Natural measures and statistical properties of non-statistical maps]Natural measures and statistical properties of
non-statistical maps with multiple neutral fixed points

Douglas Coates
Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Brazil

Ian Melbourne
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
https://homepages.warwick.ac.uk/ maslaq/research

Amin Talebi
Department of Mathematical Sciences, Sharif University of Technology, Iran,

Institute for Research in Fundamental Sciences (IPM), Iran

Douglas Coates was partially supported by the São Paulo Research Foundation (FAPESP) grant 2022/16259-2.

Amin Talebi is partially supported by INSF grant no. 4001845.

In this article we show that a large class of infinite measure preserving dynamical systems that do not admit physical measures nevertheless exhibit strong statistical properties. In particular, we give sufficient conditions for existence of a distinguished natural measure \( \nu \) such that the pushforwards of any absolutely continuous probability measure converge to \( \nu \). Moreover,
we obtain a distributional limit law for empirical measures.
We also extend existing results on the characterisation of the set of almost sure limit points for empirical measures.
Our results apply to various intermittent maps with multiple neutral fixed points preserving an infinite $\sigma$-finite absolutely continuous measure.

§ INTRODUCTION

In this article, we show that a large class of infinite measure preserving dynamical systems that do not admit physical measures nevertheless exhibit strong statistical properties.
Our results apply to a class of intermittent maps <cit.> such as those considered by Thaler <cit.> and Coates et al. <cit.>. Specifically, we study maps that preserve an infinite $\sigma$-finite absolutely continuous measure and have several “equally sticky” neutral fixed points.

For the sake of clarity, we first introduce our results for Thaler maps with two neutral fixed points <cit.> before discussing more general situations.
Let $f : [0,1] \to [0,1]$ be an interval map with the following properties for some $\alpha\in(0,1]$, $b_1$, $b_2 > 0$, $c\in(0,1)$:

\( 0 , 1 \) are neutral fixed points: \( f ' (0 ) = f' (1) = 1 \);

* $f|_{(0,c)}$ and $f|_{(c,1)}$ are $C^2$ diffeomorphisms onto $(0,1)$ admitting $C^2$ extensions onto $[0,1]$;

* $f'(x) > 1$ for all $x\in (0,1)$ and
\( f \) is convex (resp. concave) on a neighbourhood of \( 0 \) (resp. \( 1 \));

$fx - x \sim b_1 x^{1 + 1/\alpha}$ as $x \to 0$ and $fx - x \sim  b_2 (1-x) ^{ 1 + 1/\alpha}$ as $x \to 1$.

By <cit.>, $f$ is conservative and ergodic with a unique (up to scaling)  invariant $\sigma$-finite absolutely continuous measure $\mu$, and moreover $\mu([0,1]) = \infty$. However, it was shown in <cit.> that for almost every $x\in[0,1]$, the sequence of empirical measures

\begin{equation} \label{eq:en}
e_n (x) \coloneqq \frac{1}{n} \sum_{ j =0 }^{ n - 1 } \delta_{ f ^j x }
= \frac{1}{n} \sum_{ j =0 }^{ n - 1 } f^j_* \delta_{  x },
\end{equation}

does not converge in the weak-$*$ topology.
Indeed, the set of limit points of $e_n(x)$ is almost surely equal to the set

\[
\cS \coloneqq \{ \nu_p \coloneqq p \delta_{0} + (1 - p )\delta_1: p\in [0,1] \}
\]

of all convex combinations of the Dirac masses at the neutral fixed points.

Here and throughout, convergence of sequences of measures is with respect to the weak-$*$ topology unless otherwise specified.

Recall that an invariant probability measure $\nu$ is a physical measure if $e_n\to \nu$ with positive probability.
Since $e_n$ diverges almost everywhere for the Thaler maps described above, there exist no physical measures and such maps are said to be non-statistical, see <cit.> and references therein.

One may ask whether convergence holds when the point mass $\delta_x$ in (<ref>) is replaced by other types of probability measure $\lambda$ and
whether there is a distinguished limit measure $\nu_{\bar p}\in \cS$.
A probability measure $\nu$ is called natural <cit.> if there exists an absolutely continuous probability measure $\lambda$ such that $\lim_{n\to\infty}\frac1n\sum_{j=0}^{n-1}f_*^j\lambda=\nu$.
(By the dominated convergence theorem, physical measures are natural.)

Our first result shows that there exists a unique natural measure $\nu_{\bar p}\in\cS$.
Moreover, $\nu_{\bar p}$ attracts all absolutely continuous probability measures $\lambda$ and it is not necessary to take Cesáro averages.
(See <cit.> for similar results in the simpler symmetric setting with $a_1=a_2$, where $\bar p=\frac12$. See also <cit.> for results on nonexistence of natural measures when $f$ is badly behaved near the neutral fixed points.)

There exists a $\bar p\in[0,1]$, with corresponding measure $\nu_{\bar{p}} \in \cS$,
such that

\lim_{n\to\infty} f_{*}^n \lambda =  \nu_{ \bar{p} }
\quad\text{for every absolutely continuous probability measure $\lambda$.}

A formula for $\bar p$ is given in Section <ref>, namely $\bar p=c_1/(c_1+c_2)$ where $c_1,c_2$ are as in (<ref>).

An almost immediate consequence of Theorem <ref>
is the following decay of correlations type result.

Suppose that $\varphi:[0,1]\to\R$ is bounded, measurable, and
continuous at $0$ and $1$.

\lim_{n\to\infty} \int \psi \cdot \varphi \circ f^n \, d\Leb = \int \psi \, d\Leb \int \varphi \, d \nu_{\bar{p}}\quad
\text{for all $\psi \in L^1 (\Leb)$. }

\lim_{n\to\infty} \int \psi \cdot \varphi \circ f^n \, d \mu = \int \psi \, d \mu \int \varphi \, d \nu_{\bar{p}}$
for all $\psi \in L^1 ( \mu )$.

We now return to consideration of the sequence $e_n$ of empirical measures.
Since $e_n$ fails to converge pointwise, it is natural to consider alternative modes of convergence such as
distributional convergence.

To describe the distributional convergence of $e_n$, we recall the classical arcsine law for occupation times of <cit.> as adapted by <cit.> and then by <cit.> to the current context.
Define the sequence of occupation times

S_n \coloneqq \sum_{ j = 0 }^{ n - 1 } 1_{ B} \circ f^j,

where $B\subset [0,1)$ is a closed interval containing $0$.
Also, we define the $[0,1]$-valued random variable $Z_{\alpha, p}$ for $\alpha\in(0,1]$, $p\in[0,1]$, as follows:
When $\alpha=1$, we set $Z_{1,p}\equiv p$. When $\alpha\in(0,1)$, let $Z_{\alpha, p}$ be the random variable with continuous

\begin{equation}
\label{eq:arcsine-density}
\frac{ \hat{ p } \sin \alpha \pi }{ \pi } \frac{ t^{ \alpha }(1 - t)^{ \alpha - 1 }+t^{ \alpha - 1  }(1 - t)^{ \alpha } }{ \hat{ p }^{ 2 }t^{ 2 \alpha  } + 2\hat{ p } t^{ \alpha }(1 - t)^{  \alpha  } \cos \alpha \pi + (1 - t)^{ 2  \alpha } },
\end{equation}

where $\hat p= p^{-1}-1$.
Thaler <cit.> proved a distributional limit law for occupation times

\[
\tfrac1n S_n \text{ converges strongly in distribution to }  Z_{\alpha,\bar p}.
\]

When $\alpha\in(0,1)$, this
means that $\lambda(\frac1n S_n\in I)\to \PP(Z_{\alpha,\bar p}\in I)$ for all absolutely continuous probability measures $\lambda$ on $[0,1]$ and all open intervals $I\subset[0,1]$.
(When $\alpha=1$, we exclude intervals $I$ with an endpoint at $\bar p$.)
Equivalently, \( \frac1n S_{n*} \lambda \to \omega \)
where $\omega$ is the distribution $\omega(I)=\PP(Z_{\alpha,\bar p}\in I)$ of $Z_{\alpha,\bar p}$.

To each $[0,1]$-valued random variable $Z$, we can associate the $\cS$-valued random variable $\nu_Z=Z\delta_0+(1-Z)\delta_1$.

\[
e_n \text{ converges strongly in distribution to }  \nu_{Z_{\alpha,\bar p}} .
\]

As mentioned before, the set of accumulation points of $e_n(x)$ is precisely the set $\cS$. The result above describes how the $e_n(x)$ are asymptotically distributed on $\cS$, see Figure <ref>.

(A) at (0,0,0);
(B) at (2,0,0);
(C) at (1,sqrt(3),0);

[thin, fill=orange!30] (A) – (B) – (C) – cycle;

(D) at (0.5, sqrt(3)/4, 0);
(E) at (1.5, sqrt(3)/4, 0);
[thick, black] (D) – (E);

įn 1,...,41
0.5 + (-̨1) * (1 / 40)
1 / sqrt(15 * (/̨ 42 * (1 - /̨ 42)))

(P)̨ at (, , 0);
(Q)̨ at (, , );
[thin, gray] (P)̨ – (Q)̨;

ȷin 1,...,600
rand * 0.05 - 0.02
[purple] (1/2*+ 1, + 0.433012, 0) circle (0.05pt);

at (D) [left] $\delta_0$;
at (E) [below] $\delta_1$;
at (0.9, 0.8/2, 0) [below] $\cS$;
at (0.8,-.1, 0) [below] $\cM_1(X)$;

A schematic picture for the distributional limit law for empirical measures. The small dots around the set $\cS$ are empirical measures $e_n(x)$  of an ensemble of points in the phase space chosen randomly (w.r.t. an absolutely continuous probability measure) for some fixed large $n$. As $n$ increases, each individual dot oscillates between $\delta_0$ and $\delta_1$ (<cit.>), but in such a way as to make together an asymptotic distribution on $\cS$ (Theorem <ref>).

Let $\cM_1(M)$ denote the space of probability measures on a measure space $M$.
The sequence $e_n:[0,1]\to\cM_1([0,1])$ induces the pushforward sequence
$e_{n*}:\cM_1([0,1])\to\cM_1(\cM_1([0,1]))$, so, the theorem above is equivalent to \( \lim_{n\to\infty} e_{n*} \lambda = \omega \) for every absolutely continuous probability measure $\lambda$
where $\omega$ is the distribution of $\nu_{Z_{\alpha,\bar p}}$.

Since $(e_{n*}\lambda)(\cE)=\lambda(e_n\in\cE)$, we can view Theorem <ref> as a statement about the limit of
$e_{n*}\lambda$ for all absolutely continuous measures $\lambda$.

Since $e_{n*}\lambda$ is a probability measure on the convex set $\cM_1([0,1])$, we can consider its expectation $\E (e_{n*}\lambda) =\int_{\cM_1([0,1])}\omega \, de_{n*}\lambda \in \cM_1([0,1])$, which can be interpreted as a probability measure on $[0,1]$ satisfying $\E(e_{n*}\lambda) ( E ) = \int \omega(E) \, de_{n*}\lambda(\omega)$.

We have the following consequence of Theorem <ref>:

Let $\lambda$ be an absolutely continuous probability measure on $[0,1]$.

$\frac{1}{n} \sum_{ j = 0 }^{ n - 1 } f^j_*\lambda= \int e_n  \, d \lambda  =
\E(e_{n*}\lambda)$ for all $n\ge1$.

(b) The common limit of the sequences of measures in (a) is
given by
$\lim_{ n \to \infty } \E(e_{n*}\lambda)=  \nu_{ \bar{p} }$.

§.§.§ Beyond two neutral fixed points

Our results generalise to any finite number of neutral fixed points $\xi_1,\dots,\xi_d$.
The proof of Theorem <ref> requires no modifications.
The treatment of the arcsine law in <cit.> assumes $d=2$ but this restriction is removed by Sera & Yano <cit.> using a different method. We use <cit.> to generalise Theorem <ref>.
The generalisations of Theorem <ref> and <ref> are stated in Section <ref> as Theorems <ref> and <ref> respectively.

The final main result in this paper is Theorem <ref> which shows that for general $d\ge1$ and $\alpha\in(0,1)$, the set of limit points of the sequence of empirical measures $e_n$ is almost surely equal to the $(d-1)$-dimensional simplex

\[
\cS \coloneqq \{ p_1\delta_{\xi_1}+\dots+ p_d\delta_{\xi_d}:p_1,\dots,p_d\ge0,\;p_1+\dots + p_d=1\}.
\]

Previous arguments in <cit.> exploited connectedness of the set of limit points and were effective for $d\le 2$, $\alpha\in(0,1]$. Our method uses instead the full support of the arcsine law in <cit.> combined with Theorem <ref>. However, this full support property fails for $\alpha=1$ which means that Theorem <ref> remains open for $d\ge3$, $\alpha=1$.

The remainder of the paper is organised as follows.
In Section <ref>, we present an abstract setup which allows for any finite number of neutral fixed points, and we state our main results
Theorem <ref>, <ref> and <ref>.
Also, we state and prove Corollaries <ref> and <ref> which generalise Corollaries <ref> and <ref> respectively.
Theorems <ref>, <ref> and <ref>
are proved in Section <ref>.
In Section <ref>, we verify that our abstract setup includes the classes of intermittent maps in <cit.>.

We use “big O” and $\lesssim$ notation interchangeably, writing $a_n=O(b_n)$ or $a_n\lesssim b_n$ if there are constants $C>0$, $n_0\ge1$ such that $a_n\le Cb_n$ for all $n\ge n_0$. As usual, $a_n=o(b_n)$ means that $a_n/b_n\to0$ and $a_n\sim b_n$ means that $a_n/b_n\to1$.

§ ABSTRACT STATEMENT OF RESULTS

§.§ Maps with Gibbs-Markov first return maps

In this section we recall some definitions and basic properties of maps with Gibbs-Markov first return maps needed to state our main assumptions.

Let $X$ be a compact metric space with Borel probability measure $m$ and
let $f:X\to X$ be a nonsingular measurable map. We suppose that $f$ is ergodic and conservative.
(A standard reference for this material is <cit.>.)

We say that the map $f$ together with a countable partition $\cP$ of $X$ into positive measure subsets is a Markov map if the restriction of $f$ to each partition element is a measurable bijection onto a union of partition elements. The map $f$ is topologically mixing if for every $a,b \in \cP$ there exists an $N\ge1$ such that $f^n a \cap b \neq \emptyset$ for every $n \geq N$.

$Y \subset X$ be a union of partition elements. We define the first return map $F: Y \to Y$ by setting $F(y) \coloneqq f^{ \tau (y) }y$ where $\tau (y) \coloneqq \inf \{ n \geq 1 : f^ny \in Y \}$.
The map $F$ is Gibbs-Markov if there exists a refinement $\cP_Y$ of the partition $\cP$ for which $F$ is again a Markov map and the following additional properties hold:

* finite images: $\Card \{ F a : a \in \cP_{ Y } \} < \infty$;

* bounded distortion: there exists $\theta \in (0,1)$ and $C > 0$ such that the function $\log \frac{ d m } { dm \circ F }$ is $d_\theta$-Lipschitz on elements of $\cP_Y$
where $d_{\theta}$ is the metric $d_{\theta} (x,y) \coloneqq \theta^{s(x,y)}$ and
$s(x,y) \coloneqq \inf \{ n {\geq 0} : F^nx, F^ny \text{ lie in different elements of }\cP_Y \}$.

We denote by $\cB_{\theta}(Y)$ the space of real-valued functions which are Lipschitz continuous on $Y$ with respect to the metric $d_{\theta}$. We recall the following classical result about the existence of invariant measures for maps with Gibbs-Markov first return maps.

Suppose that $f : X \to X $ is a conservative ergodic Markov map with Gibbs-Markov first return map $F=f^\tau : Y \to Y $. Then $f$ preserves a unique (up to scaling) absolutely continuous $\sigma$-finite measure $\mu$, and
$\mu ( X ) < \infty$ if and only if $\tau\in L^1(m)$.

Moreover, $\mu(Y)<\infty$ and the density $h=d\mu/dm$ satisfies $h1_Y,\,h^{-1}1_Y\in\cB_\theta(Y)$.

Standard references include <cit.> and <cit.>.

§.§ Assumptions and notations

Let $(X,m)$ be a metric space with Borel probability measure $m$.
Throughout, we suppose that $f:X\to X$ is
is a conservative ergodic Markov map
with Gibbs-Markov first return map
$F=f^\tau:Y\to Y$, and
moreover that $f$ and $F$ are topologically mixing.
Let $\mu$ be the $f$-invariant absolutely continuous $\sigma$-finite measure in Lemma <ref>.
We assume that $\mu(X)=\infty$.

There exists fixed points $\xi_1,\dots,\xi_d \in X$
such that

\[
\mu\big(
X\setminus \{B_{ \eps} (\xi_1) \cup \dots \cup  B_{\eps} ( \xi_d )\}
\big)<\infty
\quad\text{ for all $\eps>0$ sufficiently small.}
\]

This assumption implies that neighbourhoods of the fixed points $\xi_1, \dots, \xi_d$ carry the infinite part of the mass.

The next assumption ensures that the inducing set $Y$ dynamically separates $\xi_1,\dots,\xi_d$ and that the excursion times from \( Y \) to neighbourhoods of the fixed points have certain tail distributions.

There exist constants $\alpha\in(0,1]$, \( c_1,\dots,c_d>0 \)
and a measurable partition \( X_1,\dots,X_d \) of \( X \setminus Y \)
such that

$\xi_k \in \Int X_k$, for $k = 1,\dots, d$;

For \( k\neq \ell \), orbits cannot pass from $X_k$ to $X_{\ell}$ without first entering $Y$.

\[
\{\tau=n\} = \bigcup_{k=1,\dots,d} \{ \tau^{ (k) } = n - 1 \}
\quad\text{for $1\le n<\infty$}
\]

where $\tau^{(k)} (y) \coloneqq \Card \{ n \leq \tau (y) : f^n y \in X_k \} $, for $y\in Y$, $k = 1, \dots, d$;

$\mu ( \tau^{(k)} > n ) \sim c_k n^{-\alpha}$ as $n\to\infty$ for $k=1,\dots,d$.

Note that

\begin{equation} \label{eq:tails}
\mu ( \tau > n ) \sim c_{\tau} n^{-\alpha}
\quad\text{where}\quad
c_{\tau} \coloneqq c_1+\dots+c_d>0.
\end{equation}

We expect that the results in this paper remain valid when the constants
\( c_{ k } \) in <ref>(c) are replaced by $c_k\ell(n)$ where $\ell(n)$ is slowly varying. However, the proof of Theorem <ref> requires further calculations.

Our third assumption is somewhat technical to state but is very mild.
Let $\cP_r$ denote the $r$'th refinement of the partition $\cP$ under $f^r$ and
let $\cP_r^*$ consist of $r+1$ cylinders $a=[a_0,\dots,a_r]\in\cP_r$ where
$a_0,\dots,a_{r-1}\not\subset Y$ and $a_r\subset Y$.
Given $\rho\in L^1(X,m)$ and $r\ge0$, define

\begin{equation} \label{eq:Q}
Q_r^\rho:Y\to\R,  \qquad
Q_r^\rho \coloneqq \sum_{a\in\cP_r^*} 1_{f^ra} \cdot \frac{\rho}{Jf^r}\circ f^r|_a^{-1}
\end{equation}

where $J\! f^r \coloneqq \frac{ d m \circ f^{ r } }{ dm }$ denotes the Jacobian of $f^r$ with respect to $m$.

There is a dense subset $K(X)\subset L^1(X,m)$ such that
$Q_r^\rho\in \cB_\theta(Y)$ for all $\rho\in K(X)$ and $r\ge0$.

Assumption <ref> is automatic if $f$ has finitely many branches and there exists $\eta>0$ such that $f$ is $C^{1+\eta}$ on $f^{-r}Y$ for each fixed $r\ge0$, which is the case in the applications considered in this paper. For such maps, we can take $K(X)=C^\eta(X)$.

Our final assumption is the following “smooth tail” estimate:

H4 If \( \alpha \in (0, 1 / 2] \), then \( \mu ( \tau = n ) = O ( n ^{ - ( 1+ \alpha  ) }) \).

§.§ Almost sure behaviour of the empirical measures

We let $\cM_1(X)$ denote the space of Borel probability measures on $X$. Define the map
$e_n : X \to \cM_1 ( X )$
which maps each point $x\in X$ to its $n^{th}$ empirical measure $e_n(x)$ as given in (<ref>).

Our first result concerns the almost sure behaviour of the empirical measures $e_n$. For $x \in X$, we let $\cL(x) \subset \cM_1(X)$ denote the set of weak-$*$ accumulation points of $e_n(x)$. Let $\cS_0$ be the simplex

\[
\cS_0 \coloneqq \{ p \in [0,1]^d : p_1 + \dots + p_d  = 1 \}.
\]

For $p \in \cS_0$, let $\nu_{p} \coloneqq p_1 \delta_{ \xi_1 }  + \dots +  p_d \delta_{ \xi_d }$ be the corresponding convex combination of Dirac masses at the fixed points, and define

\cS \coloneqq \{ \nu_{p} : p \in \cS_0 \},

to be the set of all such convex combinations.
With this notation in place we can state our first result.

Suppose that <ref>–<ref> hold and that $\alpha \in (0,1)$. Then

\cL (x) = \cS
\quad\text{for a.e.\ }x\in X.

In particular, when $d\ge2$ there are no physical measures.

The inclusion $\cL (x) \subset \cS$ is elementary, see Lemma <ref> below, so it is the reverse inclusion that is of interest.
The case $d=1$ is trivial, and the case $d=2$ is well-understood.
Indeed, there are techniques <cit.> for showing that $\delta_{\xi_k}\in\cL(x)$ a.e. for each $k$ and Theorem <ref> then follows for $d=2$ by connectedness of $\cL(x)$. This approach includes the case $\alpha=1$.

For $d\ge3$, Theorem <ref> is completely new to the best of our knowledge.
Our proof involves a different technique which works for all $d$ but only in the range $\alpha\in(0,1)$.
We expect that Theorem <ref> holds also for $d\ge3$, $\alpha=1$, but new ideas seem to be required.

§.§ Distributional convergence of the empirical measures

Our second result concerns strong distributional convergence of the sequence $e_n$ of empirical measures.

Suppose that $Z_n$ is a sequence of measurable functions on $X$ taking values in some Borel space $M$ and that $Z$ is a random variable with distribution $\omega \in \cM_1 ( M)$ also taking values in $M$ (but not necessarily defined on $X$).
Given $\lambda\in\cM_1(X)$,
we write $Z_n \to_\lambda Z$ if
$\lim_{ n \to \infty }Z_{n*} \lambda = \omega$.

Now let $\cM^{\mathrm{ac}}_1(X)$ denote the set of Borel probability measures on $X$ which are absolutely continuous with respect to $m$.
We say that
$Z_n$ converges strongly in distribution to $Z$, and write
$Z_n\to_dZ$, if $Z_n\to_\lambda Z$
for all  $\lambda \in \cM^{\mathrm{ac}}_1 (X)$.

Fix $\eps$ so that the neighbourhoods $B_{\eps}(\xi_k), \;k=1,\dots,d$ are disjoint.
As was the case for the Thaler maps in Theorem <ref>,
we first consider strong distributional convergence for occupation times
$S_n=(S_n^1,\dots,S_n^d):X\to[0,1]^d$ defined by

\begin{equation} \label{eq:Sn}
S_n^k \coloneqq \sum_{ j = 0 }^{ n - 1 } 1_{ B_{\eps}(\xi_k) } \circ f^j, \quad \text{for } k = 1, \dots, d, \; n\ge1.
\end{equation}

Following <cit.>, we consider a multiray generalisation of the classical arcsine law.
For \( \alpha \in (0,1) \), $p\in\cS_0$, let \( \zeta_{1},\dots, \zeta_{d} \) be independent \( [0,\infty) \)-valued random variables defined on a common probability space with one-sided \( \alpha \)-stable distribution characterised for $k=1,\dots,d$ by

\begin{equation}\label{eq:def-zeta}
\E \exp( - t \zeta_k ) = \exp ( -t^{\alpha} p_k ), \quad t > 0 .
\end{equation}

Define the $\cS_0$-valued random variable

\begin{equation}
\label{eq:def-Z}
Z_{\alpha, p} \coloneqq \frac{1}{\zeta_1+\dots+\zeta_d} ( \zeta_1, \dots, \zeta_d).
\end{equation}

When $\alpha=1$, we define $Z_{1, p}\equiv  p$.

In the special case that \( d = 2 \), one has that \( Z_{\alpha, p}^{(2)} = 1 - Z_{\alpha, p}^{(1)} \) and for $\alpha\in(0,1)$ the distribution of \( Z_{\alpha, p}^{(1)} \) admits the continuous density in (<ref>)
(identifying $ p$ with $ p_1$).

The first moment of the multiray arcsine law
$\E Z_{\alpha,p}$ is given by $\E Z_{\alpha,p}=p$.
This can be verified using the
double Laplace formula <cit.>,

\[
\int_0^\infty e^{-qt} \E\Big( \exp\Big\{-t\sum_{k=1}^d \lambda_k\zeta_k\Big\}\Big)\,dt
=\frac{\sum_{k=1}^d p_k(q+\lambda_k)^{\alpha-1}}
{\sum_{k=1}^d p_k(q+\lambda_k)^{\alpha}}, \quad
\lambda=(\lambda_1,\dots,\lambda_d)\in[0,\infty)^d,\; q>0,
\]

as derived in <cit.>.
Differentiating w.r.t. $\lambda_k$ and setting $\lambda=0$ yields $\E Z_{\alpha,p}^{(k)}=p_k$.

We can now recall the arcsine law of <cit.>.
Recall from (<ref>) that $c_\tau=\sum_{k=1}^d c_k$.

\[
\bar{p} = ( \bar p_1,\dots, \bar p_d)
\coloneqq c_\tau^{-1}( c_1 , \dots,  c_d) \in\cS_0.
\]

Suppose that <ref>–<ref> hold. Then
\frac1n S_n \to_d Z_{\alpha,\bar p}.

It suffices to check that <ref>–<ref> imply <cit.> as the result then follows from <cit.>.
Let \( F:Y\to Y \) be the Gibbs-Markov first return map in  <ref>. Notice that <ref>(b) yields <cit.>, and then <ref>(c) together with <cit.> gives <cit.>.
Finally, as \( F \) is topologically mixing it is exponentially continued fraction mixing (see for example <cit.>) which immediately implies <cit.>.

To each $\cS_0$-valued random variable $Z$, we can associate the $\cS$-valued random variable $\nu_Z=Z_1\delta_{\xi_1}+\dots+Z_d\delta_{\xi_d}$.
Notice that
$\E\nu_Z=\sum_{k=1}^d \E Z_k\delta_{\xi_k}
=\sum_{k=1}^d (\E Z)_k\delta_{\xi_k}
=\nu_{\E Z}$.
In particular, $\E \nu_{Z_{\alpha,p}}=\nu_p$.

Suppose that <ref>–<ref> hold. Then $ e_n \to_d \nu_{Z_{\alpha,\bar p}}$.
\(
\lim_{ n \to \infty } e_{n*} \lambda = \omega
\)
for all \( \lambda \in \cM_1^{ \mathrm{ac} } ( X )\)
where $\omega\in\cM_1(\cS)$ is the distribution of $\nu_{Z_{\alpha,\bar p}}$.

As in the introduction, we obtain the following consequence:

Let $\lambda\in \cM_1^{ \mathrm{ac} } ( X )$.

$\frac{1}{n} \sum_{ j = 0 }^{ n - 1 } f^j_*\lambda= \int e_n  \, d \lambda  =
\E(e_{n*}\lambda)$ for all $n\ge1$.

(b) The common limit of the sequences of measures in (a) is
given by
$\lim_{ n \to \infty } \E(e_{n*}\lambda)=  \nu_{ \bar{p} }$.

Let $i:\cM_1(X)\to \cM_1(X)$ be the identity map.

\[
\E(e_{n*}\lambda)=\int_{\cM_1(X)}\omega\,d (e_{n*}\lambda)(\omega)
=\int_{\cM_1(X)}i(\omega)\,d (e_{n*}\lambda)(\omega)
=\int_X i\circ e_n(x)\,d \lambda(x)
=\int_X e_n\,d \lambda.
\]

\[
\bigg(\int_X \delta_{f^jx}\,d\lambda(x)\bigg)(E)
\int_X \delta_x(E)\,df_*^j\lambda(x)
\int_X 1_E \, df_*^j\lambda=(f_*^j\lambda)(E),
\]

$\int_X \delta_{f^jx} \,d\lambda(x)=f^j_*\lambda$.

\[
\int_X e_n  \, d \lambda= \frac{1}{n} \sum_{ j = 0 }^{ n - 1 } \int_X \delta_{f^jx} \,d\lambda(x).
=\frac{1}{n} \sum_{ j = 0 }^{ n - 1 } f^j_*\lambda.
\]

Since $\E$ acts continuously from $\cM_1(\cM_1(X))$ to $\cM_1(X)$, it follows from
Theorem <ref> that
$\lim_{ n \to \infty } \E(e_{n*}\lambda)= \E \nu_{Z_{\alpha,\bar p}} = \nu_{ \bar{p} }.$

§.§ Existence and uniqueness of natural measures

Our third result concerns weak-$*$ convergence of pushforwards $f^n_*\lambda$
of absolutely continuous probability measures $\lambda\in\cM_1^{\mathrm ac}(X)$.

Suppose that <ref>–<ref> hold with $\alpha\in(0,1]$.

\lim_{ n \to \infty} f_*^n \lambda = \nu_{\bar{p}}
\quad\text{for all }\lambda\in \cM_1^{\mathrm ac}(X).

As in the introduction, we obtain the following result on decay of correlations:

Suppose $\varphi:X\to\R$ is bounded, measurable, and continuous at each $\xi_1,\dots, \xi_d$.

\lim_{n\to\infty}	\int \psi \cdot \varphi \circ f^n \, d m = \int \psi \, d m \int \varphi \, d \nu_{\bar{p}}
\quad\text{for all } \psi \in L^1 ( m  ).

\lim_{n \to \infty } \int  \psi \cdot \varphi \circ f^n \, d \mu = \int  \psi \, d \mu \int \varphi \, d \nu_{ \bar{p} }$
for all  $\psi \in L^1 ( \mu  ).

We begin by proving that the first limit holds.

By Theorem <ref>, $f_*^n\lambda\to\nu_{\bar p}$ weak-$^*$, so by definition
$\int\varphi\,df_*^n\lambda\to\int\varphi\,d\nu_{\bar p}$ for all continuous $\varphi:X\to\R$.
By <cit.>,
$\int\varphi\,df_*^n\lambda\to\int\varphi\,d\nu_{\bar p}$ for
all bounded measurable functions
${\varphi:X\to\R}$ that are continuous except on a set of $\nu_{\bar p}$-measure zero, which is precisely the class of observables in the statement of the corollary.

Fix such a $\varphi$.
Let $\psi\in L^1(m)$ with $\int\psi\,dm\neq0$ and write $\psi=\rho\int\psi\,dm$ where $\rho\in L^1(m)$ and $\int\rho\,dm=1$.
Let $d\lambda=\rho\,dm$.

(\textstyle{\int} \psi \, dm )^{-1}
\int \psi \cdot \varphi \circ f^n \, dm =
\int \rho \cdot \varphi \circ f^n \, dm =
\int \varphi \circ f^n \, d\lambda =
\int \varphi \, df_*^n\lambda \to \int\varphi\,d\nu_{\bar p}.

The case $\int\psi\,dm=0$ is dealt with by approximating, concluding the proof of the first limit.

If $\psi\in L^1(\mu)$, then $\psi h\in L^1(m)$ and the second limit follows from the first. Similarly, the first limit follows from the second.

§ PROOFS

In this section, we prove the three main theorems in this paper.
It turns out to be convenient to prove them in the order B, A, C.

§.§ Ergodic theorem

In this subsection, we recall the ergodic theorem for infinite measure systems and derive a consequence for the systems studied in this paper.

Suppose that <ref> holds.
If $\varphi \in L^1 ( \mu )$, then
$\lim_{n\to\infty} \frac{1}{n} \sum_{ j =0 }^{ n - 1 } \varphi \circ f^j  = 0$ almost everywhere.

This is an easy consequence of the Hopf ratio ergodic theorem, see for example <cit.>.

Suppose that <ref> holds.
Then $\cL (x) \subset \cS$ for almost every $x\in X$.

We provide the details for completeness (cf. <cit.>).

For $\eps>0$, define  $X^{(\eps)} \coloneqq X\setminus \{B_{ \eps} (\xi_1) \cup \dots \cup  B_{\eps} ( \xi_d )\}$.
By <ref>, $\mu(X^{(\eps)})<\infty$ for all ${\eps>0}$.
Choose $\psi:X\to[0,1]$ continuous and supported in $X^{(\eps/2)}$ such that $\psi|X^{(\eps)}\equiv1$.
In particular, $\int_X\psi\,d\mu<\infty$ so, by Lemma <ref>, there exists $X'\subset X$ with $\mu(X\setminus X')=0$ such that $\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\psi(f^jx)=0$
for all $x\in X'$.

Suppose that $x\in X'$ and $\omega\in \cL(x)$, and choose a subsequence $n_i$ such that $e_{n_i}(x)\to \omega$. Then $\frac{1}{n_i}\sum_{j=0}^{n_i-1}\psi(f^jx)\to\int_X\psi\,d\omega$,
Hence, elements of $\cL(x)$ are supported in
$B_\eps(x_1)\cup\dots\cup B_\eps(x_d)$ for all $x\in X'$.
Since $\eps>0$ can be chosen arbitrarily small, the result follows.

Suppose that $e_{n_k}\to_m Z$ for some $\cM_1(X)$-valued random variable $Z$ and some subsequence $n_k$. Then $Z$ takes values in $\cS$.

Let $\psi:\cM_1(X)\to\R$ be continuous and supported in $\cM_1(X)\setminus \cS$.  By the dominated convergence theorem,

\int_{X} \psi(e_{n_k} (x) )\,dm (x)
\to \E \, \psi(Z)

But $\psi(e_{n_k}) \to 0$ a.e. by Lemma <ref>. Applying the dominated convergence theorem once more, $\E\, \psi(Z)=0$, and the result follows.

Corollary <ref> shows that the only possible distributional limit points of the sequence $e_n$ are random variables of the form $\nu_Z$  where $Z$ is an $\cS_0$-valued random variable.

§.§ Proof of Theorem <ref>

Fix $\eps>0$ such that $B_\eps(\xi_k)$ are disjoint for $k=1,\dots,d$ and
define $S_n:X\to[0,1]^d$ as in (<ref>).

Let $Z$ be a random variable with values in $\cS_0$.
$e_n\to_m \nu_Z$ if and only if $n^{-1}S_n\to_m Z$.

\[
\pi : M_1(X)\to [0,1]^d, \qquad
\pi(\omega) \coloneqq
(\omega(B_{\eps}(\xi_1)),\dots, \omega(B_{\eps}(\xi_d)).
\]

Note that $\pi$ restricts to the natural identification $\nu_Z\mapsto Z$  between $\cS$ and $\cS_0$. Also, $\pi$ is continuous at elements in $\cS$ and satisfies $\pi ( e_n ) = \frac1n S_n$.

In particular, if $e_{n_k}\to_m \nu_Z$ for some subsequence $n_k$, then it is an immediate consequence of the continuity of $\pi$ at $\nu_Z$ and the continuous mapping theorem that
$\frac1n S_{n_k}=\pi (e_{n_k}) \to_m \pi(\nu_Z)=Z$
completing the proof in one direction.

The converse follows by a standard probabilistic argument. Suppose that $\frac1n S_n\to_m Z$. Let $\hZ$ be a distributional limit point for $e_n$, so $e_{n_k}\to_m \hZ$ for some subsequence $n_k$. By Remark <ref>, $\hZ=\nu_A$ for some random variable $A$ with values in $\cS_0$. By what we just proved, $\frac1n S_{n_k}\to_m A$. Hence, $A=Z$, so $\hZ=\nu_Z$ is the unique distributional limit point for $e_n$. This means that $e_n\to_m \nu_Z$.

By Theorem <ref>, $n^{-1}S_n\to_m Z_{\alpha,\bar p}$.
Hence by Lemma <ref>,
$e_n \to_m \nu_{Z_{\alpha,\bar p}}$.

Let \(d_W\) denote the Wasserstein distance on \(\cM_1(X)\),

\[
d_W ( \omega, \omega' ) \coloneqq
\sup_{ \varphi \in \Lip_1 } \Big| \int \varphi \, d\omega -
\int \varphi \, d\omega' \Big|,
\]

where $\Lip_1 = \{ \varphi : X \to \R : \Lip \varphi+\|\varphi\|_\infty \le1 \}$  and $\Lip \varphi$ denotes the smallest Lipschitz constant of \(\varphi:X\to\R\). Recall that \(d_W\) induces the weak-$*$ topology on \(\cM_1 ( X)\).
$ d_W ( e_n \circ f, e_n )\le \frac2n$, so
the functions $e_n : X \to \cM_1(X)$ satisfy the “asymptotic invariance” condition
$ d_W ( e_n \circ f, e_n )\to_m0$.
Hence, we may apply <cit.> to deduce from
$e_n\to_m \nu_{Z_{\alpha,\bar p}}$ that
$e_n\to_d \nu_{Z_{\alpha,\bar p}}$.

§.§ Proof of Theorem <ref>

We first show that the limiting random variable \( \nu_{Z_{\alpha,\bar p}} \) in Theorem <ref> has full support in \( \cS \) when \( \alpha \in (0,1) \).

Let \( \alpha \in (0,1) \), $p\in\cS_0$ with $p_i>0$ for all $i$. Then
\(
\PP(\nu_{Z_{\alpha, p}}\in B_{ \eps }( \nu ) )>0\)
for all  \(\eps > 0\), \(\nu \in \cS.  \)

Let \( \zeta_1,\dots,\zeta_d \) be the independent,  \( \alpha \)-stable random variables that appear in the definition (<ref>) of \( Z_{\alpha, p} \).
Recall <cit.> that as the \( \zeta_k \) are non-negative \( \alpha \)-stable random variables with Laplace transforms given by (<ref>), their distributions are fully supported and continuous on the half line \( [0, \infty) \).

For every \( q \in \cS_0\), \(\eps > 0 \) the set

\[
U \coloneqq \left\{ x \in [0,\infty)^d :
\frac{x_k}{x_1+\dots + x_d} \in (q_k -\eps, q_k + \eps ), \text{ for all } k=1,\dots, d\right\}
\]

contains a nonempty open rectangle \( \prod_{k=1}^d (a_k, b_k) \). By definition of \( Z_{\alpha, p} \) and the independence of the \( \zeta_k \),

\begin{align*}
\PP ( Z_{\alpha, p} \in B_{\eps} (q) )
&= \PP ( (\zeta_1, \dots, \zeta_d) \in U ) \\
&\geq \PP ( \zeta_k \in (a_k,b_k) : k = 1,\dots, d )
= \prod_{k=1}^d \PP ( \zeta_k \in (a_k,b_k) ) > 0.
\end{align*}

It follows that $\PP(\nu_{Z_{\alpha,p}}\in B_\eps(\nu_q))>0$.

By Lemma <ref>, \( \cL (x) \subset \cS \) for almost every \( x \). To prove the converse, let \( \nu \in \cS \) and consider the function

\[
\varphi (x) \coloneqq \liminf_{ n\to\infty } d ( e_n (x), \nu ),
\]

where \( d \) is any metric metrising the weak-\( * \) topology.
We will show that \( \varphi = 0  \) almost everywhere and so \( \nu \in \cL (x) \) for almost every \( x \).
The result then follows since \( \nu \in \cS \) is arbitrary.

Notice that \( \varphi \) is invariant for \( f \) and so, by ergodicity, must be almost everywhere equal to some constant \( c \geq 0 \).
Suppose for contradiction that \( c > 0 \), and let \( 0 < \eps < c \). It then follows that  \( m \{ x : e_n (x) \in B_{\eps} ( \nu ) \text{ for infinitely many $n$} \} = 0 \).

By Theorem <ref>, $e_n\to_m \nu_{Z_{\alpha,\bar p}}$ with
$\bar p_i>0$ for all $i$.
By the Portmanteau lemma together with Lemma <ref>,

\[
\liminf_{ n \to \infty }  m ( e_n \in B_{\eps} ( \nu ) )
\geq \PP(\nu_{Z_{\alpha,\bar p}}\in B_{\eps} ( \nu ))>0.
\]

\begin{align*}
m \{ x : e_n (x) \in B_{\eps} ( \nu ) \text{ for infinitely many $n$} \}
&= m \Big( \bigcap_{ n = 1 } \bigcup_{ \ell \geq n } \{ e_\ell \in B_{\eps} ( \nu ) \} \Big) \\
&= \lim_{n\to\infty} m \Big( \bigcup_{ \ell \geq n } \{ e_\ell \in B_{\eps} ( \nu ) \} \Big) \\
&\geq \liminf_{ n \to \infty }  m ( e_n \in B_{\eps} ( \nu ) )
> 0,
\end{align*}

where we have used the fact that the sequence of sets
$A_n = \bigcup_{\ell \geq n}^{ \infty} \{ e_\ell \in B_{\eps} ( \nu ) \}$ is decreasing: $A_{n+ 1} \subset A_n$.
This contradicts our assumption that \( c > 0 \) and so \( \varphi ( x ) = 0 \) almost everywhere.

§.§ Proof of Theorem <ref>

Let $f$ satisfy <ref>–<ref>.
If $\alpha\in(0,\frac12]$, then we suppose moreover that $\mu(\tau=n)=O(n^{-(\alpha+1)})$.

We let \( L : L^1 (X, \mu ) \to L^1 (X, \mu ) \) be the transfer operator for \( f \) defined by the relation

\[
\int_X v \cdot w \circ f \, d\mu  = \int_X L  v  \cdot w
\, d\mu, \quad \text{for all } v \in L^1 (X, \mu ), \; w \in L^{\infty}(X).
\]

Define for $n\ge0$,

\[
T_n:L^1(Y,\mu|_Y)\to L^1(Y,\mu|_Y), \qquad
T_nv=1_Y L^n(1_Yv).
\]

We recall some results from <cit.> which describe the asymptotic behaviour of \( T_n \) acting on the space $\cB_\theta(Y)$ in Section <ref>.

Let \( c_{\tau}\in(0,\infty) \) be as in (<ref>).
Then, for every \( v \in \cB_{\theta}(Y) \),

c_τ logn   T_n v      →∫_Y v   dμ,

c_τ n^1 - α  T_n v      →1π sinπα∫_Y v   d μ,             α∈(0,1)

uniformly on $Y$ as $n\to\infty$.

By assumption, $F=f^\tau:Y\to Y$ is a Gibbs-Markov map and
$\mu ( \tau > n ) \sim c_\tau n^{-\alpha}$.
Also, the underlying conservative ergodic map $f:X\to X$ is topologically mixing.

In the range $\alpha\in(\frac12,1]$ we can apply <cit.>.

In the range $\alpha\in(0,\frac12]$, we use <cit.>.
The hypotheses in <cit.> are stated slightly differently than in <cit.>. In the notation of these papers, the essential differences are as follows: (i) There is a stronger assumption on $\|R_n\|$ which holds since
$\mu(\tau=n)=O(n^{-(\alpha+1)})$; (ii) There is the requirement that $R(1)$ has no eigenvalues on the unit circle besides $1$ which holds since $F$ is topologically mixing (see for example <cit.>).

\[
Y_{ 0 } \coloneqq Y; \quand
Y_r \coloneqq f^{-r}Y\setminus\{\cup_{i=0}^{r-1}f^{-i}Y\},
\quad \text{for } r \ge 1.
\]

Recall that $h=\frac{d\mu}{dm}$ denotes the density of the invariant measure $\mu$.
Let $\lambda \in \cM^{\mathrm{ac}}_1 (X)$ with density $\rho = \frac{ d \lambda }{ dm }$.
Let $K(X)$ be the dense subset of $L^1(X,m)$ in <ref>.

If \( \rho \in K(X) \),  then $L^r(\rho h^{-1}1_{Y_r}) \in\cB_\theta(Y)$ for all $r\ge0$.

Note first that $L^r(\rho h^{-1}1_{Y_r})$ is supported in $Y$ by definition of $Y_r$.
Let $M:L^1(X)\to L^1(X)$ denote the transfer operator for the reference measure $m$, so $L=h^{-1}Mh$ and $M$ has the pointwise formula

\[
Mv=\sum_{a\in\cP}1_{fa} \cdot \frac{v}{Jf}\circ f|_a^{-1}.
\]

Hence, recalling (<ref>),

\begin{align*}
L^r(\rho h^{-1}1_{Y_r}) = h^{-1}M^r(\rho 1_{Y_r})
& = h^{-1} \sum_{a\in\cP_r}1_{f^ra} \cdot
1_{Y_r}\circ f^r|_a^{-1} \cdot
\frac{\rho}{Jf^r}\circ f^r|_a^{-1}
\\ & = h^{-1} \sum_{a\in\cP_r^*}1_{f^ra} \cdot
\frac{\rho}{Jf^r}\circ f^r|_a^{-1}
\end{align*}

By <ref>,
$Q_r^\rho\in \cB_\theta(Y)$, while $h^{-1}1_Y\in\cB_\theta(Y)$ by Lemma <ref>.  Hence $h^{-1}Q_r^\rho\in \cB_\theta(Y)$.

For \( r \ge 0 \), define $\rho_r=\rho h^{-1} 1_{\bigcup_{i=0}^r f^{-i}Y}$. Notice that

\begin{equation}\label{eq:rho-r}
\rho_{ r } = \sum_{ j = 0 }^{ r } \rho h^{ -1 } 1_{ Y_{ j } }.
\end{equation}

For all \( \rho \in K(X) \) and $r\ge0$,

c_τ logn   L^n ρ_r      →∫_X ρ_r   dμ,

c_τ n^1 - α  L^n ρ_r      →1π sinπα∫_X ρ_r   d μ,             α∈(0,1)

uniformly on $Y$ as $n\to\infty$.

We give the details in the case $\alpha=1$. The case $\alpha\in(0,1)$ is similar.

For $n\ge r\ge0$, equation (<ref>) gives

1_YL^n \rho_r
=\sum_{j=0}^r 1_YL^{n}(\rho h^{-1}1_{Y_j})
=\sum_{j=0}^r T_{n-j}L^j(\rho h^{-1}1_{Y_j}).

Using again (<ref>) and
the fact that \( \int_{ X }  \rho h^{ -1 } 1 _{  Y_{ j } } \,d \mu  = \int_{ Y }  L^{ j } ( \rho h^{ -1 } 1 _{  Y_{ j } } ) \,d \mu \), we obtain that on \( Y \)

\[
\begin{split}
&c_\tau \log n\, L^n \rho_r-\int_X \rho_r\,d\mu \\
= \sum_{j=0}^r \frac{\log n}{\log(n-j)}
\Big\{c_\tau \log(n-j)T_{n-j}L^j(\rho h^{-1}1_{Y_j})
-\int_Y L^j(\rho h^{-1}1_{Y_j})\,d\mu\Big\} \\
& \quad\quad
+\sum_{j=0}^r \Big\{\frac{\log n}{\log(n-j)}-1\Big\}\int_X \rho h^{-1}1_{Y_j}\,d\mu.
\end{split}
\]

The result follows by Theorem <ref> and
Proposition <ref>.

Now, let $X_1,\dots,X_d$ be the partition of $X \setminus Y$ in <ref>.

For all $\rho\in K(X)$, $n>r\ge0$, $k = 1,\dots, d$,

\int_X 1_{X_k} \, d f_{*}^n ( \lambda |_{\bigcup_{i=0}^r f^{-i}Y} )
= \sum_{ j = 1 }^n\int_{ \{ \tau^{(k)} \ge j  \}}  1_Y L^{ n - j }
\rho_r  \, d\mu.

Let $E_r= \bigcup_{i=0}^r f^{-i}Y$.
Suppose that $x\in E_r$ and that $f^n x \in X_k$ for some \( k = 1 ,\dots, d \). Then $x$ must have made its last return to $Y$ at some time $n - j$ for some $1 \leq j \leq n$. Moreover, by <ref>(b), $f^mx\in X_k$ for $n-j+1\le m\le n$.

\begin{align*}
\{ x \in E_r: f^n x \in X_k \} &=
\bigcup_{ j = 1 }^n \{x\in E_r: f^{n - j }x \in Y \text{ and } f^m x\in X_k,\, n-j+1\le m\le n \} \\
&= \bigcup_{ j = 1 }^n \{x\in E_r: f^{n - j }x \in Y \text{ and } \tau^{(k)} \circ f^{ n - j} \ge  j   \}.
\end{align*}

As this is a disjoint union,

\begin{align*}
\int_X 1_{ X_k }\, d f_*^n (\lambda|_{E_r})
&= \int_X 1_{ X_k }  \circ f^n \cdot 1_{E_r}\cdot \rho \cdot h ^{-1} \, d\mu \\
&=  \sum_{ j = 1 }^n \int_X (1_{ \{ \tau^{(k)} \ge j  \}}1_Y) \circ f^{ n - j } \cdot 1_{E_r} \cdot \rho \cdot h ^{-1} \, d\mu \\
&= \sum_{ j = 1 }^n \int_{ \{ \tau^{(k)} > j - 1 \} } 1_Y L^{ n - j }  \rho_r  \, d\mu.
\end{align*}

For all $\rho\in K(X)$, $r\ge0$, $k=1,\dots, d$,

\[
\lim_{ n \to \infty} \int_X 1_{ X_k } df_{*}^n (\lambda|_{\bigcup_{i=0}^r f^{-i}Y}) = \bar p_k\lambda\left(\bigcup_{i=0}^r f^{-i}Y\right).
\]

Again, we set $E_r= \bigcup_{i=0}^r f^{-i}Y$.
Note that $\int\rho_r\,d\mu=\int_{E_r}\rho h^{-1}\,d\mu=\Leb(E_r)$.

First, we consider the case $\alpha \in (0,1)$. Set

\[
\eps_{ n } \coloneqq
\sup_{ Y }\Big| L^n \rho_r   - \frac{ d_{\alpha} \lambda (E_r) }{ n^{ 1 - \alpha }}\Big|,
\]

where \( d_{\alpha} = \frac{ \sin  \pi \alpha } {\pi c_{\tau }} \).  By Corollary <ref>,
\(
\eps_n = o ( n^{  \alpha - 1} ),
\)
Hence, by Lemma <ref>,

\begin{align}
\nonumber
\int 1_{X_k} \, d f_{*}^n ( \lambda |_{E_r} )
&= \sum_{ j = 1 }^n\int_{ \{ \tau^{(k)} \ge j  \}} 1_Y L^{ n-j  } \rho_r  \, d\mu
\\
\nonumber & = \sum_{ j = 1 }^{ n-1 }\int_{ \{ \tau^{(k)} \ge j  \}} 1_Y L^{ n-j } \rho_r  \, d\mu +O(\mu(\tau^{(k)}\ge n)) \\
&= d_{\alpha} \lambda ( E_r )
\sum_{ j = 1 }^{ n - 1 } \frac{ \mu ( \tau^{(k)} \ge j  ) }{ ( n - j )^{1 - \alpha} }
+ O \Big( \sum_{ j = 1 }^{n-1} \frac{ \eps_{ n - j} \mu ( \tau^{(k)} \ge j  ) }{ ( n - j )^{1 - \alpha}} \Big)
\label{eq:push-with-Delta}
\end{align}

As \( \mu ( \tau^{(k)} \ge j ) \sim c_k j^{ -\alpha } \) we can conclude from Lemma <ref> and Lemma <ref> that

\begin{equation}
\label{eq:series-bigger-1}
\sum_{ j = 1}^{n-1} \frac{ \mu ( \tau^{(k)} \ge  j   ) }{ ( n - j )^{1 - \alpha } } \to \frac{c_k}{\alpha} \frac{ \pi } { \sin  \pi \alpha }
\quand
\sum_{ j = 1 }^{n-1} \frac{ \eps_{ n - j} \mu ( \tau^{(k)} \ge j  ) }{ ( n - j )^{1 - \alpha}} \to 0.
\end{equation}

Combining (<ref>) and (<ref>) we obtain

\[
\int_{X_k} \, d f_*^n (\lambda|_{E_r})
= \frac{ c_k }{  c_\tau } \lambda ( E_r ) + o ( 1 )
= \bar p_k \lambda ( E_r ) + o ( 1 ),
\]

concluding the result in the case that \( \alpha \in (0,1) \).

When $\alpha = 1$, we proceed in the same manner as before. Set

\[
\eps_{ n } \coloneqq
\sup_Y \Big|L^n\rho_r - \frac{ \lambda ( E_r ) }{ c_{\tau} \log n }\Big|
\]

and note by Corollary <ref> that \( \eps_{ n } = o ( 1 / \log n) \)
Hence, by Lemma <ref>,

\begin{align}
\nonumber
\int 1_{X_k} \, d f_{*}^n ( \lambda |_{E_r} )
&= \sum_{ j = 1 }^n\int_{ \{ \tau^{(k)} \ge j \}} 1_YL^{ n-j  } \rho_r \, d\mu
\\
\nonumber & = \sum_{ j = 1 }^{ n-2 }\int_{ \{ \tau^{(k)} \ge j \}} 1_YL^{ n-j  } \rho_r \, d\mu + O(\mu ( \tau^{(k)} \ge n - 1 ))
\\
&= \frac{ \lambda ( E_r ) }{ c_{\tau } }
\sum_{ j = 1 }^{ n - 2 } \frac{ \mu ( \tau^{(k)} \ge j ) }{ \log ( n - j ) } + O \Big( \sum_{ j = 1 }^{n-2} \frac{ \eps_{ n - j} \mu ( \tau^{(k)} \ge j ) }{ \log ( n - j )} \Big) + O(n^{-1}).
\label{eq:push-with-Delta-1}
\end{align}

Using Lemma <ref> and Lemma <ref> we conclude that

\begin{equation}
\label{eq:series-1}
\sum_{ j = 1}^{n-2} \frac{ \mu ( \tau^{(k)} \ge j  ) }{ \log  ( n - j ) } \to c_k
\quand
\sum_{ j = 1 }^{n-2} \frac{ \eps_{ n - j} \mu ( \tau^{(k)} \ge j ) }{ \log ( n - j ) } \to 0.
\end{equation}

Combining (<ref>) and (<ref>) we obtain
\(
\int_{X_k} \, d f_*^n (\lambda|_{E_r}) = \bar p_k \lambda ( E_r ) + o ( 1 ),
\)
concluding the proof.

Since $K(X)$ is dense in $L^1(X, m)$,
the conclusion of Proposition <ref> holds for
general $\lambda\in \cM_1^{\mathrm ac}(X)$.

Let $\omega=\lim_{i\to\infty} f_*^{n_i}\lambda$ be a subsequential limit of $f_*^n\lambda$.
By Lemma <ref>, $\omega=\nu_p$ for some $p\in \cS_0$.
Since $\nu_p(\partial X_k)=0$,

\[
\lim_{i\to\infty}f_*^{n_i}\lambda(X_k)=\nu_p(X_k)=p_k.
\]

by the Portmanteau Lemma.
Also, setting $E_r=\bigcup_{i=0}^r f^{-i}Y$, it follows from Proposition <ref>

$$\lim_{i\to\infty} f_*^{n_i}(\lambda|_{E_r})(X_k)=\bar p_k\lambda({E_r}).

Hence, $p_k\ge \bar p_k\lambda({E_r})$.
Letting $r\to\infty$, we obtain that
$p_k\ge\bar p_k$ for $k=1,\dots,d$ and so
$p=\bar p$.

We have shown that $\nu_{\bar p}$ is the unique subsequential limit of
$f_*^n\lambda$. By compactness of $\cM_1(X)$, it follows that
$f_*^n\lambda=\nu_{\bar p}$.

§ EXAMPLES

In this section, we apply our main results to intermittent maps.
Throughout, we write $|E|=\Leb(E)$ for measurable subsets $E\subset[0,1]$.

§.§ Thaler maps

We define \( \cT \) to be the class of interval maps  \( f : [0,1] \to [0,1] \)  studied in <cit.> which satisfy the following:

The exists $0=\xi_1<\xi_2<\dots<\xi_d=1$, $d\ge2$, with $f\xi_k=\xi_k$ and
$f'(\xi_k)=1$ for $k=1,\dots,d$;

There exist subintervals \( I_{ 1 }, \ldots, I_d \), $d\ge2$, with $\xi_k\in \Int I_k$, such that \( \bigcup_{ k = 1  }^d I_k = [0, 1]  \) and such that the restriction \( f|_{  I_k } \) extends to a \( C^{ 2 } \) diffeomorphism \( f_k : \bar{I}_k \to [0,1] \);

T3 \( f'(x) > 1 \) for all \( x \not\in \{ \xi_{ 1 }, \ldots, \xi_d\} \) and there exists an \( \eps > 0 \) so that \( f \) is concave (resp. convex) on the interval \( (\xi_k - \eps, \xi_k) \cap I_k \) (resp. \( (\xi_k, \xi_k + \eps) \cap I_k \));

There exist \( \alpha \in ( 0,1 ] \) and \( b_{ 1 }, \ldots, b_d > 0 \) such that for every \( k = 1, \ldots , d \),

\[
| f x - x | \sim  b_k | x - \xi_k |^{ 1 + 1/\alpha } \quad \text{as } x \to \xi_k.
\]

When \( d = 2 \), conditions <ref>–<ref> reduce to conditions <ref>–<ref> given in the Introduction.

Thaler maps $f\in\cT$ can be shown to lie in the abstract setting of Section <ref> and hence Theorems <ref>, <ref> and <ref> apply to these examples.
The verification of the hypotheses in Section <ref>
is essentially contained in <cit.> and the references therein. For completeness we recall the main steps of this argument here. Different inducing schemes are needed in the cases \( d = 2 \) and \( d \geq 3 \), and the case \( d = 2 \) will be treated in the more general setting considered in Section <ref>, so we focus here on the case \( d \geq 3 \).[There seem to be some typos in <cit.> where $x_i$ should be $x_j$ and $f_j$ should be $f_i$.]

By <cit.>, \( f \in \cT \) is conservative and ergodic with a unique (up to scaling) invariant absolutely continuous measure \( \mu \); moreover $\mu([0,1])=\infty$.
Since the branches $f_k:\overline{I}_k\to[0,1]$ are onto, it is immediate that $f$ is a topologically mixing Markov map.

\[
X_k \coloneqq I_k \cap f^{ -1 } I_k,\quad
k = 1 ,\ldots, d;
\quand
Y \coloneqq [0,1]\setminus (X_1\cup\dots\cup X_k).
\]

It is immediate from the definitions that <ref>(a,b) hold.
Below, we show:

The first return map \( F : Y \to Y \)
is a topologically mixing Gibbs-Markov map.

We denote the density of $\mu$ by $h$.
By <cit.>, it follows from Lemma <ref> and Proposition <ref> that $h$ is bounded on compact subsets of $[0,1]\setminus\{\xi_1,\dots,\xi_d\}$. Hence, <ref> is satisfied.
Also <ref> holds by Remark <ref>.
Hence it remains to prove
Proposition <ref>, to verify <ref>(c) and <ref>.

First, we describe the inducing scheme.
We begin by setting
$Y_k \coloneqq I_k \setminus X_k$. Then $Y=Y_1\dot\cup\cdots \dot\cup Y_d$.
Let \( g_k \coloneqq f_k^{ -1 } : [0,1] \to \overline{I}_k \) denote the inverse branch of \( f \) on \( \overline{I}_k \).
\( X_{ k, n } \coloneqq g_k^n Y_k \) for \( n \ge 1 \), $k=1,\dots, d$. Notice that \( \{X_{ k,n },\;n\ge 1\} \) is a partition of \( X_k \) and that the maps \( f : X_{  k , n + 1 } \to X_{ k, n }  \), \( f:X_{ k, 1 } \to Y_k \) are bijections.
For \( j \neq k \), set

\[
Y_{j,k,1} \coloneqq g_jY_k \quand  Y_{ j, k ,  n } \coloneqq g_j X_{ k, n - 1 },\; n\ge2.
\]

Then $\{Y_{j,k,n}:j\neq k,\,n\ge 1\}$ is a partition of $Y_j$ for each $j$ and hence

\[
\cP_Y \coloneqq \{ Y_{j,k,n}: j\neq k,\,n\ge 1\}
\]

is a partition of $Y$.

Let $F=f^\tau:Y\to Y$ be the first return map to $Y$.
Then $\tau=n$ on $\bigcup_{j\neq k}Y_{j,k,n}$ and
$F=f^n: Y_{ j , k,  n } \to Y_k $ is a bijection for all $j\neq k$ and all $n$.

By construction, \( F \) is Markov with respect to the partition $\cP_Y$. As \( F Y_{ j,k,n } = Y_k \) it is clear that \( F \) has finite images. Moreover, as \( F Y_k = \bigcup_{ \ell \neq k } Y_{ \ell } \) we see that \( F^{ 3 }  Y_{ j,k,n }  = Y \) and so \( F \) is topologically mixing.

For \( 0 \leq m \leq n - 1 \) and \( x ,y \in [0,1]  \),

\begin{equation}
\label{eq:dist-calc-1}
\log \frac{  (f^{ n - m })' (f^m x)  }{ (f^{ n - m})' (f^m y) }
= \sum_{ i = m }^{n-1} \log \frac{ f' ( f^i x) }{ f' ( f^i y) }
= \sum_{ i = m}^{n-1} \frac{ |f''( z_i)|  }{  f'( z _i)  } | f^i x -
f^i y |
\le |f''|_\infty \sum_{ i = m}^{n-1}  | f^i x - f^i y |,
\end{equation}

where \( z_i \in [ f^i x , f^i y ] \) is chosen by the mean value theorem.
In particular, for \( x ,y \in Y_{j, k, n} \),

\[
\log \frac{ (f^{ n - m })' (f^m x)  }{ (f^{ n - m})' (f^m y) }
\le |f''|_\infty\left( | Y_{ j,k,n } | + \sum_{ i = 1 }^{ \infty } | X_{k,i} | \right) \le |f''|_\infty.
\]

Hence for  $i = 0,\ldots, n - 1$,

\begin{equation}
\label{eq:dist-calc-2}
\frac{ | f^i x - f^i  y | }{ | X_{ k , n - i } | }
\le e^{2|f''|_\infty}\frac{ | f^{n-i} f^ix - f^{n-i}  f^iy | }{ | f^{n-i}X_{k,n-i} | }
= e^{2|f''|_\infty}\frac{ | F x - F y | }{ | Y_k | },
\end{equation}

Inserting (<ref>) into (<ref>) with \( m = 0 \), we obtain that

\begin{equation}\label{eq:dist-calc-3}
\log \Big| \frac{ F' x  }{  F'  y  } \Big|  \leq C | F x - F y |
\end{equation}

where $C=|f''|_\infty e^{2|f''|_\infty}\max_k|Y_k|^{-1}$.

Let $\lambda=\inf_Y f'>1$ and set $\theta=\lambda^{-1}$.
If \( s(x,y) = n  \), then \( 1 \geq | F^n x - F^n y|  \geq    \lambda ^{n - 1} | Fx - Fy | \). Combining this inequality with (<ref>), we obtain
\(
\log \big| \frac{ F' x  }{  F'  y  } \big|  \leq  C\theta^{n-1} =C\theta^{-1}d_\theta(x,y),
\)
which concludes the proof.

Let $e_k=1$ for $k=1,d$ and $e_k=2$ for $2\le k\le d-1$.

\[
c_k \coloneqq
e_k b_k^{ -\alpha } \alpha^\alpha
\sum_{j = 1, \ldots, d,\; j \neq k}
h ( g_j \xi_k ) g_j' (  \xi_k )  .
\]

In particular, for $d=2$, we have

\begin{equation}\label{eq:tha-2}
c_1 = b_1^{ -\alpha } \alpha^\alpha h ( g_2 0 ) g_2' ( 0)  , \quad
c_2 = b_2^{ -\alpha } \alpha^\alpha h ( g_1 1 ) g_1' ( 1)  .
\end{equation}

The remaining ingredients, namely <ref>(c)
and <ref>, follow from the next result.

\(
\mu ( \tau^{ ( k ) } = n )
\sim
\alpha c_k
\)
as $n\to\infty$ for $k=1,\dots,d$.

Note that $\{ \tau^{ (k) } = n \} = \bigcup_{ j = 1, \ldots, d,\; j \neq k   } Y_{ j, k, n - 1 }$.

Fix $j,k\in\{1,\dots,d\}$, $j\neq k$.
For \( k \neq 1, d \), the sets \( X_{ k, n } \) have two connected components \( X_{ k , n }^\pm \) which lie to the left or right of \( \xi_k \) and both accumulate at \( \xi_k \) as $n\to\infty$.
Accordingly, define \( Y_{ j,k,n }^\pm  \coloneqq g_j  X_{ k,n-1 }^\pm  \).
It suffices to show that $\mu(Y_{j,k,n}^\pm)$ for $k\neq 1,d$ and
$\mu(Y_{j,k,n})$ for $k=1,d$ have the asymptotic

\[
b_k^{- \alpha } \alpha^{ 1 +  \alpha }
h ( g_j \xi_k ) g_j' ( \xi_k ) n^{-(1+\alpha)}
\quad\text{as $n\to\infty$}
\]

We give the details for \( \mu ( Y_{ j,k,n }^+ ) \), \( k \neq 1,d \), the other cases being similar.

By <cit.>, $h$ is continuous on  $[ 0 , 1 ] \setminus \{ \xi_{ 1 }, \ldots, \xi_{ d } \}$ (in the standard topology).

\begin{equation}\label{eq:h-calc}
\mu ( Y_{ j , k, n  }^+ ) - h ( g_j\xi_k ) | Y_{ j,k,n }^+ |
\int_{ Y_{ j , k , n }^{ + } } ( h - h( g_j \xi_k ) )\,d\Leb
\leq | Y_{ j , k , n }^{ + }  |  \sup_{  Y_{j,k,n}^{ + } } | h - h( g_j \xi_k ) |
\end{equation}

It follows that
$\mu ( Y_{ j , k, n  }^+ )  \sim h ( g_j \xi_k ) | Y_{ j,k,n }^+ |$.
By the mean value theorem, there exists \( z_n \in X_{ k,n }^+ \) so that
$| Y_{ j,k,n }^+ | = g_j' (z_n) | X_{ k,n-1 }^+ | \sim g_j'( \xi_k ) | X_{ k,n-1 }^+|$. Combining these last two estimates,

\begin{equation}
\label{eq:mvt}
\mu ( Y_{ j , k, n  }^+ )  \sim h ( g_j \xi_k )
g_j'( \xi_k ) | X_{ k,n-1 }^+|.
\end{equation}

It remains to estimate \( | X_{k,n }^+ | \). We recall the following standard calculation.

[see for example <cit.>]

Suppose that \( T : [0, C] \to [0, \infty) \) is such that \( Tx > x \) for each \( x \in (0,C] \) and  \( Tx \sim x+ b x^{ 1 + p } \) as \( x \to 0 \). Let \( z_{ n  } = T  z_{ n + 1  }   \). Then

\[
z_n \sim ( p b n )^{ -1/p }
\quand
z_{ n  } - z_{  n + 1 }  \sim b^{ -1/p } ( p n  )^{ -(1+ 1/p) }
\quad \text{as } n \to \infty
\]

It follows from the assumptions that the sequence \( z_n \) is strictly decreasing with \( z_n \to 0 \). We then calculate

\[
z_{ n+1 }^{- p }  (1 + b z_{ n + 1 }^{ p } )^{ -p }
= z_{n+1}^{-p}  ( 1 - p  b z_{n+1}^{p} + o ( z_{n+1}^{2p} ))
= z_{n+1}^{- p }  - pb + o(1)
\]

as \( n\to \infty \).
Summing over $n$, this yields  \(z_{n}^{-p} = p b n + o(n)\) which gives the estimate for $z_n$. Finally, \(  z_n - z_{ n + 1 }  =  Tz_{ n+1 } - z_{ n + 1 }  \sim b z_{ n + 1 }^{ 1+p} \sim b^{ -1/p } ( p n  )^{ -(1+ 1/p) } \).

Recalling <ref>, the estimate for $| X_{ k, n }^{+} |$  reduces after a  change of coordinates to the situation in
Sublemma <ref> with \( p = 1/\alpha \) and \( b = b_k \). Hence
\( | X_{ k, n }^{+} | \sim b_k^{ -\alpha } (n/\alpha)^{ -(1 + \alpha) } \).
Combining this with (<ref>), we obtain

\[
\mu ( Y_{ j , k, n }^+ )
\sim  h ( g_j \xi_k ) g_j' ( \xi_k ) b_k^{- \alpha } (\alpha/n)^{ 1 +  \alpha }
\]

as required.

§.§ Intermittent maps with critical points and/or singularities

We now consider a class of intermittent interval maps with two branches that possibly admit a critical point and/or a singularity at the discontinuity. This class of maps include the maps in \( \cT \) with \( d = 2 \) and the maps described in <cit.>. We define \( \cF \) to be set of maps \( f : [-1,1] \to [-1,1] \) which satisfy the following conditions.

There exist a \( c \in (-1,1) \) such that the restrictions
the restrictions \(f_{-} \coloneqq f : (-1,c) \to (-1,1) \) and \(f_{+} : (c,1) \to (-1,1)\) are \(C^2\) orientation preserving diffeomorphisms with no fixed points.

For notational simplicity we will assume that \( c = 0 \). Notice also that we do not assume that the \(f_{ \pm }\) extend to \(C^2\) functions on the closure of \(I_{ \pm }\).

F1 There exist \(\ell_{+},\ell_{-} > 0\) and \(k_+,k_- > 0\) such that

\begin{equation}
\label{eq:def-clm}
f (x)
\begin{cases}
x + b_-(1 +  x )^{ 1 + \ell_{-} } + o ( (1 + x)^{ 1 + \ell_- } ), &\text{for } x \in U_{ -1 }\\
1 - a_- | x |^{ k_- },  &\text{for } x \in U_{ 0- }\\
- 1 + a_+ | x |^{ k_+ }, &\text{for } x \in U_{ 0+ }\\
x - b_+( 1 -  x )^{ 1 + \ell_{+} } + o ( (1 - x)^{ 1 + \ell_+ } ) & \text{for } x \in U_{ +1 }\\
\end{cases}
\end{equation}

whenever \( k_+, k_- \neq 1 \)
for some \(a_{ \pm }, b_{ \pm } > 0\), and some neighbourhoods \( U_{ -1 }, U_{ 0 } \) of \( -1, 0 \) in \( [-1,0] \) and some neighbourhoods \( U_{ 0+ }, U_{ +1 } \) of \( 0,1 \) in \( [0,1] \).
If \( k_+ = 1 \) and/or \( k_- = 1 \), then we replace the corresponding lines in (<ref>) with the assumption that \( f' (0  - ) = a_- > 1 \) and/or  \(f' (0+) = a_+ > 1 \) respectively.

It will be convenient to assume that \( f  U_{ 0 \pm }  \subset U_{ \pm 1 } \). Notice that this assumption posses no restriction on <ref> as \( U_{ 0 \pm } \) can be taken to be arbitrarily small.

This definition is more general than the one in <cit.> as <ref> stipulates only an asymptotic behaviour near the fixed points. However, it does not include the maps in <cit.> due to the restriction mentioned in Remark <ref>. We expect that our results hold also for the maps in <cit.>.

Suppose that \(f\) satisfies <ref> and let \(\gamma_{ \pm } \in I_{ \pm }\) be the two points of period \( 2 \) for \(f\). Define the intervals

\[
Y \coloneqq [\gamma_- , \gamma_+ ], \quad
X_{\pm, n} \coloneqq f^{ -n }_{ \pm } Y, \quand
Y_{ \pm, n + 1 }^{ \pm } \coloneqq f_{ \pm }^{ - 1} X_{ \mp, n }.
\]

By definition \( f : Y_{ \pm, n + 1 } \to X_{ \mp, n } \) and \( f : X_{ \mp, n } \to Y_{ \mp, n - 1 } \) are bijections and so \(\cP \coloneqq \{ Y_{ \pm, k }, X_{ \pm , n } : n \geq 1, k \geq 2 \}\) forms a Markov partition for \(f\). Our final condition will ensure that our maps have good expansion and distortion properties.

\( f \) is convex (resp. concave) on \( U_{ -1 } \) (resp. \( U_{ 1 } \)) and moreover

If \( f \) is not \( C^2 \) on \( \overline U_{-1} \) (resp. \( \overline U_{+1} \)), then \( f '' (x) \lesssim (1 + x)^{ \ell_- - 1 }  \) (resp. \( |f''( x )| \lesssim ( 1 - x )^{ \ell_+ - 1 }  \)).

If \( k_{ \pm } \neq 1 \), then there exists a \( \lambda > 1 \) such that  \( ( f^n )' (x) > \lambda  \)  for each \( x \in Y_{ \pm , n} \) and each \( Y_{ \pm , n} \not\subset U_{ 0\pm } \).

Notice that <ref>.<ref> is only assumed when the map is not \( C^{ 2 } \) at the fixed points and that <ref>.<ref>  is only an assumption about the map outside of the neighbourhoods \( U_{ 0\pm }, U_{ \pm 1 } \) and is trivially satisfied if \( f ' (x) > 1 \) for each \( x \not\in \{ -1, 1 \} \).

We let \( \alpha_+ \coloneqq 1 / \ell_+k_- \), \( \alpha_- \coloneqq 1 / \ell_- k_+ \) and define

\[
c_{ 1 } \coloneqq
h(0) a_-^{ - 1/k_-} ( \ell_+ b_+ )^{ - 1 / \alpha_- } \alpha_-^{2},
\quand
c_{ 2 } \coloneqq
h(0) a_+^{ - 1/k_+} ( \ell_- b_- )^{ - 1 / \alpha_+ } \alpha_+^{2}.
\]

We will assume that \( \alpha_{ + } = \alpha_{ - } = \alpha \in (0,1) \).

Suppose that \( f \in \cF \) with \(  \alpha \in (0,1)\). Then <ref>–<ref> hold.

Throughout this section we fix

\[
Y \coloneqq [\gamma_-, \gamma_+ ], \quad
X_-\coloneqq [0, \gamma_-], \quad
X_+ \coloneqq [\gamma_+, 0],
\]

and let \( \tau : Y \to \N \) denote the first return time to \( Y \) and \( \tau^{ (\pm) } (x) \coloneqq \Card \{ n \leq \tau (x) : f^n (x) \in X_{ \pm } \} \). Let \( F\) be the first return map \( F : Y \to Y \). By construction

\( F|_{ Y_{ \pm , n} } = f^n \) and the interval \(Y\) dynamically separates \(X_-, X_+\).

In the remainder of this section we will show that \( f \) satisfies assumptions <ref>–<ref> and thus conclude the proof of Theorem <ref>. Notice that \( X_{ \pm, n } \) and \( Y_{ \pm , n} \) here play the same roles as \( \Delta_n^{ \pm  } \)  and \( \delta_n^{ \pm } \) respectively in <cit.>. The next Lemma shows that the asymptotic behaviour of the sizes of these partition elements remains exactly the same as in <cit.>.

\[
| X_{ \pm, n } | \sim b_{ \pm }^{ - 1 / \ell_{ \pm }} ( \ell_{ \pm }  n )^{ -1 - 1 / \ell_{ \pm } }
\quand
\Leb ( \tau^{(\pm)} = n )
= | Y_{ \pm , n} |
\sim a_{ \pm }^{ - 1/k_{ \pm }} ( \ell_{ \mp } b_{ \mp } )^{ - \alpha_{ \pm } } \alpha_{ \pm } n^{-1 -  \alpha_{ \pm }}.
\]

We will only explicitly prove the estimates for \( Y_{ + , n} \) and \( X_{ - , n } \) as the other estimates follow in the same way.
Let \(\gamma_n \coloneqq f^{-n}_- \gamma\) so that  \( X_{ - , n } = [\gamma_{n}, \gamma_{n-1}]\).
As in the previous section we can apply Sublemma <ref> to the sequence \( z_{n} \coloneqq 1 + \gamma_{n} \) with \( b = b_- \) and \( p = \ell_- \) to obtain

\begin{equation}
\label{eq:clm-bound-1-gamma}
z_n \sim (b_- \ell_- n )^{ - 1 / \ell_-},
\quand
| X_{ \pm, n } | \sim b_{ \pm }^{ - 1 / \ell_{ \pm }} ( \ell_{ \pm }  n )^{ -1 - 1 / \ell_{ \pm } }.
\end{equation}

Now, notice that as \(y \downarrow 0\) we have by definition \( f_{+}^{-1} y = \big(\frac{ y + 1 }{a_+} \big)^{1/k_+} \). It follows that \(f_+^{-1}  \gamma_n  = \big(\frac{1 + \gamma_n}{a_+}\big)^{1/k_+}\).
From (<ref>) we obtain  \(f^{ -1 }_+\gamma_n \sim a_+^{ - 1/k_+} ( \ell_- b_- n)^{-1/ (\ell_- k_+)} \) which yields the claimed asymptotics for \( |Y_{ \pm , n} | \).

Having established Lemma <ref> the argument that \( F \) is Gibbs-Markov follows essentially verbatim from <cit.> replacing the roles of \( \delta_n^{ \pm } \) with \( Y_{ \pm , n} \) and \( \Delta_n^{ \pm } \) with \( Y_n^{ \pm } \). For completeness we include the main steps of this argument.

There exists \( \lambda > 1 \) such that \( F'(y) > \lambda \) for all \( y \in Y \).

We follow the proof of <cit.>.
We only consider the case \( y \in Y_{ +, n} \) as the case that \( y \in Y_{ - , n} \) is the same. Define the function \( \phi \coloneqq f_+ \circ f_- \circ f_+ \) and notice that \( \phi : Y_{ + , n + 1} \to Y_{ + , n } \) bijectively.

Let \( y \in Y_{ + , n} \) where \( n \) is such that \( Y_{ \pm , n} \subset U_{ 0- } \), then
\(
\dfrac{ f' (y) f' (f y) }{ f' ( \phi (y)) }  > 1.
\)

If \( k_+ \in (0,1) \) then \( f' \) is decreasing on \( U_{0+} \) and so, as \( y < \phi(y) \), we have that \( f ' ( y ) / f ' ( \phi (y ) ) > 1 \). By construction \( f(y) \in U_{1} \) for every \( y \in U_{0+} \), so \( f' ( f y ) > 1 \) and we are finished.

Assume now that \( k_+ > 1 \) and to ease notation set \( k = k_+, a = a_+, b = b_-, \ell = \ell_- \). Setting \( x = fy \) we recall that \( x \in U_{ -1} \) by Remark <ref> and using the convexity of \( f \) on \( U_{ -1 } \) we obtain that
\(
f  x   \leq f( -1 ) + f ' ( x ) ( x + 1 )
= -1 + f'(x)  ay^k\)
which in turn implies that
$\phi (y)
\leq f' (x)^{ 1/k } y$.
As \( y< \phi(y)\) and as \( ( f '(y) )/ ( f' ( \phi( y)) ) = ( y / \phi (y) )^{ k - 1 } \),

\[
\frac{f ' (y)}{ f' ( \phi (y)) } f' (fy)
\geq \Big( \frac{\phi(y)}{y} \Big) \Big( \frac{ f' (x)^{ 1/k } y }{ y } \Big)^{ -k } f'(x)
\geq 1.
\]

Let \( m_+ = \min \{ m : Y_{ +,  m } \subset U_{ 0+ } \} \). Condition <ref>
implies that \( F ' y \geq \lambda \)  for all  \( y \in Y_{ +, m }\) whenever \( m \leq m_+ \).
The Sublemma above allows us to conclude

\[
(f^{n+1}) ' (y) = f'(y) \cdot f' (f y) \cdots f' (f^{m} y)
= \frac{ f'(y) \cdot f' (f y) }{ f' ( \phi (y)) }  (f^{m})' ( \phi (y))
\geq (f^{n})' ( \phi (y)),
\]

and so \( F' y > F ( \phi (y)) \). Proceeding inductively we obtain \( F ' (y) \geq F ' ( \phi^{ n + 1 - m^+ } (y)) > \lambda\).

There exists a \( C > 0 \) and a \( \theta \in (0,1) \) such that

\[
\left| \log \frac{ F ' (x) }{ F'(y) } \right| \leq C \theta^{ s ( x, y ) }.
\]

We only consider \( x,y \in Y_{ + , n} \) as the argument for \( x,y\in Y_{ -,  n } \) is the same. Calculating as in (<ref>) one finds

\begin{align}
\left| \log \frac{ ( f^{ n - j } )' ( f^{j} x) }{ ( f^{ n - j } ) ' ( f^{j} y) } \right|
\leq
\label{eq:clm-dist-bound-2}
\left| \frac{ f''  u_k  }{ f' u_k } \right| | Y_{ + , n} |
+ \sum_{k=1}^{ n - 1} \left| \frac{ f''  u_k }{ f'  u_k } \right| | X_{ - , n - k } |
\end{align}

Notice the \( f'' ( u_{ 0 } ) / f' ( u_{ 0 } ) \lesssim u_{ 0 }^{ -1 } \lesssim  (f^{ -1 } \gamma_n)^{ -1 } \lesssim ( 1 + \gamma_n )^{ -1/k_+ } \) and (<ref>)  gives that \( ( 1 + \gamma^-_n )^{ 1 / k_+  } \lesssim n^{ -\alpha_- } \). So, as \( |Y_{ + , n}| \lesssim n^{ -1 - \alpha_- } \)
we find that the first term in (<ref>) is uniformly bounded in \( n \). If \( f \) is \( C^{ 2 } \) on \( \overline U_{ -1 } \) the second term in (<ref>) is summable and whence uniformly bounded. Otherwise, by assumption <ref>, (<ref>) and Lemma <ref> we have that \( f''( u_k ) | Y_{ + , n } | \lesssim ( 1 + \gamma_{ n - k }^- )^{ \ell_- - 1 } |Y_{ + , n - k }| \lesssim ( n - k )^{ -1 + 1/\ell_- } ( n - k )^{ -1 - 1/\ell_{ -} } \lesssim ( n - k )^{ -2 } \) which is summable.
Thus, (<ref>) is uniformly bounded  and we can calculate as in (<ref>) to find \( \log | F'x / F'y | \leq C | F x  - F y | \). Thus, proceeding in the same way as in the proof of Proposition <ref> one concludes the result with \( \theta \coloneqq \tilde{ \lambda }^{ - 1 } \).

Lemma <ref> and Proposition <ref> imply that \( F : Y \to Y \) is Gibbs-Markov, which yields <ref>. Remark <ref> yields <ref> and so it only remains to show <ref> and <ref>. By Lemma <ref> \( F \) preserves an absolutely continuous invariant measure \( \mu \) with density \( h \). We claim that \( h \) is continuous on \( Y \). Given the validity of this claim we can apply the same argument as in (<ref>) to conclude from Lemma <ref> that

\[
\mu ( \tau^{ \pm } = n ) = \mu( Y_{ \pm, n } )
\sim h(0) | Y_{ \pm,  n } |
\sim  h(0) a_{ \pm }^{ - 1/k_{ \pm }} ( \ell_{ \mp } b_{ \mp } )^{ - \alpha_{ \pm } } \alpha_{ \pm } n^{-1 -  \alpha_{ \pm }}.
\]

Together with Lemma <ref>, this concludes <ref> and <ref>.

Let \( \cP _{ Y }^{ ( n ) } \) denote the \( n^{ th } \) refined partition of \( F \) and let \( a \in \cP_{ Y }^{ (n) } \).
From the uniform expansion of \( F \) and the mean-value theorem we know that for \( j \leq n \), \( | F^{ j } x - F^{ j }y | \leq \lambda^{ -( n - j ) } | F^{ n } x - F^{ n } y | \) for every \( x,y \in a \). Moreover,  recalling that \( \log | F' x / F' y | \leq C | F x - F y | \) for \( x, y \in Y \) it follows that for any \( x,y \in a \)

\begin{equation}\label{eq:dist-calc-again}
\log \left| \frac{ ( F^{ n } )' x  }{ ( F^{ n } )' y } \right|
\leq \sum_{ j = 0 }^{ n - 1 } \left| \frac{ ( F^{ j } )' x  }{ ( F^{ j } )' y } \right|
\leq \frac{ C }{ \lambda - 1 } | F^{ n } x - F^{ n } y |,
\end{equation}

and so
\( | ( F^{ n }) ' x / ( F^{ n } )'  y ) - 1 | \lesssim | F^{ n } x - F^{ n } y |  \).
Moreover, (<ref>) implies \( 1 / [ ( F^{ n } )' y ] \lesssim | a | \) for all \( y \in a \).
Using the standard pointwise formula for the transfer operator \( L_{ Y } \)  for \( F \), we obtain

\begin{align*}
| L_{ Y }^{ n } 1 (x) - L_{ Y }^{ n } 1 (y) |
&= \bigg| \sum_{  \tilde x \in F^{ -n } x, \, \tilde y \in F^{ -n  } y}
\frac{ 1 } { (F^{ n }) ' \tilde x } - \frac{ 1 } { (F^{ n }) ' \tilde y }
\bigg|\\
&\leq \sum_{ \tilde x \in F^{ -n } x, \, \tilde y \in F^{ -n  }  }
\left| \frac{ (F^{ n }) ' \tilde y }{  ( F^{ n }) ' \tilde x } - 1 \right| \left| \frac{1}{  ( F^{ n } ) ' \tilde y } \right|\\
&\lesssim  \sum_{ a \in \cP_{ Y }^{ ( n ) } } | x - y | | a |
\lesssim | x - y |.
\end{align*}

Similarly, one can check that \( L_{ Y }^{ n } \) is uniformly bounded. The Arzelà-Ascoli Theorem then yields that \( \frac{ 1 }{ n } \sum_{ j = 1 }^{ n - 1 }  L_{ Y }^{ j } 1 \) has a subsequence which is uniformly convergent to some Lipschitz \( \tilde h  : Y \to \R \). As \( \tilde h \) is necessarily the density of an invariant measure it follows that \( h 1_{ Y }  = \tilde h \) yielding that \( h \) is Lipschitz (and hence continuous). This establishes our claim and concludes the proof.

§ SERIES ESTIMATES

In this appendix, we carry out some calculations required in Section <ref>.

Let $\alpha\in(0,1)$. Then

The function $(n-x)^{\alpha-1} x^{-\alpha}$ has one critical point at
$x=\alpha n$. Hence, by approximating integrals by Riemann sums on the intervals
$[0,\alpha n]$ and $[\alpha n,n]$,

\[
\sum_{j=1}^{n-1}(n-j)^{\alpha-1}j^{-\alpha}
\int_0^n (n-x)^{\alpha-1}x^{-\alpha}\,dx + O(n^{-\alpha})+O(n^{\alpha-1}).
\]

Using standard properties of Beta and Gamma functions, we obtain

\begin{align*}
\int_0^n (n-x)^{\alpha-1}x^{-\alpha}\,dx
& = \int_0^1 (1-x)^{\alpha-1}x^{-\alpha}\,dx
\\ & =\Gamma(\alpha)\Gamma(1-\alpha)
\end{align*}

as required.

Let $\alpha\in(0,1)$. Suppose that $g:\N\to\R$ satisfies $\lim_{n\to\infty}g(n)=0$.

Set $C=\max_n |g(n)|$.
Let $\eps\in(0,\frac12)$.

\begin{align*}
\sum_{j=1}^{n-1}j^{-\alpha}g(j)\,(n-j)^{\alpha-1}
\sum_{1\le j<\eps n}j^{-\alpha}g(j)(n-j)^{\alpha-1}
+\sum_{\eps n\le j< n}j^{-\alpha}g(j)(n-j)^{\alpha-1}\\
& \le
C2^{1-\alpha} n^{\alpha-1}\sum_{1\le j<\eps n}j^{-\alpha}
+(\eps n)^{-\alpha}\max_{k\ge \eps n}g(k)\sum_{\eps n\le j< n}(n-j)^{\alpha-1} \\
& \lesssim
n^{\alpha-1}(1+(\eps n)^{1-\alpha})
+(\eps n)^{-\alpha}\max_{k\ge \eps n}g(k) n^{\alpha} \\
& \lesssim
+\eps^{-\alpha}\max_{k\ge \eps n}g(k).
\end{align*}

\[
\limsup_{n\to\infty}\sum_{j=1}^{n-1}j^{-\alpha}g(j)(n-j)^{\alpha-1}
\lesssim \eps^{1-\alpha}.
\]

The first limit follows since $\eps$ is arbitrarily small.
Replacing $\alpha$ by $1-\alpha$, we obtain the second limit.

Suppose that $g(x)=o(1)$ as $x\to\infty$.
Then $\sum_{j=1}^n \frac{g(j)}{j} =o(\log n)$.

For $\eps>0$, choose $n_0\ge 1$ such that $|g(x)|<\eps$ for $x>n_0$.

\[
\sum_{j=1}^n \frac{g(j)}{j}\le
\sum_{j=1}^{n_0} \frac{g(j)}{j}+
\eps \sum_{j=n_0}^n \frac{1}{j}
\le \sum_{j=1}^{n_0} \frac{g(j)}{j}+
\eps \log n.
\]

$\limsup_{n\to\infty}(\log n)^{-1}\sum_{j=1}^n \frac{g(j)}{j}\le\eps$.

Let $g_1,g_2:[0,\infty)\to\R$ with $\lim_{x\to\infty} g_i(x)=1$ for $i=1,2$.

\[
\lim_{n\to\infty}\sum_{j=1}^{n-2} \frac{g_1(j)}{j}\frac{g_2(n-j)}{\log(n-j)}=1.
\]

\[
\sum_{j=1}^{n-2} \frac{g_1(j)}{j}\frac{g_2(n-j)}{\log(n-j)}=S_1+S_2
\]

\[
S_1=\sum_{1\le j\le n/2} \frac{g_1(j)}{j}\frac{g_2(n-j)}{\log(n-j)}, \qquad
S_2=\sum_{n/2<j\le n-2} \frac{g_1(j)}{j}\frac{g_2(n-j)}{\log(n-j)}.
\]

Now, $|S_2|\le |g_1|_\infty|g_2|_\infty\frac{2}{n}\sum_{2\le j\le n}\frac{1}{\log j}$.

\[
\sum_{2\le j\le n}\frac{1}{\log j}\lesssim \frac{n}{\log n}
+\sum_{n/\log n\le j\le n}\frac{1}{\log j}
\le \frac{n}{\log n}+\frac{1}{\log(n/\log n)}n\lesssim \frac{n}{\log n},
\]

so $|S_2|\lesssim \frac{1}{\log n}$.

By Lemma <ref>, $\sum_{j=1}^n \frac{g_1(j)}{j}\sim\log n$. Hence,

\[
S_1\le \sup_{k\ge n/2}g_2(k)\frac{1}{\log n/2}\sum_{1\le j\le n/2}\frac{g_1(j)}{j}\sim \frac{1}{\log n/2}\log n/2 = 1.
\]

\[
S_1\ge \inf_{k\ge n/2}g_2(k)\frac{1}{\log n}\sum_{1\le j\le n/2}\frac{g_1(j)}{j}\sim \frac{1}{\log n}\log n/2 \sim 1.
\]

Hence $S_1+S_2\sim S_1\sim 1$ as required.

§.§.§ Acknowledgment
We are very grateful to Stefano Luzzatto for important discussions that led to this paper and for the hospitality of ICTP where part of this research was carried out.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

American Mathematical Society, Providence, RI

seriesMathematical Surveys and Monographs
titleAn Introduction to Infinite Ergodic Theory

titleLocal Limit Theorems for Partial Sums of Stationary Sequences
Generated by Gibbs-Markov Maps
journaltitleStochastics and Dynamics

titleErgodic Theory for Markov Fibred Systems and Parabolic
Rational Maps
journaltitleTransactions of the American Mathematical Society

titleOccupation Times of Sets of Infinite Measure for Ergodic
journaltitleErgodic Theory and Dynamical Systems

John Wiley & Sons, Inc., New York

seriesWiley Series in Probability and Mathematical Statistics
titleProbability and Measure

shorttitleMulticomponent Dynamical Systems
titleMulticomponent Dynamical Systems: SRB Measures and Phase

For any C1C1Cˆ1-expanding map fff of the circle we study
the equilibrium states for the potential
f'. We formulate a C1C1Cˆ1
generalization of Pesin's entropy formula that holds for all the
Sinai-Ruelle-Bowen (SRB) measures if they exist, and for all the (necessarily
existing) SRB-like measures. In the C1C1Cˆ1-generic case
Pesin's entropy formula holds for a unique SRB measure which is not
absolutely continuous with respect to Lebesgue. The result also stands in the
non-generic case for which no SRB measure exists.

titleEquilibrium States and SRB-like Measures of
$C^1$-Expanding Maps of the Circle
journaltitlePortugaliae Mathematica. Nova Série

We study a class $$\widehat{{\mathfrak
{F}}}$$of one-dimensional full branch maps introduced in Coates et al.
(Commun Math Phys 402(2):1845–1878, 2023), 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.

titlePersistent Non-Statistical Dynamics in One-Dimensional Maps
journaltitleCommunications in Mathematical Physics

We study a class of one-dimensional full branch maps admitting two
indifferent fixed points as well as critical points and/or unbounded
derivative. Under some mild assumptions we prove the existence of a unique
invariant mixing absolutely continuous probability measure, study its rate of
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titleDoubly Intermittent Full Branch Maps with Critical Points and
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titleEmpirical Measures of Partially Hyperbolic Attractors
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titleCorrelation Asymptotics from Large Deviations in Dynamical
Systems with Infinite Measure
journaltitleColloquium Mathematicum

We give a complete description of relations between observable and natural
measures in connection with invariance, ergodicity and absolute continuity.

titleRelations between Natural and Observable Measures

We combine some results from the literature to give examples of completely
mixing interval maps without limit measure.

titleCompletely Mixing Maps without Limit Measure
journaltitleColloquium Mathematicum

titleAn Occupation Time Theorem for a Class of Stochastic
journaltitleTransactions of the American Mathematical Society

titleSur certains processus stochastiques homogènes
journaltitleCompositio Mathematica

titleOperator Renewal Theory and Mixing Rates for Dynamical
Systems with Infinite Measure
journaltitleInventiones Mathematicae

American Mathematical Society (AMS)

booktitleAlgebraic and Topological Dynamics. Proceedings of
the Conference, Bonn, Germany, May 1–July 31, 2004
titleErgodic Natural Measures
Providence, RI

titleDoubly Intermittent Maps with Critical Points, Unbounded
Derivatives and Regularly Varying Tail

Springer, Cham

seriesSpringer Series in Operations Research and Financial
titleUnivariate Stable Distributions: Models for Heavy Tailed

We study some simple dissipative dynamical systems exhibiting a transition
from a stable periodic behavior to a chaotic one. At that transition, the
inverse coherence time grows continuously from zero due to the random
occurrence of widely separated bursts in the time record.

titleIntermittent Transition to Turbulence in Dissipative
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IOP Publishing

titleFunctional Limit Theorem for Occupation Time Processes of
Intermittent Maps

We prove that the joint distribution of the occupation time ratios for
ergodic transformations preserving an infinite measure converges to a
multidimensional version of Lamperti's generalized arcsine distribution, in
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interval maps and Markov chains. We adopt the double Laplace transform
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titleMultiray Generalization of the Arcsine Laws for Occupation
Times of Infinite Ergodic Transformations
journaltitleTransactions of the American Mathematical Society

\textit{Non-statistical dynamics} are those for which a
set of points with positive measure (w.r.t. a reference probability measure
which is in most examples the Lebesgue on a manifold) do not have a
convergent sequence of empirical measures. In this paper, we show that behind
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property: \textit{statistical instability}. To this aim, we
present a general formalization of the notions of statistical stability and
instability and introduce sufficient conditions on a subset of dynamical
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titleStatistical Instability and Non-Statistical Dynamics

We present a unified approach to the Darling-Kac theorem and the arcsine
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titleMixing Limit Theorems for Ergodic Transformations
journaltitleJournal of Theoretical Probability