus
belong to the set $\Omega$, recall (9). It therefore just remains to show that
for any $p\in[0,1]$, the measure $\nu_{p}$ is a limit point of the sequence
$\mu_{n}(x)$.

In view of Lemma 5.10, it is sufficient to show that for every $p\in[0,1]$
there exists a subsequence

(49) $n_{k}^{(p)}\to\infty\quad\text{ such that
}\quad\frac{S_{n_{k}^{(p)},\varepsilon}^{+}(x)}{n_{k}^{(p)}}\to p$

for small $\varepsilon>0$. By (31) this is true for $p=0,1$ i.e. there exist
subsequences $n_{k}^{(0)},n_{k}^{(1)}$ such that

(50) $\frac{S_{n_{k}^{(0)},\varepsilon}^{+}(x)}{n_{k}^{(0)}}\to 0\quad\text{
and }\quad\frac{S_{n_{k}^{(1)},\varepsilon}^{+}(x)}{n_{k}^{(1)}}\to 1.$

Now notice that by definition of the sequence $S^{+}_{n,\varepsilon}(x)$,
recall (25), we have that

$S^{+}_{n+1,\varepsilon}(x)-S^{+}_{n,\varepsilon}(x)=\begin{cases}1&\text{ if
}g^{n}(x)\in U_{\varepsilon}^{+}\\\ 0&\text{ if }g^{n}(x)\notin
U_{\varepsilon}^{+}\end{cases}$

In particular, for any $n\geq 1$ we have
$S^{+}_{n+1,\varepsilon}(x)-S^{+}_{n,\varepsilon}(x)\leq 1$ and therefore

(51)
$\left|\frac{S^{+}_{n+1,\varepsilon}(x)}{n+1}-\frac{S^{+}_{n,\varepsilon}(x)}{n}\right|\leq\frac{1}{n}.$

This means that for sufficiently large $n$, the difference between two
consecutive terms of the sequence ${S^{+}_{n,\varepsilon}(x)}/{n}$ is
arbitrarily small. By (50) this sequence oscillates infinitely often between
values arbitrarily close to 0 and arbitrarily close to 1 and therefore, by
(51), for any $p\in(0,1)$ and any $\delta>0$ the sequence
${S^{+}_{n,\varepsilon}(x)}/{n}$ must enter the interval $(p-\delta,p+\delta)$
infinitely often. Since $\delta$ is arbitrary, there must be a subsequence
converging to $p$, giving (49) and completing the proof. ∎

##  6\. Proof of Theorem 4

Throughout this section we fix $f\in\widehat{\mathfrak{F}}$ and let and let
$\ell_{1},\ell_{2},k_{1},k_{2},a_{1},a_{2},b_{1},b_{2},\iota$ be the
corresponding parameters as in (3). Then, given _arbitrary constants_
$\tilde{\ell}_{1}.\tilde{\ell}_{2}\geq 0$, in Section 6.1 we will give a quite
explicit construction of a map $g$ which, in Section 6.2, we will show belongs
to our class $\mathfrak{F}$, with parameters
$\tilde{\ell}_{1},\tilde{\ell}_{2},k_{1},k_{2},a_{1},a_{2},\tilde{b}_{1},\tilde{b}_{2},\tilde{\iota}$,
for appropriately chosen constants
$\tilde{b}_{1},\tilde{b}_{2},\tilde{\iota}$. In Section 6.3 we estimate the
distance between $f$ and $g$ in an appropriate topology and finally, in
Section 6.4, we apply these estimates to the various cases required by Theorem
4 and thus complete the proof.

###  6.1. Construction of $g$

In Section 6.1.1 we describe the general construction of $g$ and introduce the
other parameters and functions on which $g$ will depend. In Section 6.1.2 we
then make some specific choices of the various parameters and functions
involved in the construction.

####  6.1.1. General strategy for constructing the perturbation

Let $f\in\widehat{\mathfrak{F}}$ and let the corresponding parameters as in
(3) be $\ell_{1},\ell_{2}\geq 0$, $k_{1},k_{2},a_{1},a_{2},b_{1},b_{2}>0$. For
any two constants $\tilde{\ell}_{1}>0$ and $\tilde{\ell}_{2}>0$, and any two
compact intervals

$[\tilde{x}_{1},x_{1}]\subset U_{-1}\quad\text{ and
}\quad[x_{2},\tilde{x}_{2}]\subset U_{1},$

we define functions $h_{1}:U_{-1}\to[-1,1]$ and $h_{2}:U_{1}\to[-1,1]$ as
follows. If $\tilde{\ell}_{1}=\ell_{1}$ or $\tilde{\ell}_{2}=\ell_{2}$ then we
let $h_{1}=f|_{U_{-1}}$ and $h_{2}=f|_{U_{1}}$ respectively. Otherwise, we let

(52) $h_{1}(x)\coloneqq x+\tilde{b}_{1}(1+x)^{1+\tilde{\ell}_{1}}\quad\text{
and }\quad h_{2}(x)\coloneqq x-\tilde{b}_{2}(1-x)^{1+\tilde{\ell}_{2}},$

where $\tilde{b}_{1},\tilde{b}_{2}>0$ are any constants such that

(53) $h_{1}(x)\leq f(x)\quad\text{ on }\quad[x_{1},\tilde{x}_{1}],\quad\text{
and }\quad h_{2}(x)\geq g(x)\text{ on }\quad[\tilde{x}_{2},x_{2}],$

for every integer $k\leq\lceil\ell_{1}\rceil$ and $k\leq\lceil\ell_{2}\rceil$
respectively. Note that if $\tilde{\ell}_{1,2}\geq\ell_{1,2}$ then we can take
$\tilde{b}_{1,2}=b_{1,2}$ and the corresponding line of (53) will hold.
Moreover, regardless of the relative values of $\ell_{1},\ell_{2}$ and
$\tilde{\ell}_{1},\tilde{\ell}_{2}$, we (53) will always hold for all
$\tilde{b}_{1},\tilde{b}_{2}>0$ sufficiently small as we are only asking for
the inequalities in (53) to be satisfied for $x$ in compact intervals away
from $-1$ and $1$. We let

$\xi_{1}:[\tilde{x}_{1},x_{1}]\to[0,1]\quad\text{ and
}\quad\xi_{2}:[x_{2},\tilde{x}_{2}]\to[0,1]$

be $C^{\infty}$ _monotone increasing bijections_ and define $g$ on the
intervals $[-1,0)$ and $[0,1]$ by

(54) $g|_{[-1,0]}(x)\coloneqq\begin{cases}h_{1}(x)&\text{if
}x\in[-1,-\tilde{x}_{1}]\\\ h_{1}(x)+\xi_{1}(x)(f(x)-h_{1}(x))&\text{if
}x\in(\tilde{x}_{1},x_{1})\\\ f(x)&\text{if }x\in[x_{1},0).\end{cases}$

and

(55) $g|_{[0,1]}(x)\coloneqq\begin{cases}f(x)&\text{if }x\in[0,x_{2}]\\\
f(x)+\xi_{2}(x)(h_{2}(x)-f(x))&\text{if }x\in(x_{2},\tilde{x}_{2})\\\
h_{2}(x)&\text{if }x\in[\tilde{x}_{2},1].\end{cases}$

Notice that $g$ is equal to $f$ outside $U_{-1}$ and $U_{1}$ and, apart from
$\tilde{\ell}_{1},\tilde{\ell}_{2}$, depends on the two intervals
$[\tilde{x}_{1},x_{1}],[x_{2},\tilde{x}_{2}]$, the constants
$\tilde{b}_{1},\tilde{b}_{2}$, and the functions $\xi_{1},\xi_{2}$, which we
explain below how to choose.

####  6.1.2. Choosing
$x_{1},x_{2},\tilde{x}_{1},\tilde{x}_{2},\tilde{b}_{1},\tilde{b}_{2},\xi_{1},\xi_{2}$

We explain how to make a specific choice of the constants
$\tilde{x}_{1},\tilde{x}_{2},\tilde{b}_{1},\tilde{b}_{2}$ and the functions
$\xi_{1},\xi_{2}$ depending on arbitrary $x_{1}\in U_{-1}$ and $x_{2}\in
U_{1}$. First of all we define the _affine orientation preserving bijections_
$\eta_{1}:[\tilde{x}_{1},x_{1}]\to[0,1]$ and
$\eta_{2}:[x_{2},\tilde{x}_{2}]\to[0,1]$ by

$\eta_{1}(x)\coloneqq\frac{x-\tilde{x}_{1}}{x_{1}-\tilde{x}_{1}}\quad\text{
and }\quad\eta_{2}(x)\coloneqq\frac{x-x_{2}}{\tilde{x}_{2}-x_{2}}.$

Then we define a $C^{\infty}$ map $\xi:[0.1]\to[0,1]$ by

(56) $\xi(x)\coloneqq\begin{cases}0&\text{if }x=0\\\
\exp\left\\{1-\frac{1}{1-(x-1)^{2}}\right\\}&\text{if }x\in(0,1].\end{cases}$

and let

$\xi_{1}(x):=\xi\circ\eta_{1}(x)\quad\text{ and
}\quad\xi_{2}(x):=\xi\circ\eta_{2}(x)$

Notice that $\xi$ is monotone increasing, $\xi(0)=0$, $\xi(1)=1$, and
$D^{k}\xi(0)=D^{k}\xi(1)=0$ for every $k\geq 1$ and therefore
$\xi_{1},\xi_{2}$ are also in particular $C^{\infty}$ and flat at the
endpoints. We now fix $\tilde{x}_{1}$ to be the mid-point between $-1$ and
$x_{1}$, and $\tilde{x}_{2}$ to be the midpoint between $x_{2}$ and $1$, i.e.

$\tilde{x}_{1}\coloneqq\frac{1}{2}x_{1}-\frac{1}{2},\quad\text{ and
}\quad\tilde{x}_{2}\coloneqq\frac{1}{2}x_{2}+\frac{1}{2}.$

######  Remark 6.1.

The fundamental reason for these choices is that, by using the explicit forms
of $\eta_{1},\eta_{2}$ and $\xi$, and the definition of $\xi_{1},\xi_{2}$, we
can verify by repeated use of the chain rule that there exists a $C>0$ such
that for every $x_{1}\in U_{-1}$, every $x\in[x_{1},\tilde{x}_{1}]$ and for
every $k\leq\lceil\ell_{1}\rceil$ we have

(57)
$D^{k}\xi_{1}(x)=D^{k-1}\left(\frac{2}{1+x_{1}}\xi^{\prime}(\eta_{1}(x))\right)=\left(\frac{2}{1+x_{1}}\right)^{k}D^{k}\xi(\eta_{1}(x))\leq
C(1+x_{1})^{-k}.$

and, similarly, for every $x_{2}\in U_{1}$, every $x\in[x_{2},1]$ and every
$k\leq\lceil\ell_{2}\rceil$, we have

$D^{k}\xi_{2}(x)\leq C(1-x_{2})^{-k}.$

Next, we will fix $\tilde{b}_{1},\tilde{b}_{2}>0$. These constants play no
role in the statistical properties of the map but will need to be chosen
carefully to ensure that $g$ indeed satisfies -(A2). We have already that
$\tilde{b}_{1},\tilde{b_{2}}$ are small enough so that (53) holds, and now we
also require the following conditions:

(58) $\displaystyle\text{if }\ell_{1}>0,\text{then }$ $\displaystyle
b_{1}\ell_{1}(1+x)^{\ell_{1}}-\tilde{b}_{1}\tilde{\ell}_{1}(1+x)^{\tilde{\ell}_{1}}$
$\displaystyle\geq 0$ $\displaystyle\text{ for all
}x\in[\tilde{x}_{1},x_{1}],$ $\displaystyle\text{if }\ell_{1}=0,\text{then }$
$\displaystyle 1-\tilde{b}_{1}\tilde{\ell}(1+x)^{\tilde{\ell}_{1}}$
$\displaystyle\geq 0$ $\displaystyle\text{ for all
}x\in[\tilde{x}_{1},x_{1}],$

and

(59) $\displaystyle\text{if }\ell_{2}>0,\text{then }$ $\displaystyle
b_{2}\ell_{2}(1+x)^{\ell_{2}}-\tilde{b}_{2}\tilde{\ell}_{2}(1+x)^{\tilde{\ell}_{2}}$
$\displaystyle\geq 0$ $\displaystyle\text{ for all
}x\in[x_{2},\tilde{x}_{2}],$ $\displaystyle\text{if }\ell_{2}=0,\text{then }$
$\displaystyle 1-\tilde{b}_{1}\tilde{\ell}_{2}(1+x)^{\tilde{\ell}_{2}}$
$\displaystyle\geq 0$ $\displaystyle\text{ for all
}x\in[x_{2},\tilde{x}_{2}].$

Note that it is always possible to find $\tilde{b}_{1},\tilde{b}_{2}>0$ small
enough so that the above hold.

This concludes our definition of the map $g$ which depends only $f$ and the
parameters $\tilde{\ell}_{1},\tilde{\ell}_{2}>0$ and $x_{1},x_{2}$. Notice
that choice of $\tilde{\ell}_{1},\tilde{\ell}_{2}>0$ is completely arbitrary
and the only restriction on the choice of $x_{1},x_{2}$ is that they have to
lie in the neighbourhoods $U_{-1},U_{1}$, in particular notice that
$x_{1},x_{2}$ can be chosen arbitrarily close to the fixed points $-1,1$.

###  6.2. The map $g$ is in the class $\widehat{\mathfrak{F}}$

######  Lemma 6.2.

$g$ belongs to the class $\widehat{\mathfrak{F}}$ with parameters
$\tilde{\ell}_{1},\tilde{\ell}_{2},k_{1},k_{2},a_{1},a_{2},\tilde{b}_{1},\tilde{b}_{2}$.

######  Remark 6.3.

Note that all of the parameters of $g$ are fixed by the map $f$ _except_ for
$\tilde{\ell}_{1},\tilde{\ell}_{2}$ which are _arbitrary_ positive constants,
which means that we are free to choose $\tilde{\ell}_{1},\tilde{\ell}_{2}$ so
that $g$ is in any one of the three distinct subclasses
$\mathfrak{F},\mathfrak{F}_{\pm},\mathfrak{F}_{*}$ defined in (8).

###### Proof.

It follows immediately from the construction that the map is full branch and
$C^{2}$ and that  is therefore satisfied. The neighbourhoods $U_{\pm 1}$ and
$U_{0\pm}$ no longer have the required form as in  but we can shrink them and
define the intervals

(60)
$\begin{split}\widetilde{U}_{-1}\coloneqq[-1,\tilde{x}_{1}),\quad\widetilde{U}_{1}\coloneqq(\tilde{x}_{2},1],\quad\widetilde{U}_{0-}=g^{-1}(\widetilde{U}_{1}),\quad\widetilde{U}_{0+}\coloneqq
g^{-1}(\widetilde{U}_{-1}).\end{split}$

It then follows that $g$ satisfies  with respect to these neighbourhoods for
the required parameters
$\tilde{\ell}_{1},\tilde{\ell}_{2},k_{1},k_{2},a_{1},a_{2},\tilde{b}_{1},\tilde{b}_{2}$.
Since we have shrunk the neighbourhoods and modified the map in the regions
$U_{-1}\setminus\widetilde{U}_{-1}$ and $U_{1}\setminus\widetilde{U}_{1}$ it
is no longer immediate that the expansivity condition (A2) continues to hold
outside these new neighbourhoods and we therefore just need to check (A2).

Let $\tilde{\delta}_{n}^{\pm},\tilde{\Delta}_{n}^{\pm}$ and
$\delta_{n}^{\pm},\Delta_{n}^{\pm}$ denote the partitions corresponding to $g$
and $f$ respectively. We know from the construction of $g$ that $g^{n}\equiv
f^{n}$ on $\delta_{n}^{\pm}$, and $\tilde{\delta}_{n}^{\pm}=\delta_{n}^{\pm}$
for every $n\leq n^{\pm}\coloneqq\min\\{x:\delta_{n}\subset U_{0\pm}(f)\\}$.
As $f$ satisfies (A2) we know that

(61)
$(g|_{\delta_{n}^{\pm}}^{n})^{\prime}(x)=(f|_{\delta_{n}^{\pm}}^{n})^{\prime}(x)>\lambda>1\text{
for every }n\leq n^{\pm}.$

So, in order to verify that $g$ satisfies (A2) it remains to check that

(62) $(g^{n})^{\prime}(x)>\lambda>1,\text{ for every
}x\in\tilde{\delta}_{n}^{\pm}\text{ and for every }n^{\pm}+1\leq
n\leq\tilde{n}^{\pm},$

for some (possibly different) $\lambda>1$, where
$\tilde{n}^{\pm}\coloneqq\min\\{x:\delta_{n}\subset\tilde{U}_{0\pm}(f)\\}$. We
will follow the argument given [CoaLuzMub22]*Section 3.3 and show that
$(g^{n+1})^{\prime}(x)\geq(g^{n})^{\prime}(x)$ for every $n\geq n^{+}$ and
every $x\in\tilde{\delta}_{n}^{+}$. The argument for $n\geq n^{-}$ and
$x\in\tilde{\delta}_{n}^{-}$ then follows in the same way. Note that if
$k_{1},\in(0,1]$ then there is nothing to prove, so we will assume throughout
this proof that $k_{1}>1$. As in [CoaLuzMub22]*Section 3.3 we define

(63) $\varphi\coloneqq(g|_{U_{0+}})^{-1}\circ g|_{U_{-1}}\circ g|_{U_{0+}}$

and claim that the conclusion of [CoaLuzMub22]*Lemma 3.7 holds for $g$, namely
we claim that

(64)
$\frac{(g^{2})^{\prime}(x)}{g^{\prime}(\varphi(x))}=\frac{g^{\prime}(x)}{g^{\prime}(\varphi(x))}g^{\prime}(g(x))>1,\text{
for every }x\in\tilde{\delta}_{n+1}^{+}\text{ and every }n\geq n^{+}.$

Notice that if $g(x)\in[x_{1},0)$, or if $g(x)\in[-1,\tilde{x}_{1}]$ then we
can apply[CoaLuzMub22]*Lemma 3.7 to obtain
$(g^{2})^{\prime}(x)/g^{\prime}(\varphi(x))>1$. If instead
$g(x)\in(\tilde{x}_{1},x_{1})$, then [CoaLuzMub22]*Lemma 3.7 cannot be applied
directly, but its proof can be adapted to our setting as show below. So, let
us assume that $x\in\delta_{n}^{+}$ for some $n$ and that
$g(x)\in(\tilde{x}_{1},x_{1})$. Since we are working only with
$x\in\delta_{n}^{+}$ we will drop the subscripts on the parameters to ease
notation, specifically we will let $\ell=\ell_{1}$,
$\tilde{\ell}=\tilde{\ell}_{1}$, $b=b_{1}$, $)\tilde{b}=\tilde{b}_{1}$,
$k=k_{2}$ and $a=a_{2}$. By the definition of $g$ in $U_{0+}$ given in  we
have

(65)
$\frac{g^{\prime}(x)}{g^{\prime}(\varphi(x))}=\left(\frac{x}{\varphi(x)}\right)^{k-1}=\left(\frac{\varphi(x)}{x}\right)^{1-k}$

Recall that $k>1$ and $x<\varphi(x)$ and so the ratio above is strictly less
than $1$. Let us fix

(66) $y\coloneqq g(x)=-1+ax^{k},$

We will compute $(\varphi(x)/x)^{k}$ and $g^{\prime}(g(x))$ in case that
$\ell=0$ and the case that $\ell>0$ separately. First we note that regardless
the value of $\ell$ we find that inserting the definition of $g$ into (63)
yields

(67)
$\varphi(x)=\left(\frac{1}{a}\right)^{1/k}(1+g(y))^{1/k}=\left(\frac{1}{a}\right)^{1/k}[1+h_{1}(y)+\xi(y)(f(y)-h_{1}(y))]^{1/k}.$

and, using (53) we have that

(68)
$g^{\prime}(g(x))=h_{1}^{\prime}(y)+\xi(y)(f^{\prime}(y)-h_{1}^{\prime}(y))+\xi^{\prime}(y)(f(y)-h(y))\geq
h_{1}^{\prime}(y)+\xi(y)(f^{\prime}(y)-h_{1}^{\prime}(y))$

1) Suppose that $\ell>0$. Inserting (66) into the definitions of $f$ and
$h_{1}$ we find
$f(y)-h_{1}(y)=ax^{k}(ba^{\ell}x^{\ell}-\tilde{b}a^{\tilde{\ell}}x^{\tilde{\ell}})$,
and we note for later use that (53) ensures that

(69) $ba^{\ell}x^{\ell}-\tilde{b}a^{\tilde{\ell}}x^{\tilde{\ell}}\geq 0.$

Thus, from (67),

$\displaystyle\left(\frac{\varphi(x)}{x}\right)^{k}$
$\displaystyle=\left(\frac{1}{ax^{k}}\right)\left[ax^{k}+\tilde{b}a^{\tilde{\ell}+1}x^{k(\tilde{\ell}+1)}+\xi(y)ax^{k}\left(ba^{\ell}x^{\ell}-\tilde{b}a^{\tilde{\ell}}x^{\tilde{\ell}}\right)\right]$
$\displaystyle=1+\tilde{b}a^{\tilde{\ell}}x^{k\tilde{\ell}}+\xi(y)\left(ba^{\ell}x^{\ell}-\tilde{b}a^{\tilde{\ell}}x^{\tilde{\ell}}\right)$

Next, using (58), (68) and (69), we obtain

$\displaystyle g^{\prime}(g(x))$
$\displaystyle>1+\tilde{b}x^{k\tilde{\ell}}+\xi(y)(f^{\prime}(y)-h_{1}^{\prime}(y))$
$\displaystyle=1+\tilde{b}x^{k\tilde{\ell}}+\xi(y)((1+\ell)ba^{\ell}x^{k\ell}-(1+\tilde{\ell})\tilde{b}a^{\tilde{\ell}}x^{k\tilde{\ell}})$
(70) $\displaystyle\geq
1+\tilde{b}x^{k\tilde{\ell}}+\xi(y)\left(ba^{\ell}x^{k\ell}-\tilde{b}a^{\tilde{\ell}}x^{k\tilde{\ell}}\right)=\left(\frac{\varphi(x)}{x}\right)^{k}.$

So, from (65) and (70) we get

$\frac{g^{\prime}(x)}{g^{\prime}(\varphi(x))}g^{\prime}(g(x))>\left(\frac{\varphi(x)}{x}\right)^{k-1}\left(\frac{\varphi(x)}{x}\right)^{k}>1,$

Which proves our claim (64), in the case that $\ell>0$.

2) If $\ell=0$, then we proceed as before and insert the definitions of $f$
and $h_{1}$ into (63)

$\displaystyle\varphi(x)=\left(\frac{1}{a}\right)^{1/k}\left[ax^{k}+\tilde{b}a^{\ell+1}x^{k(\tilde{\ell}+1)}+\xi(y)\left(-1+\eta(y)+bax^{k}-\tilde{b}a^{1+\tilde{\ell}}x^{k(1+\tilde{\ell})}\right)\right],$

and so

$\left(\frac{\varphi(x)}{x}\right)^{k}=1+\tilde{b}a^{\tilde{\ell}}x^{k\tilde{\ell}}+\xi(y)\left(\frac{-1+\eta(y)}{ax^{k}}+b-\tilde{b}a^{\tilde{\ell}}x^{k\tilde{\ell}}\right).$

We recall from [CoaLuzMub22]*Equation(13) that
$\eta^{\prime\prime}(y)\implies\eta^{\prime}(y)\geq\eta(y)/(1+y)>(\eta(y)-1)/(1+y)$
for every $y\in U_{-1}$. So, inserting the expressions for $f^{\prime}$ and
$h_{1}^{\prime}$ into (68)

$\displaystyle g^{\prime}(y)$ $\displaystyle\geq
1+\tilde{b}(1+\tilde{\ell})(1+y)^{\tilde{\ell}}+\xi(y)\left(\eta^{\prime}(y)+1+b-\tilde{b}(1+\tilde{\ell})(1+y)^{\tilde{\ell}}\right)$
$\displaystyle>1+\tilde{b}a^{\tilde{\ell}}x^{k\tilde{\ell}}+\xi(y)\left(\frac{-1+\eta(y)}{ax^{k}}+b-\tilde{b}a^{\tilde{\ell}}x^{k\tilde{\ell}}+1-\tilde{b}\tilde{\ell}a^{\tilde{\ell}}x^{k\tilde{\ell}}\right)\geq\left(\frac{\varphi(x)}{x}\right)^{k}.$

This concludes (64) in the case that $\ell=0$. We can then proceed as in
[CoaLuzMub22]*Corollary 3.9, to get

$(g^{n+1})^{\prime}(x)=g^{\prime}(x)g^{\prime}(g(x))\cdots
g^{\prime}(g^{n}(x))=\frac{g^{\prime}(x)g^{\prime}(g(x))}{g^{\prime}(\varphi(x))}(g^{n})^{\prime}(\varphi(x))>(g^{n})^{\prime}(\varphi(x)).$

This, together with (61), implies (62) and allows us to conclude that $g$
satisfies (A2). ∎

###  6.3. $g$ is $C^{r}$ close to $f$

Let $f\in\widehat{\mathfrak{F}}$. In Sections 6.1 and 6.2 we constructed a map
$g\in\widehat{\mathfrak{F}}$ ultimately depending only on the choice of two
arbitrary constants $\tilde{\ell}_{1},\tilde{\ell}_{2}>0$ and two points
$x_{1}\in U_{-1}$, $x_{2}\in U_{1}$. We now show that $g$ can be chosen
arbitrarily close to $f$, by choosing the points $x_{1},x_{2}$ sufficiently
close to the fixed points $-1,1$ respectively, in a topology determined by the
constants

(71) $r_{1}\coloneqq\min\\{\ell_{1},\tilde{\ell}_{1}\\},\quad\text{ and }\quad
r_{2}\coloneqq\min\\{\ell_{1},\tilde{\ell}_{2}\\}.$

More precisely, we have the following result.

######  Lemma 6.4.

There exists a $C>0$ such that

$\|f-g\|_{C^{\lceil r_{1}\rceil}([-1,0])}\leq C(1+x_{1})^{1+r_{1}-\lceil
r_{1}\rceil},\quad\text{ and }\quad\|f-g\|_{C^{\lceil r_{2}\rceil}([0,1])}\leq
C(1-x_{2})^{1+r_{2}-\lceil r_{2}\rceil}.$

###### Proof.

We will only give an explicit proof of the bound for $\|f-g\|_{C^{\lceil
r_{1}\rceil}}$ as the argument for $\|f-g\|_{C^{\lceil r_{2}\rceil}}$ is the
same. If $\ell_{1}=\tilde{\ell_{1}}$ we obtain trivially that $f-g\equiv 0$.
So, let us assume $\ell_{1}\neq\tilde{\ell_{1}}$ and consider the subcases
$\ell_{1}=0$, $\ell_{1}>0$ separately. If $\ell_{1}=0$, then $r_{1}=0$ and
using the definition (52) of $h_{1}$ near $-1$, the fact that $\xi$ is
bounded, and the definition of $g$ one finds that

$\|g-f\|_{C^{0}}\leq 2\|h_{1}-f\|_{C^{0}}<C(1+x)\leq C(1+x_{1}).$

This proves the result in the case that $\ell_{1}=0.$ Let us now suppose
throughout the remainder of the proof that $\ell_{1}>0$. We begin by
establishing the following sublemma.

######  Sublemma 6.5.

There exists a $C>0$ so that for every $1\leq k\leq\lceil r_{1}\rceil$

(72) $\begin{split}&|D^{k}f(x)-D^{k}h_{1}(x)|\leq
C(1+x_{1})^{1+r_{1}-k}\quad\forall x\in[-1,x_{1}]\\\ \end{split}$

###### Proof of Sublemma 6.5.

Suppose that $0<r_{1}=\ell_{1}<\tilde{\ell}_{1}$, let $1\leq k\leq\lceil
r_{1}\rceil$ and note that for some constants $c_{k},\tilde{c}_{k}$

$\displaystyle|D^{k}f(x)-D^{k}h_{1}(x)|$
$\displaystyle=|c_{k}(1+x)^{1+\ell_{1}-k}+\tilde{c}_{k}(1+x)^{1+\tilde{\ell}_{1}-k}|=(1+x)^{1+\ell_{1}-k}|c_{k}-\tilde{c}_{k}(1+x)^{\tilde{\ell}_{1}-\ell_{1}}|.$

As $(1+x)^{\tilde{\ell}_{1}-\ell_{1}}\to 0$ as $x\to-1$, and as we are
considering only finitely many $k$, we see that there exists some constant
$C>0$, independent of $k$ such that (72) holds. Suppose now that
$0<\tilde{\ell}_{1}<\ell_{1}$. Repeating the calculation above with $\ell_{1}$
and $\tilde{\ell}_{1}$ exchanged we conclude the proof. ∎

We now continue the proof of Lemma 6.4. The bound (72) immediately implies

$\|f-g\|_{C^{\lceil r_{1}\rceil}[-1,\tilde{x}_{1}]}=\|f-h_{1}\|_{C^{\lceil
r_{1}\rceil}[-1,\tilde{x}_{1}]}\leq C(1+x_{1})^{1+r_{1}-\lceil r_{1}\rceil}$

for some $C$ which depends on $\lceil r_{1}\rceil$, but not on $x_{1}$. For
$x\in[\tilde{x}_{1},x_{1}]$ we find by repeated applications of the product
rule that

(73)
$D^{k}g(x)=D^{k}h_{1}(x)+\sum_{j=0}^{n}\binom{k}{j}D^{j}\xi_{1}(x)\cdot(D^{k-j}f(x)-D^{k-j}h_{1}(x)).$

Using (57) and using (72), we obtain

$\displaystyle\left|\sum_{j=0}^{n}\binom{k}{j}D^{j}\xi_{1}\cdot(D^{k-j}f(x)-D^{k-j}h_{1}(x))\right|$
$\displaystyle\leq
C\sum_{j=0}^{n}\binom{k}{j}(1+x_{1})^{-j}x_{1}^{1+r_{1}-k+j}$ (74)
$\displaystyle=C(1+x_{1})^{1+r_{1}-k}.$

Finally, combining (72), (73) and (74) we find that for any $k=0,\ldots,\lceil
r_{1}\rceil$ we find that $|D^{k}f(x)-D^{k}g(x)|\leq C(1+x_{1})^{1+r_{1}-k}$,
for every $x\in[-1,-1+x_{1}]$. So,

(75) $\|f-g\|_{C^{\lceil r_{1}\rceil}[-1,0]}\leq C(1+x_{1})^{1+r_{1}-\lceil
r_{1}\rceil},$

for some $C$ which does not depend on $x_{1}$. ∎

###  6.4. Concluding the proof of Theorem 4

###### Proof.

Let $f\in\widehat{\mathfrak{F}}$ and let $\varepsilon>0$. For each of the
classes $\mathfrak{F},\mathfrak{F}_{\pm},\mathfrak{F}_{*}$ we will choose
$\tilde{\ell}_{1},\tilde{\ell}_{2}>0$ so that the corresponding map $g$
constructed in Section 6.1 belongs to the chosen class, and then, by Lemma
6.4, we can choose $x_{1}\in U_{-1},x_{2}\in U_{1}$ so that $g$ is
$\varepsilon$-close to $f$ in the appropriate topology.

We illustrate this process in detail in a couple of cases and then give some
tables to show that choices in all cases. Suppose first that
$f\in\mathfrak{F}$ (so that $\beta\in[0,1)$ and $f$ has a physical measure
equivalent to Lebesgue) and let us approximate $f$ by some map
$g\in\mathfrak{F}_{*}$, ( so that $\beta\geq 1$ and $g$ has no physical
measure). Set $\tilde{\ell}_{1}=1/k_{2},\tilde{\ell}_{2}=1/k_{1}$ so that
$\tilde{\ell}_{1}k_{2}=\tilde{\ell}_{2}k_{1}=1$, which ensures that
$g\in\mathfrak{F}_{*}$. Notice that we could choose
$\tilde{\ell}_{1}=t/k_{2},\tilde{\ell}_{2}=t/k_{1}$ for any $t\geq 1$ and that
for any such choice we have $\tilde{\ell}_{1}>\ell_{1}$ and
$\tilde{\ell}_{2}>\ell_{2}$ because by assumption we have
$\ell_{1}k_{2},\ell_{2}k_{1}\in[0,1)$. Once
$\tilde{\ell}_{1},\tilde{\ell}_{2}$ have been chosen we immediately get the
regularity of the approximation from (71) which in this case is given by
$r=\lceil\min\\{\ell_{1},\tilde{\ell}_{1},\ell_{2},\tilde{\ell}_{2}\\}\rceil=\lceil\min\\{\ell_{1},\ell_{2}\\}\rceil=r_{*}(f)$,
thus proving the Theorem in this case.

For a second example, suppose that $f\in\mathfrak{F}_{\pm}$ with
$0<\beta^{+}<1\leq\beta^{-}$ (so that $f$ has a physical measure supported on
the fixed point $-1$), and let us construct a $\tilde{g}\in\mathfrak{F}$ (so
that $g$ has a physical measure equivalent to Lebesgue) that is close to $f$.
Recall that $\tilde{g}\in\mathfrak{F}$ if and only if
$\beta^{+},\beta^{-}\in[0,1)$ and so we can leave unchanged the value of
$\beta^{+}$, and therefore let $\tilde{\ell}_{2}=\ell_{2}$, but we need to
lower the value of $\beta^{-}$ to something less than 1. We can do this by
letting $\tilde{\ell}_{1}=1/k_{2}-\gamma$ for any $0<\gamma<1/k_{2}$, which
gives
$\beta^{-}(g)=\tilde{\ell}_{2}k_{2}=(1/k_{2}-\gamma)k_{2}=1-k_{2}\gamma<1$,
and therefore $g\in\mathfrak{F}$ as required. To estimate the distance between
$f$ and $g$, and to choose the appropriate metric for this distance, notice
that $g|_{[0,1]}=f|_{[0,1]}$ and therefore we only need to worry about the
distance between $f$ and $g$ on $[-1,0]$. Therefore, by (71) we get
$r=\lceil\min\\{\ell_{1},\tilde{\ell}_{1}\\}\rceil=\lceil\ell_{1}\rceil=\lceil
1/k_{2}-\gamma\rceil$ and by Lemma 6.4 we can construct $g$ arbitrarily close
to $f$ in the $d_{r}$ metric with $r=\lceil 1/k_{2}-\gamma\rceil$ for any
$\gamma>0$ arbitrarily small. Notice however that if $\gamma$ is sufficiently
small then $\lceil 1/k_{2}-\gamma\rceil=\lceil 1/k_{2}\rceil$ and therefore
$r=\tilde{r}(f)$ as claimed in the Theorem.

All the cases can be obtained by a simple reasoning as illustrated in the two
examples above, from which we deduce the choices for
$\tilde{\ell}_{1},\tilde{\ell}_{2}$ and the corresponding regularity of the
approximation given in the Table 1 below, thus completing the proof.

Parameters of $f$ | $\tilde{\ell}_{1}$ | $\tilde{\ell}_{2}$
---|---|---
$1\leq\beta^{+}<1\leq\beta^{-}$ | $1/k_{2}-\gamma$ | $\ell_{2}$
$1\leq\beta^{-}<1\leq\beta^{+}$ | $\ell_{1}$ | $1/k_{1}-\gamma$
$\beta^{+},\beta^{-}\geq 1$ | $1/k_{2}-\gamma$ | $1/k_{1}-\gamma$
(a)

Parameters of $f$ | $\tilde{\ell}_{1}$ | $\tilde{\ell}_{2}$
---|---|---
$\ell_{1}\geq\ell_{2}$ and $\beta^{-}<1$ | $1/k_{2}$ | $\ell_{2}$
$\ell_{1}\geq\ell_{2}$ and $\beta^{-}\geq 1$ | $\ell_{1}+\gamma$ | $\ell_{2}$
$\ell_{1}<\ell_{2}$ and $\beta^{+}<1$ | $\ell_{1}$ | $1/k_{1}$
$\ell_{1}<\ell_{2}$ and $\beta^{+}\geq 1$ | $\ell_{1}$ | $\ell_{2}+\gamma$
(b)

Parameters of $f$ | $\tilde{\ell}_{1}$ | $\tilde{\ell}_{2}$
---|---|---
$\beta\in[0,1)$ | $1/k_{2}$ | $1/k_{1}$
$\beta^{+}<1\leq\beta^{-}$ | $\ell_{1}$ | $\beta^{-}/k_{1}$
$\beta^{-}<1\leq\beta^{+}$ | $\beta^{+}/k_{2}$ | $\ell_{2}$
$\beta^{+},\beta^{-}\geq 1$ and $k_{1}\geq k_{2}$ | $\beta^{+}/k_{2}$ | $\ell_{2}$
$\beta^{+},\beta^{-}\geq 1$ and $k_{1}<k_{2}$ | $\ell_{1}$ | $\beta^{-}/k_{1}$
(c)

Table 1. Choosing $\tilde{\ell}_{1},\tilde{\ell}_{2}$ to complete the proof of
Theorem 4.

∎

## References