# Almost sure asymptotic behaviour of Birkhoff sums for infinite measure-
preserving dynamical systems

Claudio Bonanno  Dipartimento di Matematica, Università di Pisa, Largo Bruno
Pontecorvo 5, 56127 Pisa, Italy<EMAIL_ADDRESS>and  Tanja I.
Schindler  Centro di Ricerca Ennio De Giorgi, Scuola Normale Superiore, Piazza
dei Cavalieri 3, 56126 Pisa, Italy<EMAIL_ADDRESS>
###### Abstract.

We consider a conservative ergodic measure-preserving transformation $T$ of a
$\sigma$-finite measure space $(X,{\mathcal{B}},\mu)$ with $\mu(X)=\infty$.
Given an observable $f:X\to\mathbb{R}$, we study the almost sure asymptotic
behaviour of the Birkhoff sums $S_{N}f(x):=\sum_{j=1}^{N}\,(f\circ
T^{j-1})(x)$. In infinite ergodic theory it is well known that the asymptotic
behaviour of $S_{N}f(x)$ strongly depends on the point $x\in X$, and if $f\in
L^{1}(X,\mu)$, then there exists no real valued sequence $(b(N))$ such that
$\lim_{N\to\infty}S_{N}f(x)/b(N)=1$ almost surely. In this paper we show that
for dynamical systems with strong mixing assumptions for the induced map on a
finite measure set, there exists a sequence $(\alpha(N))$ and $m\colon
X\times\mathbb{N}\to\mathbb{N}$ such that for $f\in L^{1}(X,\mu)$ we have
$\lim_{N\to\infty}S_{N+m(x,N)}f(x)/\alpha(N)=1$ for $\mu$-a.e. $x\in X$.
Instead in the case $f\not\in L^{1}(X,\mu)$ we give conditions on the induced
observable such that there exists a sequence $(G(N))$ depending on $f$, for
which $\lim_{N\to\infty}S_{N}f(x)/G(N)=1$ holds for $\mu$-a.e. $x\in X$.

###### Key words and phrases:

Infinite ergodic theory; almost sure limits for Birkhoff sums; $\psi$-mixing;
trimmed sums

###### 2020 Mathematics Subject Classification:

37A40, 37A25, 60F15

The authors are partially supported by the PRIN Grant 2017S35EHN$\\_$004
“Regular and stochastic behaviour in dynamical systems” of the Italian
Ministry of University and Research (MUR), Italy. This research is part of the
authors’ activity within the UMI Group “DinAmicI” www.dinamici.org and of the
first author’s activity within the Gruppo Nazionale di Fisica Matematica,
INdAM, Italy. The second author acknowledges the support of the Centro di
Ricerca Matematica Ennio de Giorgi and of UniCredit Bank R&D group for
financial support through the ‘Dynamics and Information Theory Institute’ at
the Scuola Normale Superiore.

##  1\. Introduction

Let $T:X\to X$ be a conservative ergodic measure-preserving transformation of
a $\sigma$-finite measure space $(X,{\mathcal{B}},\mu)$ and consider an
observable $f$, that is a measurable function $f:X\to\mathbb{R}$. The almost
sure asymptotic behaviour of the _Birkhoff sums of $f$_

$S_{N}f(x):=\sum_{j=1}^{N}\,(f\circ T^{j-1})(x)$

is the main result of the famous Birkhoff’s Ergodic Theorem. It states that if
$\mu$ is a probability measure and $f\in L^{1}(X,\mu)$, then $S_{N}f(x)$ is
asymptotic to $N\int_{X}f\,d\mu$ as $N\to\infty$ for $\mu$-a.e. $x\in X$. On
the other hand, if $f\not\in L^{1}(X,\mu)$ the almost sure asymptotic
behaviour of $S_{N}f$ is not described by any sequence $(b(N))$. In
particular, letting $f:X\to\mathbb{R}_{\geq 0}$, one can apply [A97, Cor.
2.3.4] to show that given a sequence $(b(N))$ of positive numbers with
$b(N)/N\to\infty$, for $\mu$-almost every (a.e.) $x\in X$ either
$\limsup_{N}S_{N}f(x)/b(N)=\infty$ or $\liminf_{N}S_{N}f(x)/b(N)=0$.

The situation is different if the measure $\mu$ is $\sigma$-finite but
infinite, that is $\mu(X)=\infty$. In this case if $f\in L^{1}(X,\mu)$ then
$S_{N}f(x)=o(N)$ for $\mu$-a.e. $x\in X$, but again the almost sure asymptotic
behaviour of $S_{N}f$ is not described by any sequence $(b(N))$ slower than
$N$. It is the content of [A97, Thm. 2.4.2] that given a sequence $(b(N))$ of
positive numbers, even with $b(N)/N\to 0$, for $\mu$-a.e. $x\in X$ either
$\liminf_{N}S_{N}f(x)/b(N)=0$ for all $f\in L^{1}(X,\mu)$, or there exists a
sequence $(N_{k})$ such that $\lim_{k}S_{N_{k}}f(x)/b(N_{k})=\infty$ for all
$f\in L^{1}(X,\mu)$. For infinite measure-preserving dynamical systems, an
information about the almost sure asymptotic behaviour of the Birkhoff sums of
summable observables is given in Hopf’s Ratio Ergodic Theorem (see [A97, Thm
2.2.5]). It is shown that for $f,g\in L^{1}(X,\mu)$ with $g\geq 0$ and
$\int_{X}g\,d\mu>0$, the ratio $S_{N}f(x)/S_{N}g(x)$ converges to
$\int_{X}f\,d\mu/\int_{X}g\,d\mu$ as $N\to\infty$ for $\mu$-a.e. $x\in X$.
That is, the almost sure asymptotic behaviour of $S_{N}f$ is the same for all
$f\in L^{1}(X,\mu)$, and its variability which leads to the negative results
we have recalled above, depends on the initial condition $x\in X$. As far as
we know, the almost sure asymptotic behaviour of Birkhoff sums has not been
studied for non-summable observables in the context of infinite ergodic
theory, with the sole exception of [LM18].

In this paper we restrict our attention to the case that $\mu$ is infinite and
study the almost sure asymptotic behaviour of Birkhoff sums for both summable
and non-summable observables with respect to sequences. We consider dynamical
systems with strong mixing assumptions, namely _$\psi$ -mixing_ (see
Definition 2.1), for the induced map on a finite measure subset $E\subset X$
(see Section 2 for the detailed assumptions), and prove two main results.
First we prove that for $f\in L^{1}(X,\mu)$ there exist sequences of positive
real numbers $(\alpha(N))$ and $(m(N,E,x))$, the second sequence depending
additionally on $x$, such that $S_{N+m(N,E,x)}f(x)$ is asymptotic to
$\alpha(N)\int_{X}f\,d\mu$ as $N\to\infty$ for $\mu$-a.e. $x\in X$. That is to
obtain the same asymptotic behaviour for $\mu$-a.e. $x\in X$ we have to change
the number of terms to consider in the Birkhoff sums (see Theorem 2.3). Then
we prove that for a non-summable observable $f:X\to\mathbb{R}_{\geq 0}$, under
suitable assumptions on its induced version $f^{E}$ (see (2.12)) it is
possible to find a sequence $(G(N))$ which depends on $f$ such that
$S_{N}f(x)\sim G(N)$ as $N\to\infty$ for $\mu$-a.e. $x\in X$ (see Theorem
2.7). The main difference is that in this non-summable case the properties of
$f$ play an important role.

The proofs rely on the method of _trimmed sums_ (see (2.8)), that is sums from
which a number of largest entries is deleted. The method to trim sums of
independent random variables to prove (pointwise and distributional) limit
theorems which fail to hold for the untrimmed sum is long established and most
results were developed in the 80th and 90th of the last century. One generally
differentiates between different strengths of trimming, according to the
number $r$ of largest entries deleted. In the light trimming case the number
of deleted entries $r$ is constant for all $n$, while in the intermediate
trimming case the number of deleted entries $r(n)$ depends on $n$ and tends to
infinity, but with $r(n)=o(n)$.

The first result using trimming in the dynamical systems context was given by
Diamond and Vaaler [DV86] providing a strong law of large numbers for the
continued fraction digits under the use of light trimming. More precisely, if
$(a_{n}(x))$ are the coefficients of the continued fraction expansion of a
point $x\in[0,1]$, it is proved in [DV86] that
$\lim_{n\to\infty}(\sum_{k=1}^{n}a_{k}(x)-\max_{1\leq k\leq n}a_{k}(x))/(n\log
n)=\log 2$ holds for Lebesgue-a.e. $x\in[0,1]$. Hence in this case it is
enough to trim the sum of the coefficients $(a_{n}(x))$ by deleting only the
largest entry, so $r=1$. These results were generalised by Aaronson and Nakada
[AN03] giving strong laws of large numbers under light trimming for
sufficiently fast $\psi$-mixing random variables, thus giving the analog to
some statement for independent random variables from [KM92]. Haynes further
quantified in [H14] the results by Aaronson and Nakada by giving precise error
terms.

Kesseböhmer and the second author of this paper proved strong laws of large
numbers under intermediate trimming using a spectral gap property of the
transfer operator, see [KS19b] and [KS20a] for an application of these results
to subshifts of finite type and [HM87, H93, KS19a] for these and further
reaching results in the independent case. See further [KS20b] for a
convergence in mean result concerning the same intermediately trimmed sums.

The paper is organised as follows. In Section 2 we describe the setting we
consider and state the main results, Theorems 2.3 and 2.7, whose proofs are
collected in Section 4. In Section 3 we describe two classes of dynamical
systems to which our results are applicable, and discuss the assumptions on
the observables for Theorem 2.7. We also study in more detail our results for
the Farey map, a well-known dynamical system on the interval, also in relation
to the results in [LM18] which are shown to be a particular case of ours for
this system. Finally, two appendices complete the paper. In particular
Appendix B contains definitions and results on slowly varying functions, an
important class of functions used in the statement of Theorem 2.7.

We conclude with some comments and a discussion on possible directions of
future research. As we have stated above, our main results hold for dynamical
systems for which there exists a finite measure set $E$ on which the induced
map satisfies a $\psi$-mixing condition. It would be interesting to study if
this $\psi$-mixing condition could be relaxed. This seems to be highly
unlikely in general. Aaronson and Nakada already gave in [AN03] an example of
a mixing though not $\psi$-mixing system which does not fulfil a strong law of
large numbers after trimming, even though a sufficiently fast $\psi$-mixing
system with the same distribution functions would fulfil a trimmed strong law.
Haynes gave another simple example with even stronger mixing properties which
does not fulfil a lightly trimmed strong law, see [H14, Thm. 4]. However, this
particular system still fulfils an intermediately trimmed strong law, see
[S18]. Thus, there might be possibilities to generalise Theorems 2.3 and 2.7
in this way. In particular, in this paper we have considered assumptions on
the system $(X,T,\mu)$ for which light trimming is enough to get the results,
actually it has been enough to trim the sums by deleting only the largest
entry. As shown in Section 3.1 for maps of the interval with indifferent fixed
points, this is related with the order with which the derivative of the map
converges to 1 in the indifferent fixed points. Using light trimming we have
been able to consider only the so-called “barely infinite” situation. It is
possible that we can obtain similar results for systems with a different
“order of infinity” by using intermediate trimming.

##  2\. The setting and main results

Let $T:X\to X$ be a conservative ergodic measure-preserving transformation of
the measure space $(X,{\mathcal{B}},\mu)$ with $\mu$ a $\sigma$-finite measure
with $\mu(X)=\infty$. Let $E\in{\mathcal{B}}$ with $\mu(E)=1$, and denote by
$\varphi_{{}_{E}}$ the _first return time_

$\varphi_{{}_{E}}:E\to\mathbb{N}\,,\qquad\varphi_{{}_{E}}(x):=\min\left\\{k\geq
1\,:\,T^{k}(x)\in E\right\\}.$

The first return time is finite for $\mu$-a.e. $x\in E$, so we define
$T_{{}_{E}}(x):=T^{\varphi_{{}_{E}}(x)}(x)$ to be the _induced map_
$T_{{}_{E}}:E\to E$, which is an ergodic measure-preserving transformation of
the probability space $(E,{\mathcal{B}}|_{E},\mu)$. The first return time
generates two families of sets, the level sets

(2.1) $A_{n}:=\left\\{x\in E\,:\varphi_{{}_{E}}(x)=n\right\\}$

which are a measurable partition of $E$, and the super-level sets

(2.2) $A_{>n}:=\left\\{x\in
E\,:\,\varphi_{{}_{E}}(x)>n\right\\}=\bigsqcup_{k>n}\,A_{k}.$

We also use the notation $A_{\geq n}=A_{n}\cup A_{>n}$. Applying Kac’s Theorem
one has

(2.3) $\sum_{k\geq 1}\,k\,\mu(A_{k})=\sum_{n\geq
0}\,\mu(A_{>n})=\mu(X)=\infty.$

For this reason the vanishing order of the sequence $(\mu(A_{>n}))$ is called
the _order of infinity_ of the system $(X,T,\mu)$. It is known that this order
is independent of the finite measure subset $E$ we have fixed.

A useful notion is the _longest excursion out of $E$ beginning in the first
$N$-steps_ defined for $\mu$-a.e. $x\in X$ as

(2.4) $m(N,E,x):=1+\max\left\\{k\geq
1\,:\,\exists\,\ell\in\\{1,\dots,N+1\\}\text{ s.t. }T^{\ell+j}(x)\not\in
E,\,\forall\,j=0,\dots,k-1\right\\}.$

If we denote by $R_{{}_{E,N}}(x)$ the _number of visits to $E$ up to time $N$_
along the orbit of a point $x$, that is

(2.5) $R_{{}_{E,N}}(x):=\sum_{k=1}^{N+1}\,(\mathbbm{1}_{{}_{E}}\circ
T^{k-1})(x)$

it is immediate to verify that for $\mu$-a.e. $x\in E$

(2.6) $m(N,E,x)=\max\left\\{(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,:\,k=1,\dots,R_{{}_{E,N}}(x)\right\\}.$

A notion related to $m(N,E,x)$ is that of the _longest excursion out of $E$
seen up to time $N$,_ defined for $\mu$-a.e. $x\in E$ by

(2.7) $w(N,E,x):=1+\max\left\\{k\geq
1\,:\,\exists\,\ell\in\\{1,\dots,N-k\\}\text{ s.t. }T^{\ell+j}(x)\not\in
E,\,\forall\,j=0,\dots,k-1\right\\}.$

Notice that $w(N,E,x)$ can correspond to an excursion which does not return to
$E$ up to time $N$.

In this paper we are interested in studying the pointwise asymptotic behaviour
of the Birkhoff sums

$S_{N}f(x):=\sum_{n=1}^{N}\,(f\circ T^{n-1})(x)$

for summable and non-summable observables $f:X\to\mathbb{R}$. For $f\in
L^{1}(X,\mu)$ it is a straightforward consequence of Hopf’s Ratio Ergodic
Theorem [A97, Thm 2.2.5] that $\mu$-a.s. $S_{N}f(x)=o(N)$. Moreover, the exact
asymptotic pointwise behaviour of $S_{N}f$ cannot be recovered for all $f\in
L^{1}(X,\mu)$ by changing the normalising sequence due to Aaronson’s Ergodic
Theorem [A97, Thm 2.42], which basically states that for any sequence of
positive real numbers the growth rate of the Birkhoff sums of a non-negative
summable observable will almost surely be either over- or under-estimated
infinitely often. In this paper we obtain some results for the pointwise
asymptotic behaviour of $S_{N}f$ under strong mixing assumptions for the
induced system $(E,T_{{}_{E}},\mu)$.

We now state the definitions and the results we need in the general case of
random variables. We refer to [B05] for more definitions.

######  Definition 2.1.

Let $(Y_{n})$ be a sequence of random variables on a probability space
$(Y,\mathbb{P})$, and let ${\mathcal{F}}_{h}^{k}$, for $0\leq h<k\leq\infty$,
be the $\sigma$-field generated by $(Y_{n})_{h\leq n\leq k}$. The sequence
$(Y_{n})$ is _$\psi$ -mixing_ if

$\psi(n):=\sup\left\\{\Big{|}\frac{\mathbb{P}(B\cap
C)}{\mathbb{P}(B)\mathbb{P}(C)}-1\Big{|}\,:\,B\in{\mathcal{F}}_{0}^{j},\,C\in{\mathcal{F}}_{j+n}^{\infty},\,\mathbb{P}(B)>0,\,\mathbb{P}(C)>0,\,j\in\mathbb{N}\right\\}$

satisfies $\psi(n)\to 0$ as $n\to\infty$.

We now recall the notion of _lightly trimmed sums_. Given a sequence $(Y_{n})$
of random variables on a probability space $(Y,\mathbb{P})$ and a point $y\in
Y$, for each $N\in\mathbb{N}$ we choose a permutation $\pi$ of
$\\{1,2,\dots,N\\}$ such that $Y_{\pi(1)}(y)\geq Y_{\pi(2)}(y)\geq\dots\geq
Y_{\pi(N)}(y)$. For a given $r\in\mathbb{N}_{0}$, the lightly trimmed sum of
the $(Y_{n})$ is defined by

(2.8) $S_{N}^{r}(y):=\sum_{n=r+1}^{N}\,Y_{\pi(n)}(y)$

that is the sum of the first $N$ random variables trimmed by the largest $r$
entries.

The next result is adapted from [AN03] as shown in Appendix A.

######  Lemma 2.2.

Let $(Y_{n})$ be a sequence of non-negative, identically distributed random
variables on a probability space $(Y,\mathbb{P})$ which is $\psi$-mixing with
coefficient $\psi(n)$ fulfilling $\sum_{n\geq 1}\psi(n)/n<\infty$. Let $F$ be
the distribution function of the $(Y_{n})$, that is $F(y)=\mathbb{P}(Y_{1}\leq
y)$, and let for some $y_{0}>0$

(2.9) $\displaystyle
W\coloneqq\min\left\\{r\in\mathbb{N}\colon\int_{y_{0}}^{\infty}\left(\frac{y\left(1-F(y)\right)}{\int_{0}^{y}\left(1-F(t)\right)\mathrm{d}t}\right)^{r+1}\frac{1}{y}\,\mathrm{d}y<\infty\right\\},$

where we set the $\min$ as $\infty$ if such an $r$ does not exist. Then there
exists a sequence $(d(N))$ such that

$\displaystyle\lim_{N\to\infty}\frac{S_{N}^{W}(y)}{d(N)}=1,\quad\text{$\mathbb{P}$-a.s.}$

If we set $a(y):=y/\int_{0}^{y}(1-F(t))\mathrm{d}t$, then $d(n)$ can be set as
the inverse function of $a(n)$.

Furthermore, if we denote by $M^{r}_{N}(y)$ the $r$-th maximum in
$\\{Y_{1},\ldots,Y_{N}\\}$ then

(2.10)
$\lim_{N\to\infty}\frac{M^{r}_{N}(y)}{d(N)}=0,\quad\text{$\mathbb{P}$-a.s.}$

for all $r>W$.

We now state our main results. We first consider the Birkhoff sums of an
observable $f\in L^{1}(X,\mu)$. We prove that $S_{N}f$ converges almost surely
to the spatial mean of the observable if we consider the average with respect
to a sequence $\alpha(n)$ growing slower than $n$ and consider more iterates
for the observable. The dependence on the point $x$ is given by the number
$m(N,E,x)$, as defined in (2.4), of iterates for the observable that we have
to add. We remark that the sequence $\alpha(n)$ coincides with that found in
the Weak Law of Large Numbers for dynamical system which are pointwise dual
ergodic and have wandering rate given by a slowly varying sequence (see [A97,
Chapter 3]).

######  Theorem 2.3.

Let $T:X\to X$ be a conservative ergodic measure-preserving transformation of
the measure space $(X,{\mathcal{B}},\mu)$ with $\mu$ a $\sigma$-finite measure
with $\mu(X)=\infty$, and let $E\in{\mathcal{B}}$ with $\mu(E)=1$. Let’s
assume that the first return time to $E$ satisfies:

  * (i)

the sequence of random variables $(\varphi_{{}_{E}}\circ T_{{}_{E}}^{n-1})$
with $n\geq 1$ defined on $(E,\mu)$ is $\psi$-mixing with coefficient
$\psi(n)$ fulfilling $\sum_{n\geq 1}\psi(n)/n<\infty$;

  * (ii)

the super-level sets of $\varphi_{{}_{E}}$ satisfy

$\sum_{n\geq
1}\frac{n\,(\mu(A_{>n}))^{2}}{(\sum_{j=0}^{n-1}\mu(A_{>j}))^{2}}<\infty.$

Then for all $f\in L^{1}(X,\mu)$ and for $\mu$-a.e. $x\in X$

$\lim_{N\to\infty}\,\frac{1}{\alpha(N)}\,\sum_{k=1}^{N+m(N,E,x)}\,(f\circ
T^{k-1})(x)=\int_{X}\,f\,\mathrm{d}\mu$

where

(2.11) $\alpha(n):=\frac{n}{\sum_{j=0}^{n-1}\,\mu(A_{>j})}\,.$

In the following we also give a slightly different formulation of Theorem 2.3
which uses $w(N,E,x)$. The main difference is that the number of iterates for
the observable is fixed and the dependence on $x$ is found in the
renormalising sequence. Thus, this second formulation is more similar to
Hopf’s Ratio Ergodic Theorem in the spirit.

######  Theorem 2.4.

Under the same assumptions as in Theorem 2.3, for all $f\in L^{1}(X,\mu)$ and
for $\mu$-a.e. $x\in X$ it holds

$\lim_{N\to\infty}\,\frac{1}{\alpha(N-w(N,E,x))}\,\sum_{k=1}^{N}\,(f\circ
T^{k-1})(x)=\int_{X}\,f\,\mathrm{d}\mu$

where $w(N,E,x)$ is defined in (2.7) and $\alpha(n)$ is given in (2.11).

######  Remark 2.5.

The asymptotic behaviour of the sequence $(w(N,E,x))$ has been considered in
[GHPZ] for some intermittent maps of the interval with infinite measure.
Following the same argument and using the relations between $m(N,E,x)$ and
$w(N,E,x)$ one can prove that under the conditions of the two previous
theorems

$\lim_{N\to\infty}\,\frac{\log m(N,E,x)}{\log N}=\lim_{N\to\infty}\,\frac{\log
w(N,E,x)}{\log N}=1$

for $\mu$-a.e. $x\in X$.

We have shown that in the case of summable observables the assumptions only
depend on the set $E$ and on its first return time. This is in accordance with
what one could expect with Hopf’s Ratio Ergodic Theorem in mind. We now study
the Birkhoff sums of non-negative observables $f\not\in L^{1}(X,\mu)$. In this
case we need assumptions also on the observable itself.

Given $T:X\to X$ and $E\in{\mathcal{B}}$ with $\mu(E)=1$ as above, for a
measurable function $f:X\to\mathbb{R}$, a standard approach is to introduce
the induced observable $f^{E}$ defined on the full measure set of points $x\in
E$ with finite first return time by

(2.12) $f^{E}(x):=\sum_{k=1}^{\varphi_{{}_{E}}(x)}\,(f\circ T^{k-1})(x)\,.$

The induced observable simply gives the contribution to the Birkhoff sums
$S_{N}f$ of the excursions out of $E$ for the orbit of a point $x$. For
$\mu$-a.e. $x\in E$ let’s define the time $\tau_{{}_{E,x}}(N)$ of the $N$-th
return to $E$, which corresponds to the Birkhoff sum of $\varphi_{{}_{E}}$ for
the system $(E,T_{{}_{E}})$. We have

(2.13) $\tau_{{}_{E,x}}(N):=\sum_{k=1}^{N}\,(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,.$

Recalling now the notion $R_{{}_{E,N}}(x)$ of the number of visits to $E$ up
to time $N$ along the orbit of $x$ defined in (2.5), the last visit to $E$ for
$x$ up to time $N$ is $T_{{}_{E}}^{R_{{}_{E,N}}(x)-1}(x)$ and the time of this
last visit is then given by $\tau_{{}_{E,x}}(R_{{}_{E,N}}(x)-1)\leq N$. Then
the following relation follows for $\mu$-a.e. $x\in E$

(2.14) $S_{N}f(x)=\sum_{n=1}^{R_{{}_{E,N}}(x)-1}\,(f^{E}\circ
T^{n-1}_{{}_{E}})(x)+\sum_{k=\tau_{{}_{E,x}}(R_{{}_{E,N}}(x)-1)+1}^{N}\,(f\circ
T^{k-1})(x)$

with the standard convention that a summation vanishes if the upper index is
smaller than the lower index. Moreover, using that $f\geq 0$ and
$\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))>N$, being $\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))$
the time of the first visit to $E$ after time $N$, we immediately obtain

$\sum_{n=1}^{R_{{}_{E,N}}(x)-1}\,(f^{E}\circ T^{n-1}_{{}_{E}})(x)\leq
S_{N}f(x)\leq\sum_{n=1}^{R_{{}_{E,N}}(x)}\,(f^{E}\circ T^{n-1}_{{}_{E}})(x)$

for $\mu$-a.e. $x\in E$. Thus the pointwise convergence of the Birkhoff sums
of $f$ could in principle be obtained by the behaviour of the Birkhoff sums of
$f^{E}$ for the induced system $(E,T_{{}_{E}},\mu)$. However $f\not\in
L^{1}(X,\mu)$ implies $f^{E}\not\in L^{1}(E,\mu)$ (see for examples [Z09,
Lemma 2]), hence we cannot use Birkhoff’s Theorem to obtain the pointwise
convergence for the Birkhoff sums of $f^{E}$. A solution is to apply Lemma 2.2
to the sequence of random variables $(f^{E}\circ T_{{}_{E}}^{n-1})$.

The simplest case to consider is when $f^{E}$ is constant on the level sets
$A_{n}$, which corresponds to $f$ constant on the level sets of the _hitting
time function_

(2.15) $h_{{}_{E}}:X\to\mathbb{N}_{0}\,,\quad h_{{}_{E}}(x):=\min\\{k\geq
0\,:\,T^{k}(x)\in E\\}$

which is defined and finite $\mu$-a.e. on $X$. Setting

(2.16) $E_{n}:=\\{x\in X\,:\,h_{{}_{E}}(x)=n\\}$

with $E_{0}=E$, for the level sets of $h_{{}_{E}}$, if $f(x)=f_{k}$ for all
$x\in E_{k}$ then

(2.17) $f^{E}(x)=\sum_{k=0}^{n-1}\,f_{k}\,,\quad\forall\,x\in A_{n}\,.$

We recall that $\mu(A_{>n})=\mu(E_{n})$ for all $n\geq 0$ (see [Z09, Lemma
1]).

In the proof of the main result of this section, we use the following property
for the waiting times between two consecutive returns to $E$.

######  Lemma 2.6.

Let us consider the function $q:\mathbb{R}_{>0}\to\mathbb{N}$ given by

$q(t):=\min\left\\{j\in\mathbb{N}\,:\,\mu(A_{>j})\leq\frac{1}{t}\right\\}$

for the super-level sets $(A_{>n})$ defined in (2.2). Then

$\mu\left(\left\\{x\in E\,:\,\max_{1\leq k\leq
R_{{}_{E,n}}(x)}\,(\varphi_{{}_{E}}\circ T_{{}_{E}}^{k-1})(x)\geq
n\,\xi(n)\quad\text{for infinite $n$}\right\\}\right)=0$

where

$\xi(n):=\frac{1}{n}\,q\Big{(}\alpha(n)\,\log(\alpha(n))\log^{2}(\log(\alpha(n)))\Big{)}$

and $\alpha(n)$ is given in (2.11).

######  Theorem 2.7.

Let $T:X\to X$ be a conservative ergodic measure-preserving transformation of
the measure space $(X,{\mathcal{B}},\mu)$ with $\mu$ a $\sigma$-finite measure
with $\mu(X)=\infty$, and let $E\in{\mathcal{B}}$ with $\mu(E)=1$. Let
$f:X\to\mathbb{R}_{\geq 0}$ be a non-negative measurable function such that
$f\not\in L^{1}(X,\mu)$ and let $f^{E}$ be its induced version on $E$ defined
in (2.12). We assume that:

  * (i)

the sequence of random variables $(\varphi_{{}_{E}}\circ T_{{}_{E}}^{n-1})$
with $n\geq 1$ defined on $(E,\mu)$ is $\psi$-mixing with coefficient
$\psi(n)$ fulfilling $\sum_{n\geq 1}\psi(n)/n<\infty$;

  * (ii)

there exist $g_{1},g_{2}:X\to\mathbb{R}_{\geq 0}$ constant on the level sets
$(E_{n})$ of the hitting time function to $E$ such that:

    * (ii)-(a)

$g_{1}(x)\leq f(x)\leq g_{2}(x)$ for $\mu$-a.e. $x\in X$;

    * (ii)-(b)

the induced functions satisfy $g_{i}^{E}(n):=g_{i}^{E}|_{A_{n}}=n\,L_{i}(n)$
for $i=1,2$, where $L_{i}(n)$ are normalised slowly varying function (see
(B.2)), and $L_{1}(n)\sim L_{2}(n)$ as $n\to\infty$;

    * (ii)-(c)

given the function $\xi(n)$ defined in Lemma 2.6, for all functions
$\tilde{\xi}(n)$ with $c\leq\tilde{\xi}(n)\leq\xi(n)$ for some constant $c>0$,
the functions $L_{i}$ for $i=1,2$ satisfy $L_{i}(n\,\tilde{\xi}(n))\sim
L_{i}(n)$ as $n\to\infty$;

    * (ii)-(d)

if $\Gamma_{i}(n):=\min\\{k\in\mathbb{N}\,:\,g_{i}^{E}(k)>n\\}$ for $i=1,2$,
then

(2.18)
$\int_{0}^{\infty}\frac{y\,\left(\mu(A_{\geq\Gamma_{i}(y)})\right)^{2}}{\left(\int_{0}^{y}\,\mu(A_{\geq\Gamma_{i}(t)})\mathrm{d}t\right)^{2}}\mathrm{d}y<\infty\,;$

    * (ii)-(e)

if $\ell_{i}(n):=\sum_{k=0}^{n}\,\mu(A_{\geq\Gamma_{i}(k)})$, then
$\ell_{i}(n)$ for $i=1,2$ is super-slowly varying at infinity with rate
function itself (see Definition B.5) and

(2.19) $\ell_{i}\Big{(}\alpha(N)\Big{)}\sim
L_{i}(N)\,\sum_{k=0}^{N}\,\mu(A_{>k})\,,$

as $N\to\infty$ for $i=1,2$, where $\alpha(n)$ is defined in (2.11).

Then there exists a function $G:\mathbb{N}\to\mathbb{R}_{\geq 0}$ which is
asymptotically equivalent to $f^{E}$, that is there exists a set
$\tilde{E}\subset E$ with $\mu(\tilde{E})=1$ such that for every sequence
$(x_{n})$ with $x_{n}\in A_{n}\cap\tilde{E}$ for all $n\in\mathbb{N}$ we have
$f^{E}(x_{n})\sim G(n)$ as $n\to\infty$, and for $\mu$-a.e. $x\in X$

$\lim_{N\to\infty}\,\frac{1}{G(N)}\,\sum_{k=1}^{N}\,(f\circ T^{k-1})(x)=1\,.$

In Section 3.3 we discuss the assumptions of Theorem 2.7 and consider
particular cases for the observable $f$ and specific dynamical systems.

##  3\. Examples and applications

We first describe two classes of dynamical systems to which our results apply.
Then we discuss the assumptions and the result of Theorem 2.7. Some proofs of
statements in this section are collected in Section 5 to ease the reading of
the paper.

###  3.1. Maps of the interval with an indifferent fixed point

This class of maps is considered the easiest example of non-uniformly
hyperbolic systems, and thus the easiest model of a system with “anomalous”
dynamical phenomena. Here we use the general approach introduced in [Z00].

Let $T:[0,1]\to[0,1]$ be a piecewise monotonic $C^{2}$ map with respect to a
partition ${\mathcal{P}}$ of open subintervals, that is ${\mathcal{P}}$ is a
finite or countable family of open subintervals $(P_{j})_{j\in J}$ such that
$m([0,1]\setminus\cup_{j}P_{j})=0$, being $m$ the Lebesgue measure, and
$T|_{P_{j}}$ is $C^{2}$ and strictly monotonic for all $j\in J$.

######  Definition 3.1.

A map $T:[0,1]\to[0,1]$ which is piecewise monotonic and $C^{2}$ with respect
to a partition ${\mathcal{P}}=(P_{j})_{j\in J}$ of open subintervals defines a
_basic AFN-system_ if it is conservative and ergodic with respect to $m$ and
if

  * (A)

$T^{\prime\prime}/(T^{\prime})^{2}$ is bounded on $\cup_{j}P_{j}$;

  * (F)

$T{\mathcal{P}}=\\{T(P_{j})\,:\,j\in J\\}$ is a finite collection;

  * (N)

there exists a non-empty finite collection
$\tilde{\mathcal{P}}\subset{\mathcal{P}}$ such that each
$P_{j}\in\tilde{\mathcal{P}}$ has an _indifferent fixed point_ $x_{j}$, that
is

$\lim_{x\to x_{j},\,x\in P_{j}}\,T(x)=x_{j}\quad\text{and}\quad
T^{\prime}(x_{j}):=\lim_{x\to x_{j},\,x\in P_{j}}\,T^{\prime}(x)=1\,,$

which is a _one-sided regular source_ , that is $T^{\prime}$ decreases on
$P_{j}\cap(0,x_{j})$ and increases on $P_{j}\cap(x_{j},1)$. Moreover $T$ is
uniformly expanding away from the indifferent fixed points, that is for all
$\epsilon>0$ there exists $\rho_{\epsilon}>1$ such that

$|T^{\prime}(x)|\geq\rho_{\epsilon}\,,\quad\forall\,x\in[0,1]\setminus\bigcup_{P_{j}\in\tilde{\mathcal{P}}}\,P_{j}\cap(x_{j}-\epsilon,x_{j}+\epsilon)\,.$

By [Z00, Thm A] a basic AFN-system admits an absolutely continuous invariant
measure $\mu$ with a lower semicontinuous density $h(x)$ of the form

$h(x)=c(x)\,g(x)\,,\quad\text{with}\quad
g(x)=\left\\{\begin{array}[]{ll}\frac{x-x_{j}}{x-(T|_{P_{j}})^{-1}(x)}\,,&\text{for
}\,x\in P_{j}\in\tilde{\mathcal{P}}\\\\[5.69046pt] 1\,,&\text{for }\,x\in
P_{j}\not\in\tilde{\mathcal{P}}\end{array}\right.$

and $c(x)$ satisfying $0<C^{-1}\leq c(x)\leq C$ for some constant $C>0$.

Examples of basic AFN-systems are the Pomeau-Manneville maps, the Liverani-
Saussol-Vaienti maps and the Farey map. They all share the following
structure. Let ${\mathcal{T}}$ be the family of maps of $[0,1]$ for which
there is a partition ${\mathcal{P}}=\\{P_{0},P_{1}\\}$ of $[0,1]$ into two
open intervals with respect to which they are piecewise monotonic and $C^{2}$,
and $T(P_{0})=T(P_{1})=(0,1)$. Moreover the collection $\tilde{\mathcal{P}}$
contains only the set $P_{0}=(0,\bar{x})$, and the maps have an indifferent
fixed point at $x_{0}=0$ at which they satisfy

(3.1) $T(x)=x+C\,x^{1+p}+o(x^{1+p})\,,\quad\text{for $x\in P_{0}$ and $x\to
0^{+}$}$

with $p\geq 1$ for some constant $C>0$, and are uniformly expanding away from
$x_{0}$.

By [Z00, Lemma 8] there is a set $E\subset[0,1]$ with $\mu(E)<\infty$ such
that the induced map $T_{{}_{E}}$ is piecewise monotonic and $C^{2}$ with
respect to a partition ${\mathcal{P}}_{E}$, satisfies conditions (A) and (F),
and is uniformly expanding (U). It is then called an AFU-system. Finally
[AN05, Thm 1] gives that if $T_{{}_{E}}$ is weakly mixing and $\inf_{n\geq
1,Q\in{\mathcal{P}}^{n}_{E}}m(T_{{}_{E}}^{n}(Q))>0$, then the sequence of
random variables $(\varphi_{{}_{E}}\circ T_{{}_{E}}^{n-1})$ with $n\geq 1$
defined on $(E,\mu)$ is $\psi$-mixing with coefficient $\psi(n)$ fulfilling
$\psi(n)\leq K\,\theta^{n}$ for some constants $K>0$ and $\theta\in(0,1)$.
Thus assumption (i) of Theorems 2.3 and 2.7 is satisfied.

For the maps in the family ${\mathcal{T}}$ one can choose $E=P_{1}$ to obtain
an exact AFU-system $T_{{}_{E}}$ which satisfies the assumptions of [AN05, Thm
1].

Moreover we need information on the super-level sets $(A_{>n})$.

######  Definition 3.2.

We say that a basic AFN-system $T$ admits _regularly varying nice expansions_
if for all $P_{j}\in\tilde{\mathcal{P}}$ there are $p_{j}\geq 1$ and $L_{j}$
slowly varying functions at infinity (see Appendix B) such that

$T(x)\sim
x+\left|x-x_{j}\right|^{1+p_{j}}\,\left(L_{j}\left(1/\left|x-x_{j}\right|\right)\right)^{-1}\,,\quad\text{for
$x\in P_{j}$ and $x\to x_{j}$.}$

Using then computations similar to those in [T83, Lemma 2], one can show that
a basic AFN-system with regularly varying nice expansions satisfies assumption
(ii) of Theorem 2.3 if $p:=\max_{P_{j}\in\tilde{\mathcal{P}}}p_{j}=1$ and
$L(y):=\max_{P_{j}\in\tilde{\mathcal{P}}}|L_{j}(y)|$ satisfies

(3.2)
$\int_{1}^{\infty}\,\left(\frac{L(y)}{\int_{1}^{y}\frac{L(t)}{t}\mathrm{d}t}\right)^{2}\,\frac{1}{y}\,\mathrm{d}y<\infty\,.$

Indeed if $p=1$ the super-level sets satisfy $\mu(A_{>n}\cap
P_{j})\sim|L_{j}(n)|/n$ as $n\to\infty$ for all $P_{j}\in\tilde{\mathcal{P}}$,
so that by (3.2) assumption (ii) is satisfied.

Maps in ${\mathcal{T}}$ have $L_{0}(y)\equiv C$, so that the previous
condition is easily satisfied.

###  3.2. Fibred systems related to multi-dimensional continued fraction
expansions

The class of systems in dimension greater than one for which the $\psi$-mixing
property has been studied includes the so-called _fibred systems_ on compact
and connected subsets $E$ of $\mathbb{R}^{d}$.

The notion of fibred system was introduced in [S95] in relation to the theory
of real number expansions. Here we use results from [NN03]. We first recall
the definition of a fibred system.

######  Definition 3.3.

Let $E$ be a compact and connected subset of $\mathbb{R}^{d}$, with the Borel
$\sigma$-algebra ${\mathcal{B}}$ and let $m$ denote the $d$-dimensional
normalised Lebesgue measure on $E$. Let $V$ be a measurable map of $E$ onto
itself. The pair $(E,V)$ is called a _fibred system_ if it satisfies the
following properties.

  * (h1)

There exists a finite or countable measurable partition
${\mathcal{C}}=\\{C_{j}\\}_{j\in J}$ of $E$ into open sets such that the
restriction of $V$ to $C_{j}$ is injective for all $j\in J$.

  * (h2)

The map $V$ is measurable and non-singular. To simplify the next conditions we
assume that $V|_{C_{j}}$ is differentiable for all $j\in J$.

For $j\in J$, we denote by $\phi_{j}$ the inverse of the restriction
$V|_{C_{j}}$. Given the iterated partition
${\mathcal{C}}^{n}=\bigvee_{k=0}^{n-1}\,V^{-k}{\mathcal{C}}$, we denote by
$\phi_{j_{1},\,\dots,\,j_{n}}$ the local inverse of $V^{n}$ restricted to
$C_{j_{1},\,\dots,\,j_{n}}\in{\mathcal{C}}^{n}$.

  * (h3)

There exists a sequence $(\sigma(n))_{n\geq 0}$ with $\sigma(n)\rightarrow 0$
as $n\rightarrow\infty$ and such that

$\sup_{(j_{1},\,\dots,\,j_{n})\in
J^{n}}\,\operatorname*{diam}C_{j_{1},\,\dots,\,j_{n}}\leq\sigma(n).$

  * (h4)

There exist a finite number of measurable subsets $U_{1},\,\ldots,\,U_{N}$ of
$E$ such that for any cylinder $C_{j_{1},\,\dots,\,j_{n}}$ of positive
measure, there exists $U_{i}$ with $1\leq i\leq N$ such that
$V^{n}(C_{j_{1},\,\dots,\,j_{n}})=U_{i}$ up to measure-zero sets.

  * (h5)

There exists a constant $\lambda\geq 1$ such that

$\operatorname*{ess\,sup}_{V^{n}(C_{j_{1},\,\dots,\,j_{n}})}|D\phi_{j_{1},\,\dots,\,j_{n}}|\leq\lambda\,\operatorname*{ess\,inf}_{V^{n}(C_{j_{1},\,\dots,\,j_{n}})}|D\phi_{j_{1},\,\dots,\,j_{n}}|$

where $D\phi_{j_{1},\dots,j_{n}}$ denotes the Jacobian determinant of
$\phi_{j_{1},\,\dots,\,j_{n}}$.

  * (h6)

For any $1\leq i\leq N$, $U_{i}$ contains a proper cylinder.

  * (h7)

There is a constant $r_{1}>0$ such that

$\left|D\phi_{j_{1},\,\dots,\,j_{n}}(p_{1})-D\phi_{j_{1},\,\dots,\,j_{n}}(p_{2})\right|\leq
r_{1}m(C_{j_{1},\,\dots,\,j_{n}})\|p_{1}-p_{2}\|$

for any $p_{1},p_{2}\in U_{i}$ and all $i$.

  * (h8)

There is a constant $r_{2}>0$ such that

$\left\|\phi_{j_{1},\,\dots,\,j_{n}}(p_{1})-\phi_{j_{1},\,\dots,\,j_{n}}(p_{2})\right\|\leq
r_{2}\sigma(n)\|p_{1}-p_{2}\|$

for any $p_{1},p_{2}\in U_{i}$ and all $i$.

  * (h9)

Let $\mathcal{F}$ be a finite partition generated by $U_{1},\,\dots,\,U_{N}$
and denote by ${\mathcal{F}}^{c}_{\ell}$ the cylinders in
${\mathcal{C}}^{\ell}$ that are not contained in any element of $\mathcal{F}$.
Then, as $\ell\rightarrow\infty$

$\gamma(\ell)\coloneqq\sum_{C(j_{1},\,\dots,\,j_{\ell})\in{\mathcal{F}}^{c}_{\ell}}\,m(C(j_{1},\,\dots,\,j_{\ell}))\rightarrow
0.$

By [NN03, Thm 2], a fibred system $(E,V)$ is $\psi$-mixing, with respect to
the iterated partition ${\mathcal{C}}^{n}$, and there exist constants $K>0$
and $\theta\in(0,1)$ such that

(3.3) $\psi(n)\leq
K\,\left(\theta^{\sqrt{n}}+\sigma(\sqrt{n})+\gamma(\sqrt{n})\right)\,,$

where $\sigma(k)$ and $\gamma(k)$ are defined in (h3) and (h9).

Examples of fibred systems are obtained in the study of multi-dimensional
continued fractions algorithms. Here we consider the two-dimensional system
studied in [BDM21], which is related to the two-dimensional continued
fractions algorithm introduced in [G01]. Let

$\triangle\coloneqq\left\\{(x,y)\in\mathbb{R}^{2}\,:\,1\geq x\geq y\geq
0\right\\}$

and define the partition $\\{\Gamma_{0},\Gamma_{1}\\}$ of $\triangle$ by

$\Gamma_{0}\coloneqq\left\\{(x,y)\in\mathbb{R}^{2}\,:\,1\geq x\geq
y>1-x\right\\},$

and

$\Gamma_{1}\coloneqq\left\\{(x,y)\in\mathbb{R}^{2}\,:\,1-y\geq x\geq y\geq
0\right\\}.$

We consider the map $S:\triangle\rightarrow\triangle$ defined as

$S(x,y)\coloneqq\begin{cases}\left(\frac{y}{x},\,\frac{1-x}{x}\right)&\text{if
}(x,y)\in\Gamma_{0}\\\\[5.69046pt]
\left(\frac{x}{1-y},\,\frac{y}{1-y}\right)&\text{if
}\,(x,y)\in\Gamma_{1}\end{cases}.$

In [BDM21] it is proved that $S$ admits an absolutely continuous invariant
measure $\mu$ with density $h(x,y)=\frac{1}{xy}$, so that
$\mu(\triangle)=\infty$, and that the system $(\triangle,S,\mu)$ is ergodic
and conservative.

The next result shows that there exists a set $E\subset\triangle$ for which
assumption (i) of Theorem 2.3 holds. The proof of the proposition is in
Section 5.

######  Proposition 3.4.

Let $E\subset\triangle$ be the triangle with vertices
$Q_{1}=\left(\frac{1}{2},\frac{1}{2}\right)$,
$Q_{2}=\left(\frac{2}{3},\frac{1}{3}\right)$ and $Q_{3}=(1,1)$, with the sides
$Q_{1}Q_{2}$ and $Q_{2}Q_{3}$ not included. Then $\mu(E)<\infty$, the induced
map $S_{{}_{E}}$ on $E$ is $\psi$-mixing with respect to the level sets
$(A_{n})$ of the return time function $\varphi_{{}_{E}}$, and $\sum_{n\geq
1}\,\psi(n)/n<\infty$.

Finally, assumption (ii) of Theorem 2.3 is indirectly studied in [BDM21,
Appendix B] but not proved to hold. Its validity implies the wandering rate
$w_{n}(E):=\sum_{k=0}^{n}\,\mu(A_{>n})$ of $E$ to be slowly varying as
discussed in [AN03]. This property of $w_{n}(E)$ is studied in [BDM21] where
it is proved that $w_{n}(E)\asymp(\log n)^{2}$, from which one could
conjecture $\mu(A_{>n})=O(\frac{\log n}{n})$, and assumption (ii) would
follow. Unfortunately we don’t have a proof of this asymptotic behaviour for
$\mu(A_{>n})$.

###  3.3. Discussions on Theorem 2.7

As previously remarked, in the case of non-summable observables the pointwise
asymptotic behaviour of the Birkhoff sums depends on some properties of the
observable itself. We have been able to show that for $f\geq 0$ it holds
$S_{N}f(x)\sim G(N)$ for $\mu$-a.e. $x\in X$, where $G(n)$ is the asymptotic
behaviour of the induced observable $f^{E}$ on the level sets $(A_{n})$, if
$f$ can be bounded by functions $g_{1}$ and $g_{2}$ with induced functions
$g_{i}^{E}$ of the form $nL_{i}(n)$ for suitable normalised slowly varying
functions (see Appendix B for definitions and details on slowly varying
functions). The assumptions on the functions $L_{i}(n)$ concern their
asymptotic behaviour, as in assumptions (ii)-(b) and (c), and their link with
the first return time function to the set $E$, as in assumptions (ii)-(d) and
(e).

First we discuss assumptions (ii)-(b) and (c), showing sufficient conditions
for $f$ and for a slowly varying function to satisfy them. Then we discuss
assumptions (ii)-(d) and (e) in a specific example, and consider the necessity
of (ii)-(e). Further we consider the existing literature on this problem.

######  Proposition 3.5.

Let $f:X\to\mathbb{R}_{\geq 0}$ be constant on the level sets $(E_{n})$ of the
hitting time function (2.15) and assume that the induced function $f^{E}$
satisfies $f^{E}|_{A_{n}}=n\,L(n)$ for all $n\geq 0$, with $L$ slowly varying.
Then there exist $g_{1},g_{2}:X\to\mathbb{R}_{\geq 0}$ satisfying (ii)-(a) and
(b) of Theorem 2.7.

###### Proof.

Apply Lemma B.3 to $f^{E}$ to find $g_{1}^{E}$ and $g_{2}^{E}$. Then $g_{1}$
and $g_{2}$ are obtained using relation (2.17) between a function which is
constant on the level sets $(E_{n})$ and its induced function. ∎

We can thus reduce our attention to the case of a function $f$ with induced
function which is asymptotically equivalent to a function $G(n)=n\,L(n)$ with
$L$ normalised slowly varying. Hence using normalised slowly varying functions
does not reduce generality.

Let us now consider (ii)-(c). First notice that if $\xi(n)$ is bounded then
assumption (ii)-(c) is easily obtained by the properties of slowly varying
functions. Moreover, if $h(n)$ is a function for which $\xi(n)\leq h(n)$ and
assumption (ii)-(c) is satisfied for $h$, then it is immediately satisfied
also for $\xi$.

Fixed a slowly varying function $h$, in Lemma B.4 we give a sufficient
condition on slowly varying functions $L$ to satisfy $L(n\,\tilde{h}(n))\sim
L(n)$ for all $\tilde{h}$ such that $c\leq\tilde{h}(n)\leq h(n)$. Notice that
in Lemma B.4 we assume that $h$ is bounded below from 1. If $h$ does not
satisfy this assumption, simply apply the lemma to
$k(n):=\min\\{1+h_{0},h(n)\\}$ for some $h_{0}>0$, which is still slowly
varying.

To study assumption (ii)-(c) using Lemma B.4 for the function $\xi$ it is thus
sufficient to show that it is slowly varying or that there exists a slowly
varying function $h$ such that $\xi(n)\leq h(n)$. It follows from computations
in Proposition 3.7 that $\xi(n)\sim\log^{2}(\log n)$, and therefore is slowly
varying, for the class ${\mathcal{T}}$ of maps of the interval considered in
Section 3.1. For the map studied in Section 3.2 instead, it is shown in
[BDM21] that $\mu(A_{>n})=O(\frac{\log^{2}n}{n})$ and
$\alpha(n)=O(n/\log^{2}n)$, hence in Lemma 2.6 one finds
$q(t)=O(t\,\log^{2}t)$, and

$\xi(n)=O\Big{(}\log n\,\log^{2}(\log\,n)\Big{)}\,.$

Thus we estimate $\xi$ with a slowly varying function and Lemma B.4 gives a
sufficient condition on the slowly varying function $L$ to satisfy assumption
(ii)-(c). Hence if the measure of the super-level sets $(A_{>n})$ is bounded
by a regular varying function with index -1, and the sequence $\alpha(n)$
defined in (2.11) is bounded by a regular varying function with index 1, then
$\xi$ is bounded by a slowly varying function. In Remark 4.6 we show that it
is the case if (2.18) holds.

Finally we discuss the relation between the observable $f$ and the first
return time function to $E$. In particular we show that assumptions (ii)-(d)
and (e) simplify when the two limiting functions $g_{1}$ and $g_{2}$ have
induced functions which increase asymptotically linearly. The proof of the
following proposition is in Section 5.

######  Proposition 3.6.

Let $T$, $(X,{\mathcal{B}},\mu)$ and $E$ be defined as in Theorem 2.7. Let the
sequence of random variables $(\varphi_{{}_{E}}\circ T_{{}_{E}}^{n-1})$ with
$n\geq 1$ satisfy assumption (i) of Theorem 2.7. Let $f:X\to\mathbb{R}_{\geq
0}$ be a non-negative measurable function such that $f\not\in L^{1}(X,\mu)$
and let $f^{E}$ be its induced version on $E$ defined in (2.12). We assume
that:

  * (c1)

there exist $g_{1},g_{2}:X\to\mathbb{R}_{\geq 0}$ constant on the level sets
$(E_{n})$ of the hitting time function to $E$ such that $g_{1}(x)\leq f(x)\leq
g_{2}(x)$ for $\mu$-a.e. $x\in X$, and the induced functions satisfy
$g_{i}^{E}(n)\sim n$ for $i=1,2$;

  * (c2)

the super-level sets of $\varphi_{{}_{E}}$ satisfy

$\sum_{n\geq
1}\frac{n\,(\mu(A_{>n}))^{2}}{(\sum_{j=0}^{n-1}\mu(A_{>j}))^{2}}<\infty;$

  * (c3)

if $\ell(n):=\sum_{k=0}^{n}\,\mu(A_{\geq k})$, then
$\ell(N/\ell(N))\sim\ell(N)$ as $N\to\infty$.

Then for $\mu$-a.e. $x\in X$ it holds

$\lim_{N\to\infty}\,\frac{1}{N}\,\sum_{k=1}^{N}\,(f\circ T^{k-1})(x)=1\,.$

######  Corollary 3.7.

Let us consider a map of the interval $T\in{\mathcal{T}}$ with only one
indifferent fixed point at $0$ satisfying (3.1) with $p=1$ and $C=1$, and let
$\mu$ be the infinite absolutely continuous $T$-invariant measure. If
$f:X\to\mathbb{R}_{\geq 0}$ is constant on the level sets $(E_{n})$ of the
hitting time function (2.15) and the induced function $f^{E}$ satisfies
$f^{E}|_{A_{n}}\sim n$ as $n\to\infty$ then

(3.4) $\lim_{N\to\infty}\,\frac{1}{N}\,\sum_{n=1}^{N}\,(f\circ
T^{n-1})(x)=1\,,\quad\text{for $\mu$-a.e.\ $x\in[0,1]$.}$

###### Proof.

This is a special case of Proposition 3.6. Assumption (i) of Theorem 2.7 holds
as discussed in Section 3.1, and (c1) follows by definition of the observable
$f$. Finally for $T\in{\mathcal{T}}$ with $p=1$ and $C=1$ it holds
$\mu(A_{>n})\sim\frac{1}{n}$. Thus $\ell(n)\sim\log n$ and (c2)-(c3) hold
true. ∎

In order to not only considering the trivial case of $G(n)\sim n$, we give a
more involved example generalising Corollary 3.7.

######  Example 3.8.

Let $(X,\mathcal{B},\mu)$ and $E\in\mathcal{B}$ be a system fulfilling (i) of
Theorem 2.7 with $\mu(A_{>n})\sim\log^{b}(\log n)/n$ as $n\to\infty$ for
$b\in\mathbb{R}$. Furthermore we assume $f:X\to\mathbb{R}_{\geq 0}$ to be
constant on the level sets $(E_{n})$ of the hitting time function (2.15) and
let the induced function $f^{E}$ satisfy $f^{E}|_{A_{n}}\sim n\,\log^{c}(\log
n)$ as $n\to\infty$ with $c\in\mathbb{R}$, then

(3.5) $\lim_{N\to\infty}\,\frac{1}{N\,\log^{c}(\log
N)}\,\sum_{n=1}^{N}\,(f\circ T^{n-1})(x)=1\,,\quad\text{for $\mu$-a.e.\ $x\in
X$.}$

We will use Theorem 2.7 to verify this statement. Obviously, conditions (i),
(ii)-(a), and (ii)-(b) are already fulfilled by assumption. To verify
condition (ii)-(c) we notice that $q(n)\sim n\,\log^{b}(\log n)$ (using
[BGT87, Cor. 2.3.4]). Furthermore $\alpha(n)\sim n/(\log n\,\log^{b}(\log n))$
implies $\xi(n)\sim n\,\log^{2}(\log n)$ and
$\log^{c}(\log(n\,\xi(n)))\sim\log^{c}(\log n)$ gives condition (ii)-(c).

To verify condition (ii)-(d) we notice that $\Gamma(n)\sim n/\log^{c}(\log n)$
(also by applying [BGT87, Cor. 2.3.4]) and so
$\mu(A_{>\Gamma(n)})\sim\log^{b+c}(\log n)/n$. Thus

$\displaystyle\int_{0}^{\infty}\frac{y\,\left(\mu(A_{\geq\Gamma(y)})\right)^{2}}{\left(\int_{0}^{y}\,\mu(A_{\geq\Gamma(t)})\mathrm{d}t\right)^{2}}\mathrm{d}y$
$\displaystyle\sim\int_{0}^{\infty}\frac{\log^{2(b+c)}(\log
y)}{y\left(\int_{0}^{y}\,\log^{b+c}(\log
t)/t)\mathrm{d}t\right)^{2}}\mathrm{d}y\sim\int_{0}^{\infty}\frac{1}{y\,\log^{2}y}\mathrm{d}y<\infty.$

Finally, we are left to verify condition (ii)-(e). We have $\ell(n)\sim\log
n\,\log^{b+c}(\log n)$ which is super-slowly varying with rate function itself
and $\ell(\alpha(n))\sim\ell(n)$. On the other hand

$L(n)\,\sum_{k=1}^{n}\mu(A_{>k})\sim\Big{(}\log^{c}(\log
n)\Big{)}\,\Big{(}\log n\,\log^{b}(\log n)\Big{)}\sim\ell(n)$

giving all properties of Theorem 2.7 and (3.5) holds.

In the last examples we have seen that not only the trivial case $G(n)\sim n$
and $\mu(A_{>n})\sim\frac{1}{n}$ is applicable on Theorem 2.7. However, as we
will see in the next proposition, condition (ii)-(e) can be quite restrictive
and, as we will see, we will not be able to drop it.

######  Proposition 3.9.

Let $T$, $(X,{\mathcal{B}},\mu)$ and $E$ be defined as in Theorem 2.7. Let the
sequence of random variables $(\varphi_{{}_{E}}\circ T_{{}_{E}}^{n-1})$ with
$n\geq 1$ satisfy assumption (i) of Theorem 2.7. Let $f:X\to\mathbb{R}_{\geq
0}$ be a non-negative measurable function such that $f\not\in L^{1}(X,\mu)$
and let $f^{E}$ be its induced version on $E$ defined in (2.12). We assume
that:

  * (w1)

there exist $g_{1},g_{2}:X\to\mathbb{R}_{\geq 0}$ constant on the level sets
$(E_{n})$ of the hitting time function to $E$ such that $g_{1}(x)\leq f(x)\leq
g_{2}(x)$ for $\mu$-a.e. $x\in X$, and the induced functions satisfy
$g_{i}^{E}(n)\sim n\,\log n$ for $i=1,2$;

  * (w2)

the super-level sets of $\varphi_{{}_{E}}$ satisfy
$\mu(A_{>n})\sim\frac{1}{n}$.

Then there exist $u_{1}<u_{2}$ such that for $\mu$-a.e. $x\in X$ it holds

$\limsup_{N\to\infty}\,\frac{1}{N\,\log N}\,\sum_{k=1}^{N}\,(f\circ
T^{k-1})(x)>u_{2}\quad\text{ and }\quad\liminf_{N\to\infty}\,\frac{1}{N\,\log
N}\,\sum_{k=1}^{N}\,(f\circ T^{k-1})(x)<u_{1}\,.$

We will first assure ourselves that Proposition 3.9 fulfils all assumptions of
Theorem 2.7 with $G(n)\sim n\,\log n$ except for (ii)-(e). It is clear that
assumptions (ii)-(a) and (b) are satisfied by (w1). To verify (ii)-(c), we
note that (w2) implies $\alpha(n)\sim n/\log n$ from which we obtain
$\xi(n)\sim\log^{2}(\log n)$ and $\log(n\,\xi(n))\sim\log n$ verifying
(ii)-(c). To verify (ii)-(d) we first notice that $\Gamma(n)\sim n/\log n$ is
an asymptotic inverse of $G(n)$ which we either get by direct calculation or
by using e.g. [BGL18, Cor. 2.3.4]. This implies that
$\mu(A_{>\Gamma(n)})\sim\log n/n$ and by an easy calculation we obtain that
(2.18) is fulfilled. The proof of Proposition 3.9 is in Section 5.

Finally, we discuss the existing literature. The pointwise convergence of
Birkhoff averages for non-summable observables and infinite measure-preserving
dynamical systems has been recently discussed in [LM18]. The authors of [LM18]
have proved that (3.4) holds under strong assumptions on the observable $f$
and very weak assumptions on the system. Basically they considered
$L^{\infty}$ observables for which it is possible to control in a very strong
way the contributions to the Birkhoff sums from the excursions out of a finite
measure set $E$, the set on which we have considered the induced map. In this
paper we have less stringent assumptions on the observables but have
considered more particular dynamical systems, those for which an induced map
exists which is $\psi$-mixing. We finish this section by discussing the
relations of our results with those in [LM18] for the Farey map, a system in
the class ${\mathcal{T}}$.

######  Remark 3.10.

Let us consider the particular case of the Farey map $F:[0,1]\to[0,1]$ which
belongs to the class ${\mathcal{T}}$ considered in Section 3.1 and is defined
as

$F(x)=\left\\{\begin{array}[]{ll}\frac{x}{1-x}\,,&\text{if
$x\in\left[0,\frac{1}{2}\right]$,}\\\\[5.69046pt] \frac{1-x}{x}\,,&\text{if
$x\in\left[\frac{1}{2},1\right]$.}\end{array}\right.$

It is well known that the absolutely continuous invariant measure is
$d\mu(x)=\frac{1}{x\log 2}\,dx$. If $E=(\frac{1}{2},1)$, for which $\mu(E)=1$,
it holds $E_{n}=(\frac{1}{n+1},\frac{1}{n})$. Thus
$\mu(E_{n})=\mu(A_{>n})=\log_{2}(1+\frac{1}{n})\sim\frac{1}{n\log 2}$.

A particular class of observables studied in the context of infinite mixing
theory is that of _global observables_. For maps in ${\mathcal{T}}$, we recall
that a global observable is a function $f\in L^{\infty}([0,1],\mu)$ such that
the limit

$\lim_{a\to 0^{+}}\,\frac{1}{\mu(a,1)}\,\int_{a}^{1}\,f\,d\mu$

exists and is finite (see [BGL18, BL21] for more details). We first show that
observables $f\in L^{\infty}(X,\mu)$ which are constant on the level sets
$(E_{n})$ with $f_{n}:=f|_{E_{n}}$ and for which the induced function
satisfies $f^{E}|_{A_{n}}\sim n$ as $n\to\infty$, are global observables. The
definition of global observable for the Farey map and for observables constant
on the sets $E_{n}$ reduces to show that the limit

(3.6)
$\lim_{n\to\infty}\,\frac{1}{\mu\left(\frac{1}{k+1},1\right)}\,\int_{\frac{1}{k+1}}^{1}\,f(x)\,d\mu=\lim_{n\to\infty}\,\frac{1}{\sum_{k=1}^{n}\,\log_{2}(1+\frac{1}{k})}\,\sum_{k=1}^{n}\,f_{k}\,\log_{2}\left(1+\frac{1}{k}\right)$

exists and is finite. To prove this, it is enough to recall that
$\sum_{k=1}^{n}\,\log_{2}(1+\frac{1}{k})\sim\log_{2}n$ and to apply Abel’s
summation formula to get

$\sum_{k=1}^{n}\,f_{k}\,\log_{2}\left(1+\frac{1}{k}\right)\sim\left(\sum_{k=1}^{n}\,f_{k}\right)\,\log_{2}\left(1+\frac{1}{n}\right)+\frac{1}{\log
2}\int_{1}^{n}\,\left(\sum_{k=1}^{t}\,f_{k}\right)\,\frac{1}{t^{2}+t}\,dt\sim$
$\sim
f^{E}|_{A_{n}}\,\log_{2}\left(1+\frac{1}{n}\right)+\int_{1}^{n}\,\frac{f^{E}|_{A_{t}}}{\log
2}\,\frac{1}{t^{2}+t}\,dt\sim
n\,\log_{2}\left(1+\frac{1}{n}\right)+\frac{1}{\log
2}\int_{1}^{n}\,\frac{t}{t^{2}+t}\,dt\sim\log_{2}(n+1)\,.$

On the other hand let’s consider a global observable $f:X\to\mathbb{R}_{\geq
0}$ which is constant on the level sets $(E_{n})$ with $f_{n}:=f|_{E_{n}}$. By
rescaling we can assume that the limit in (3.6) exists and is equal to 1. In
general it is not possible to conclude about the asymptotic behaviour of
$f^{E}|_{A_{n}}$, however some sufficient conditions can be obtained by the
following argument (see [M06]). With
$\ell_{n}:=\sum_{k=1}^{n}\,\log_{2}(1+\frac{1}{k})$ let

$\tau_{n}:=\frac{1}{\ell_{n}}\,\sum_{k=1}^{n}\,f_{k}\,\log_{2}\left(1+\frac{1}{k}\right)\,.$

Then $f_{1}=\ell_{1}\tau_{1}$ and

$f_{n}=\frac{1}{\log_{2}\left(1+\frac{1}{n}\right)}\,(\ell_{n}\tau_{n}-\ell_{n-1}\tau_{n-1})\,,\quad\text{for
$n\geq 2$,}$

from which

$f^{E}|_{A_{n}}=\sum_{k=1}^{n}\,f_{k}=\frac{\ell_{n}\tau_{n}}{\log_{2}\left(1+\frac{1}{n}\right)}-\sum_{k=1}^{n-1}\,\ell_{k}\tau_{k}\left(\frac{1}{\log_{2}\left(1+\frac{1}{k+1}\right)}-\frac{1}{\log_{2}\left(1+\frac{1}{k}\right)}\right)\,.$

Since $\ell_{n}\sim\log_{2}n$, it follows for example that if $\tau_{n}\sim
1+o((\log_{2}n)^{-1})$ then $f^{E}|_{A_{n}}\sim n$. Hence we can check if
Proposition 3.7 is applicable to a given global observable.

Let us now compare the conditions on $f$ in Proposition 3.7 with the statement
in [LM18, Theorem 2.5]. In our setting and with proper norming, the
assumptions in [LM18, Theorem 2.5] can be stated as follows: $f\in
L^{\infty}(X,\mu)$ and for all $\epsilon>0$ there exist $N,K\in\mathbb{N}$
such that for all $x\in\bigcup_{k\geq K}E_{k}$ we have that

(3.7) $\displaystyle\left|\frac{\sum_{j=1}^{N}\left(f\circ
T^{j-1}\right)(x)}{N}-1\right|<\epsilon.$

First we show that if the assumptions for an observable $f$ in [LM18, Theorem
2.5] are satisfied, then we obtain a good asymptotic behaviour for the induced
observable $f^{E}$, so that we are basically in a good situation to apply
Theorem 2.7.

######  Lemma 3.11.

Let $f:[0,1]\to\mathbb{R}_{\geq 0}$ be in $L^{\infty}(X,\mu)$ and such that
for all $\epsilon>0$ there exist $N,K\in\mathbb{N}$ such that for all
$x\in\bigcup_{k\geq K}E_{k}$ we have that (3.7) holds. Then
$f^{E}|_{A_{k}}\sim k$ for $k\to\infty$.

###### Proof.

Let $\epsilon>0$ be given and consider $N,K$ fixed as in (3.7). We set
$f_{N}(x)\coloneqq\sum_{j=1}^{N}(f\circ T^{j-1})(x)$. For $x\in A_{R}$ with
$R>K$, writing $R=K+rN+s$ with $r\in\mathbb{N}_{0}$ and $0\leq s\leq N-1$, we
get

$f^{E}(x)=\sum_{j=1}^{R+1}\,(f\circ
T^{j-1})(x)=\sum_{i=1}^{r}\,f_{N}(T^{N(i-1)}(x))+\sum_{j=rN+1}^{R+1}\,(f\circ
T^{j-1})(x)\,.$

Since $T^{N(i-1)}(x)\in\bigcup_{k\geq K}E_{k}$ for all $i=1,\dots,r$, we apply
(3.7), and the fact that $f\in L^{\infty}(X,\mu)$ and $f\geq 0$, to obtain

$rN\,(1-\epsilon)\leq f^{E}(x)\leq
rN\,(1+\epsilon)+(N+K)\,\|f\|_{\infty}\,,\quad\forall\,x\in A_{R}\,.$

Since $R\sim rN\to\infty$ as $R\to\infty$, the lemma is proved. ∎

Finally we give two examples to which Theorem 2.7 is applicable, but which
fail to fulfil the conditions in [LM18, Theorem 2.5]. The first is an example
with $f\notin L^{\infty}(X,\mu)$, the second one with $f\in L^{\infty}(X,\mu)$
but not satisfying (3.7).

######  Example 3.12.

Let us consider a strictly monotonic sequence of natural numbers $\gamma_{n}$
defined by $\gamma_{1}=4$ and
$\gamma_{n+1}=\gamma_{n}+\left\lfloor\gamma_{n}^{1/2}\right\rfloor$. Then we
set $f$ to be constant on the intervals $E_{k}$ with

$\displaystyle f(x)=\begin{cases}\gamma_{n}-\gamma_{n-1}\,,&\text{if $x\in
E_{\gamma_{n}}$ for some $n\in\mathbb{N}$;}\\\\[5.69046pt]
0\,,&\text{else}.\end{cases}$

Since $\gamma_{n}-\gamma_{n-1}$ tends to infinity, $f\notin
L^{\infty}(X,\mu)$. On the other hand the induced function satisfies
$f^{E}|_{A_{k}}=\gamma_{n}$ for $\gamma_{n}\leq k<\gamma_{n+1}$, so that for
all $k\in\mathbb{N}$ and all $x\in A_{k}$ we have $k+1-\sqrt{k+1}\leq
f^{E}(x)\leq k$, implying that $f^{E}|_{A_{k}}\sim k$ as $k\to\infty$.

######  Example 3.13.

First we define a strictly monotonic sequence of even numbers $\kappa_{n}$ by
setting $\kappa_{1}=4$ and
$\kappa_{n+1}=\kappa_{n}+2\left\lfloor\kappa_{n}^{1/2}/2\right\rfloor$.
Furthermore, we set

$\displaystyle
U\coloneqq\left\\{\kappa_{n},\kappa_{n}+1,\ldots,\kappa_{n}+1/2\left(\kappa_{n+1}-\kappa_{n}\right),\,n\in\mathbb{N}\right\\}.$

We define

$\displaystyle f(x)=\begin{cases}2\,,&\text{if $x\in E_{k}$ and $k\in
U$;}\\\\[5.69046pt] 0\,,&\text{else}.\end{cases}$

Then $0\leq f(x)\leq 2$ and thus $f\in L^{\infty}(X,\mu)$. Furthermore, if
$x\in A_{k}$ then $k-2\sqrt{k}\leq f^{E}(x)\leq k+2\sqrt{k}$ implying that
$f^{E}|_{A_{k}}\sim k$ as $k\to\infty$.

On the other hand, for each $K\in\mathbb{N}$ there is a positive measure set
$A\subset\cup_{k\geq K}E_{k}$ such that for all $x\in A$ we have that
$f(x)=\ldots=\left(f\circ T^{K}\right)(x)=0$ implying that (3.7) cannot be
fulfilled.

##  4\. Proofs of the main results

###  4.1. Proof of Theorem 2.3

We prove the theorem for $\mu$-a.e. $x\in E$. The result for $\mu$-a.e. $x\in
X$ follows since $T$ is conservative and ergodic. Let $f\in L^{1}(X,\mu)$ and
let $R_{{}_{E,N,m}}(x):=R_{{}_{E,N+m(N,E,x)}}(x)$ denote the number of visits
to $E$ in the first $N+m(N,E,x)$ steps of the orbit of $x$. Thus for each $N$
we add a number of steps depending on $x$ and being non-decreasing in $N$.

######  Lemma 4.1.

For $\mu$-a.e. $x\in E$ it holds

$\lim_{N\to\infty}\,\frac{1}{R_{{}_{E,N,m}}(x)}\,\sum_{k=1}^{N+m(N,E,x)}\,(f\circ
T^{k-1})(x)=\int_{X}\,f\,\mathrm{d}\mu\,.$

###### Proof.

We apply Hopf’s Ratio Ergodic Theorem to $f(x)$ and $\mathbbm{1}_{{}_{E}}(x)$
and obtain for $\mu$-a.e. $x\in E$

$\lim_{N\to\infty}\,\frac{\sum_{k=1}^{N+m(N,E,x)}\,(f\circ
T^{k-1})(x)}{R_{{}_{E,N,m}}(x)}=$
$=\lim_{N\to\infty}\,\frac{\sum_{k=1}^{N+m(N,E,x)}\,(f\circ
T^{k-1})(x)}{(\mathbbm{1}_{{}_{E}}\circ
T^{N+m(N,E,x)})(x)+\sum_{k=1}^{N+m(N,E,x)}\,(\mathbbm{1}_{{}_{E}}\circ
T^{k-1})(x)}=$
$=\frac{\int_{X}\,f\,d\mu}{\int_{X}\,\mathbbm{1}_{{}_{E}}\,d\mu}=\int_{X}\,f\,\mathrm{d}\mu$

where we have used that $\sum_{k=1}^{N+m(N,E,x)}\,(\mathbbm{1}_{{}_{E}}\circ
T^{k-1})(x)$ is divergent. ∎

Let us now recall for $\mu$-a.e. $x\in E$ the definition of the time
$\tau_{{}_{E,x}}(N)$ of the $N$-th return to $E$ given in (2.13). Then as in
(2.8), we consider its trimmed version

(4.1) $\tau^{1}_{{}_{E,x}}(N):=\tau_{{}_{E,x}}(N)-\max_{1\leq k\leq
N}\,(\varphi_{{}_{E}}\circ T_{{}_{E}}^{k-1})(x)$

where we are trimming $\tau_{{}_{E,x}}$ by deleting the largest entry.

######  Lemma 4.2.

For $\mu$-a.e. $x\in E$ it eventually holds

$N<\tau^{1}_{{}_{E,x}}(R_{{}_{E,N,m}}(x))\leq N+M^{2}_{{}_{N,m}}(x)$

where $M^{2}_{{}_{N,m}}(x)$ denotes the 2-nd maximum in
$\\{(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,:\,k=1,\dots,R_{{}_{E,N,m}}(x)\\}$ which in case of
equality might coincide with the maximum.

###### Proof.

We first give some remarks on $\tau_{{}_{E,x}}$ and $R_{{}_{E,N}}(x)$. Given
$N\geq 1$, $R_{{}_{E,N}}(x)$ is the number of visits to $E$ up to time $N$,
hence $R_{{}_{E,N}}(x)\leq N+1$ and $T_{{}_{E}}^{R_{{}_{E,N}}(x)-1}(x)$ is the
last visit to $E$ for $x$ up to time $N$. It follows that

$\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))=\sum_{k=1}^{R_{{}_{E,N}}(x)}\,(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)>N$

as the $(R_{{}_{E,N}}(x))$-th return to $E$, equivalently the
$(R_{{}_{E,N}}(x)+1)$-th visit to $E$, happens after time $N$. Moreover

$\sum_{k=1}^{R_{{}_{E,N}}(x)}\,(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)=\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N}}(x)-1}(x)+\tau_{{}_{E,x}}(R_{{}_{E,N}}(x)-1)\leq\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N}}(x)-1}(x)+N$

as the $(R_{{}_{E,N}}(x)-1)$-th return to $E$, equivalently the
$(R_{{}_{E,N}}(x))$-th visit to $E$, happens at time less than or equal to
$N$.

Using now $R_{{}_{E,N,m}}(x)$, we have proved

(4.2) $N+m(N,E,x)<\tau_{{}_{E,x}}(R_{{}_{E,N,m}}(x))\leq
N+m(N,E,x)+\varphi_{{}_{E}}\circ T_{{}_{E}}^{R_{{}_{E,N,m}}(x)-1}(x).$

We now show that

(4.3) $\max\left\\{(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,:\,k=1,\dots,R_{{}_{E,N,m}}(x)\right\\}=\max\left\\{m(N,E,x),\,\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N,m}}(x)-1}(x)\right\\}.$

From (2.6), $m(N,E,x)$ maximises the return times $(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)$ for $k=1,\dots,R_{{}_{E,N}}(x)$, that is up to time
$\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))$. Moreover, repeating the same argument as
above, the return times $(\varphi_{{}_{E}}\circ T_{{}_{E}}^{k-1})(x)$ with
$k=R_{{}_{E,N}}(x)+1,\dots,R_{{}_{E,N,m}}(x)-1$ are concerned with excursions
outside $E$ for the orbit of $x$ with time in $(N,N+m(N,E,x)]$. Then

$m(N,E,x)=\max\left\\{(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,:\,k=1,\dots,R_{{}_{E,N}}(x)-1\right\\}$

and (4.3) is proved.

Then, if

$\max\left\\{(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,:\,k=1,\dots,R_{{}_{E,N,m}}(x)\right\\}=m(N,E,x)$

it immediately follows from (4.2) that

$N<\tau_{{}_{E,x}}(R_{{}_{E,N,m}}(x))-\max_{1\leq k\leq
R_{{}_{E,N,m}}(x)}\,(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)=\tau^{1}_{{}_{E,x}}(R_{{}_{E,N,m}}(x))\,,$

otherwise, we obtain

$\tau^{1}_{{}_{E,x}}(R_{{}_{E,N,m}}(x))=\tau_{{}_{E,x}}(R_{{}_{E,N,m}}(x)-1)>N$

since $x$ necessarily visits $E$ at some time $k$ in $(N,N+m(N,E,x)]$. In both
cases we have thus obtained the first inequality to prove.

Moreover from (4.2) and (4.3) we obtain

$\tau_{{}_{E,x}}(R_{{}_{E,N,m}}(x))-\max_{1\leq k\leq
R_{{}_{E,N,m}}(x)}\,(\varphi_{{}_{E}}\circ T_{{}_{E}}^{k-1})(x)\leq
N+\min\left\\{m(N,E,x),\,\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N,m}}(x)-1}(x)\right\\}$

and by (4.3)

$\min\left\\{m(N,E,x),\,\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N,m}}(x)-1}(x)\right\\}\leq M^{2}_{{}_{N,m}}(x).$

The proof is finished. ∎

######  Lemma 4.3.

For $\mu$-a.e. $x\in E$ we have

$\lim_{N\to\infty}\,\frac{\tau^{1}_{{}_{E,x}}(N)}{d(N)}=1$

where $d(n)$ is the asymptotic inverse function of the sequence $(\alpha(n))$
defined in (2.11).

###### Proof.

By their definition (4.1), the terms $\tau^{1}_{{}_{E,x}}(N)$ are the trimmed
Birkhoff sums of $\varphi_{{}_{E}}$ for the system $(E,T_{{}_{E}})$. Thus, it
is enough to show that Lemma 2.2 can be applied to the sequence of random
variables $(\varphi_{{}_{E}}\circ T_{{}_{E}}^{n-1})$.

By assumption (i), the sequence is $\psi$-mixing with mixing coefficient
$\psi(n)$ fulfilling $\sum_{n\geq 1}\psi(n)/n<\infty$. Moreover, the
distribution function of $(\varphi_{{}_{E}}\circ T_{{}_{E}}^{n-1})$ is given
by

$F(y)=\mu(\varphi_{{}_{E}}\leq y)=\sum_{k=1}^{\lfloor
y\rfloor}\,\mu(A_{k})=1-\mu(A_{>\lfloor y\rfloor}),$

by (2.1) and (2.2). Thus, in (2.9) with $y_{0}=1$ we find

$\displaystyle\int_{1}^{\infty}\left(\frac{y\left(1-F(y)\right)}{\int_{0}^{y}\left(1-F(t)\right)\mathrm{d}t}\right)^{r+1}\frac{1}{y}\,\mathrm{d}y=\sum_{n\geq
1}\,\int_{n}^{n+1}\,\left(\frac{y\,\mu(A_{>n})}{\int_{0}^{y}\,\mu(A_{>\lfloor
t\rfloor})\mathrm{d}t}\right)^{r+1}\frac{1}{y}\,\mathrm{d}y\leq$
$\displaystyle\leq\sum_{n\geq
1}\,\frac{(n+1)^{r}\,(\mu(A_{>n}))^{r+1}}{(\int_{0}^{n}\,\mu(A_{>\lfloor
t\rfloor})\mathrm{d}t)^{r+1}}=\sum_{n\geq
1}\,\frac{(n+1)^{r}\,(\mu(A_{>n}))^{r+1}}{(\sum_{j=0}^{n-1}\,\mu(A_{>j}))^{r+1}}$

and by assumption (ii) we find $W\leq 1$. Analogously,

$\int_{1}^{\infty}\left(\frac{y\left(1-F(y)\right)}{\int_{0}^{y}\left(1-F(t)\right)\mathrm{d}t}\right)^{r+1}\frac{1}{y}\,\mathrm{d}y\geq\sum_{n\geq
1}\,\frac{n^{r}\,(\mu(A_{>n+1}))^{r+1}}{(\sum_{j=0}^{n}\,\mu(A_{>j}))^{r+1}}$

and (2.3) imply $W>0$, so that $W=1$. Thus, the proof is finished by applying
Lemma 2.2 and recalling that the Birkhoff sums of $(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{n-1})$ trimmed by the largest entry are $\tau^{1}_{{}_{E,x}}(N)$.
∎

######  Lemma 4.4.

For $\mu$-a.e. $x\in E$ it holds

$\lim_{N\to\infty}\,\frac{R_{{}_{E,N,m}}(x)}{\alpha(N)}=1\,.$

###### Proof.

Since $R_{{}_{E,N,m}}(x)$ is diverging, we obtain from Lemma 4.3 that

$\lim_{N\to\infty}\,\frac{\tau^{1}_{{}_{E,x}}(R_{{}_{E,N,m}}(x))}{d(R_{{}_{E,N,m}}(x))}=1$

for $\mu$-a.e. $x\in E$. Moreover, from Lemma 4.2 we obtain

$\frac{N}{d(R_{{}_{E,N,m}}(x))}\leq\frac{\tau^{1}_{{}_{E,x}}(R_{{}_{E,N,m}}(x))}{d(R_{{}_{E,N,m}}(x))}\leq\frac{N+M^{2}_{{}_{N,m}}(x)}{d(R_{{}_{E,N,m}}(x))}$

where we recall that $M^{2}_{{}_{N,m}}(x)$ denotes the 2-nd maximum in
$\\{(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,:\,k=1,\dots,R_{{}_{E,N,m}}(x)\\}$. Thus, by applying
Lemma 2.2 to $(\varphi_{{}_{E}}\circ T_{{}_{E}}^{n-1})$ with $W=1$ as shown
above, we use (2.10) with $r=2$ to obtain

$\lim_{N\to\infty}\,\frac{N}{d(R_{{}_{E,N,m}}(x))}=\lim_{N\to\infty}\,\frac{N+M^{2}_{{}_{N,m}}(x)}{d(R_{{}_{E,N,m}}(x))}=1$

for $\mu$-a.e. $x\in E$. The result now follows by using that $\alpha(n)$ is
the asymptotic inverse of $d(n)$. ∎

Theorem 2.3 follows by applying Lemma 4.1 and Lemma 4.4. ∎

###  4.2. Proof of Theorem 2.4

As before we prove the theorem for $\mu$-a.e. $x\in E$. The first step is the
analog of Lemma 4.1. Using Hopf’s Ratio Ergodic Theorem we have for $\mu$-a.e.
$x\in E$

(4.4) $\lim_{N\to\infty}\,\frac{\sum_{k=1}^{N}\,(f\circ
T^{k-1})(x)}{R_{{}_{E,N}}(x)}=\lim_{N\to\infty}\,\frac{\sum_{k=1}^{N}\,(f\circ
T^{k-1})(x)}{(\mathbbm{1}_{{}_{E}}\circ
T^{N})(x)+\sum_{k=1}^{N}\,(\mathbbm{1}_{{}_{E}}\circ
T^{k-1})(x)}=\int_{X}\,f\,\mathrm{d}\mu.$

Then we consider the sequence $\tau_{{}_{E,x}}$ defined in (2.13) and the
trimmed version $\tau^{1}_{{}_{E,x}}$. In Lemma 4.2 we have proved

(4.5) $N<\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))\leq N+\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N}}(x)-1}(x)$

for $\mu$-a.e. $x\in E$. For the trimmed sums

$\tau^{1}_{{}_{E,x}}(R_{{}_{E,N}}(x))=\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))-\max_{1\leq
k\leq R_{{}_{E,N}}(x)}\,(\varphi_{{}_{E}}\circ T_{{}_{E}}^{k-1})(x)$

we now show that

(4.6) $N-w(N,E,x)\leq\tau^{1}_{{}_{E,x}}(R_{{}_{E,N}}(x))\leq
N-w(N,E,x)+M^{2}_{{}_{N}}(x)$

where $w(N,E,x)$ is defined in (2.7) and $M^{2}_{{}_{N}}(x)$ denotes the 2-nd
maximum in $\\{(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,:\,k=1,\dots,R_{{}_{E,N}}(x)\\}$.

We remark that analogously to (2.6) we have for $\mu$-a.e. $x\in E$

(4.7) $w(N,E,x)=\max\Big{\\{}\left\\{(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,:\,k=1,\dots,R_{{}_{E,N}}(x)-1\right\\}\,,\,N-\tau_{{}_{E,x}}(R_{{}_{E,N}}(x)-1)\Big{\\}}$

since $\tau_{{}_{E,x}}(R_{{}_{E,N}}(x)-1)$ is the time of the
$(R_{{}_{E,N}}(x)-1)$-th return to $E$, the last before step $N+1$. We thus
consider two cases. If

$\max\left\\{(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,:\,k=1,\dots,R_{{}_{E,N}}(x)\right\\}=\max\left\\{(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,:\,k=1,\dots,R_{{}_{E,N}}(x)-1\right\\},$

by (4.7) it follows

$w(N,E,x)=\max\left\\{(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,:\,k=1,\dots,R_{{}_{E,N}}(x)-1\right\\}$

since $\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N}}(x)-1}(x)>N-\tau_{{}_{E,x}}(R_{{}_{E,N}}(x)-1)$. So
(4.6) follows from (4.5) by subtracting $w(N,E,x)$ from all terms and using
$\varphi_{{}_{E}}\circ T_{{}_{E}}^{R_{{}_{E,N}}(x)-1}(x)\leq
M^{2}_{{}_{N}}(x)$. If on the contrary

$\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N}}(x)-1}(x)=\max\left\\{(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,:\,k=1,\dots,R_{{}_{E,N}}(x)\right\\},$

then

$\tau^{1}_{{}_{E,x}}(R_{{}_{E,N}}(x))=\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))-\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N}}(x)-1}(x)=\tau_{{}_{E,x}}(R_{{}_{E,N}}(x)-1).$

Now from (4.7) we may have

$w(N,E,x)=N-\tau_{{}_{E,x}}(R_{{}_{E,N}}(x)-1)=N-\tau^{1}_{{}_{E,x}}(R_{{}_{E,N}}(x))$

in which case (4.6) is immediate. Otherwise we may have

$w(N,E,x)=\max\left\\{(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,:\,k=1,\dots,R_{{}_{E,N}}(x)-1\right\\}$

in which case

$N-\tau^{1}_{{}_{E,x}}(R_{{}_{E,N}}(x))\leq w(N,E,x)=M^{2}_{{}_{N}}(x)$

so that

$N-w(N,E,x)\leq\tau^{1}_{{}_{E,x}}(R_{{}_{E,N}}(x))\leq
N=N-w(N,E,x)+M^{2}_{{}_{N}}(x)$

where we have used (4.5) in the second inequality, and again we have proved
(4.6).

Finally, we use Lemma 4.3 to write as in the proof of Lemma 4.4

$\lim_{N\to\infty}\,\frac{\tau^{1}_{{}_{E,x}}(R_{{}_{E,N}}(x))}{d(R_{{}_{E,N}}(x))}=1$

for $\mu$-a.e. $x\in E$. Moreover, from (4.6) we obtain

$\frac{N-w(N,E,x)}{d(R_{{}_{E,N}}(x))}\leq\frac{\tau^{1}_{{}_{E,x}}(R_{{}_{E,N}}(x))}{d(R_{{}_{E,N}}(x))}\leq\frac{N-w(N,E,x)+M^{2}_{{}_{N}}(x)}{d(R_{{}_{E,N}}(x))}.$

Thus, by applying Lemma 2.2 to $(\varphi_{{}_{E}}\circ T_{{}_{E}}^{n-1})$ with
$W=1$ as above, we use (2.10) with $r=2$ to obtain

$\lim_{N\to\infty}\,\frac{N-w(N,E,x)}{d(R_{{}_{E,N}}(x))}=1$

for $\mu$-a.e. $x\in E$. Using now that $\alpha(n)$ is the asymptotic inverse
of $d(n)$, we obtain

(4.8) $\lim_{N\to\infty}\,\frac{R_{{}_{E,N}}(x)}{\alpha(N-w(N,E,x))}=1$

for $\mu$-a.e. $x\in E$. Theorem 2.4 follows from (4.4) and (4.8). ∎

###  4.3. Proof of Lemma 2.6

We first prove that

(4.9) $\displaystyle\mu\left(\left\\{x\in E\,:\,\max_{1\leq i\leq
n}\left(\varphi_{{}_{E}}\circ T_{{}_{E}}^{i-1}\right)(x)>q\left(n\,\log
n\,\log^{2}(\log n)\right)\text{for infinite $n$}\right\\}\right)=0.$

We note that for all $2^{n}\leq k<2^{n+1}$

(4.10) $\displaystyle\left\\{x\in E\,:\,\max_{1\leq i\leq
k}\left(\varphi_{{}_{E}}\circ T_{{}_{E}}^{i-1}\right)(x)>q\left(k\,\log
k\,\log^{2}(\log k)\right)\right\\}\,\subset$
$\displaystyle\subset\left\\{x\in E\,:\,\max_{1\leq i\leq
2^{n+1}}\left(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{i-1}\right)(x)>q\left(2^{n}\,\log 2^{n}\,\log^{2}(\log
2^{n})\right)\right\\}.$

Furthermore, we have by the definition of $q$ that

(4.11) $\displaystyle\mu\left(\left\\{x\in E\,:\,\max_{1\leq i\leq
2^{n+1}}\left(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{i-1}\right)(x)>q\left(2^{n}\,\log 2^{n}\,\log^{2}(\log
2^{n})\right)\right\\}\right)\leq$
$\displaystyle\leq\sum_{j=1}^{2^{n+1}}\mu\left(\left\\{x\in
E\,:\,\left(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{j-1}\right)(x)>q\left(2^{n}\,\log 2^{n}\,\log^{2}(\log
2^{n})\right)\right\\}\right)=$
$\displaystyle=\sum_{j=1}^{2^{n+1}}\frac{1}{2^{n}\,\log 2^{n}\,\log^{2}(\log
2^{n})}=\frac{2}{\log 2^{n}\,\log^{2}(\log 2^{n})}.$

Since

$\displaystyle\sum_{n=1}^{\infty}\frac{2}{\log 2^{n}\,\log^{2}(\log
2^{n})}<\infty,$

we obtain by the first Borel-Cantelli lemma and a combination of (4.10) and
(LABEL:eq:_BC_2) that (4.9) holds. From (4.8) we obtain that for $\mu$-a.e.
$x\in E$

$\displaystyle\limsup_{n\to\infty}\frac{R_{{}_{E,n}}(x)}{\alpha(n)}\leq 1.$

Hence,

$\mu\left(\left\\{x\in E\,:\,\max_{1\leq i\leq
R_{{}_{E,n}}(x)}\left(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{i-1}\right)(x)>q\left(\alpha(n)\,\log(\alpha(n))\,\log^{2}(\log(\alpha(n)))\right)\text{for
infinite $n$}\right\\}\right)=0$

and with the definition of $\xi$ the statement of the lemma follows. ∎

###  4.4. Proof of Theorem 2.7

For simplicity let us first assume that $f$ is constant on the level sets
$E_{n}$ and we can take $g_{1}=g_{2}=f\geq 0$ in the assumptions. Hence there
exists a non-decreasing $G:\mathbb{N}\to\mathbb{R}_{\geq 0}$ with
$f^{E}(x)=G(n)$ for all $x\in A_{n}$, such that $G(n)=n\,L(n)$ with $L$ a
normalised slowly varying function and that satisfies (ii)-(c), (d) and (e)
with $\Gamma_{i}(n)=\Gamma(n):=\min\\{k\in\mathbb{N}\,:\,G(k)>n\\}$. In this
case the sequence of random variables $(f^{E}\circ T_{{}_{E}}^{n-1})$ is
$\psi$-mixing since it generates the same $\sigma$-field as the sequence
$(\varphi_{{}_{E}}\circ T_{{}_{E}}^{n-1})$. Thus it satisfies the first
assumption of Lemma 2.2.

Moreover the distribution function of the random variables $(f^{E}\circ
T_{{}_{E}}^{n-1})$ is given by

$F(y)=\mu(f^{E}\leq y)=1-\mu(A_{\geq\Gamma(y)})$

so that (2.18) implies $W\leq 1$ in (2.9). Setting $G(0)=0$ we have by
definition $\Gamma(y)=k$ for all $y\in[G(k-1),G(k))$ for all $k\geq 1$. Hence

$\int_{0}^{\infty}\,\mu(A_{\geq\Gamma(y)})\mathrm{d}y=\sum_{k=1}^{\infty}\,(G(k)-G(k-1))\,\mu(A_{>k-1})=\sum_{k=1}^{\infty}\,f_{k-1}\,\mu(A_{>k-1})=\int_{0}^{\infty}\,f(x)\mathrm{d}\mu=\infty$

where we have used (2.17) with $f_{n}=f|_{E_{n}}$ for all $n\geq 0$, with
$E_{n}$ the level sets (2.16) of the hitting time function to $E$. It follows
that $W>0$, so $W=1$.

Applying Lemma 2.2 to $(f^{E}\circ T_{{}_{E}}^{n-1})$ it follows that letting
$d(n)$ be the inverse function of

(4.12) $a(y):=\frac{y}{\int_{0}^{y}\,\mu(A_{\geq\Gamma(t)})\mathrm{d}t}$

we have for $N\to\infty$

(4.13) $\sum_{n=1}^{R_{{}_{E,N}}(x)-1}\,(f^{E}\circ T^{n-1}_{{}_{E}})(x)\sim
d(R_{{}_{E,N}}(x)-1)+\max_{1\leq k\leq R_{{}_{E,N}}(x)-1}\,(f^{E}\circ
T^{k-1}_{{}_{E}})(x)$

for $\mu$-a.e. $x\in E$, since $R_{{}_{E,N}}(x)$ is diverging.

Looking at (2.14) we need now to consider the last term on the right hand
side. We recall that if $\tau_{{}_{E,x}}(R_{{}_{E,N}}(x)-1)=N$ this term does
not appear. If $\tau_{{}_{E,x}}(R_{{}_{E,N}}(x)-1)<N$ let us first write

$\sum_{k=\tau_{{}_{E,x}}(R_{{}_{E,N}}(x)-1)+1}^{N}\,(f\circ
T^{k-1})(x)=\sum_{k=\tau_{{}_{E,x}}(R_{{}_{E,N}}(x)-1)+1}^{\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))}\,(f\circ
T^{k-1})(x)-\sum_{k=N+1}^{\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))}\,(f\circ
T^{k-1})(x)$

where $\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))\geq N+1$ by definition. Then by
definition (2.12) and by (2.17)

$\Big{(}f^{E}\circ
T_{{}_{E}}^{R_{{}_{E,N}}(x)-1}\Big{)}(x)=\sum_{k=\tau_{{}_{E,x}}(R_{{}_{E,N}}(x)-1)+1}^{\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))}\,(f\circ
T^{k-1})(x)=$ $=\sum_{j=0}^{(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N}}(x)-1})(x)-1}\,f_{j}=G\Big{(}\Big{(}\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N}}(x)-1}\Big{)}(x)\Big{)}$

and by (2.17)

$\sum_{k=N+1}^{\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))}\,(f\circ
T^{k-1})(x)=\sum_{j=1}^{\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))-N}\,f_{j}=G\Big{(}\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))-N+1\Big{)}-f_{0}\,.$

Since $(f^{E}\circ T^{k-1}_{{}_{E}})(x)=G((\varphi_{{}_{E}}\circ
T^{k-1}_{{}_{E}})(x))$, using (4.13) in (2.14), we finally have

(4.14) $\displaystyle S_{N}f(x)\sim$
$\displaystyle\,d(R_{{}_{E,N}}(x)-1)+\max_{1\leq k\leq
R_{{}_{E,N}}(x)-1}\,G\Big{(}(\varphi_{{}_{E}}\circ
T^{k-1}_{{}_{E}})(x)\Big{)}+$
$\displaystyle+G\Big{(}\Big{(}\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N}}(x)-1}\Big{)}(x)\Big{)}-G\Big{(}\tau_{{}_{E,x}}(R_{{}_{E,N}}(x))-N+1\Big{)}$

where we have discarded the constant $f_{0}$.

Let us fix $x\in E$, a constant $\eta\in(0,\frac{1}{2})$111The bound
$\eta<\frac{1}{2}$ simplifies a step of the proof but is not necessary., and
let $(U_{j})$ be the sequence of the return times to $E$, that is
$U_{j}=\tau_{{}_{E,x}}(j)$ for $j\geq 1$ and $R_{{}_{E,U_{j}}}(x)-1=j$. Then
we consider the following subintervals of $\mathbb{N}$ depending on $x$:

  * •

the intervals corresponding to excursions outside $E$ which do not achieve a
new record for the return time

(4.15)
${\mathcal{I}}_{j}:=\left\\{N\in(U_{j},U_{j+1})\,:\,(\varphi_{{}_{E}}\circ
T^{j}_{{}_{E}})(x)\leq\max_{1\leq k\leq j}\,(\varphi_{{}_{E}}\circ
T^{k-1}_{{}_{E}})(x)\right\\}\,;$

  * •

the intervals corresponding to excursions outside $E$ which achieve a new
record for the return time

(4.16)
${\mathcal{J}}_{j}:=\left\\{N\in(U_{j},U_{j+1})\,:\,(\varphi_{{}_{E}}\circ
T^{j}_{{}_{E}})(x)>\max_{1\leq k\leq j}\,(\varphi_{{}_{E}}\circ
T^{k-1}_{{}_{E}})(x)\right\\}\,;$

we also need to consider the subintervals for which $N$ is asymptotically not
too far from $U_{j+1}$ with respect to $U_{j}$ up to $\eta$ in the following
sense

(4.17)
${\mathcal{J}}_{j}^{\eta}:=\left\\{N\in{\mathcal{J}}_{j}\,:\,U_{j+1}-N\leq\eta\,(N-U_{j})\right\\}\,.$

The sequence $(U_{j})$ and the intervals ${\mathcal{I}}_{j}$ and
${\mathcal{J}}_{j}$ depend on $x$, we have dropped this dependence in the
notation for simplicity.

We now show that for $\mu$-a.e. $x\in E$ we have

(4.18) $\lim_{N\to\infty}\,\frac{1}{G(N)}\,\sum_{k=1}^{N}\,(f\circ
T^{k-1})(x)=1\,.$

The proof follows from the following lemmas.

######  Lemma 4.5.

For $\mu$-a.e. $x\in X$ we have

(4.19) $d(R_{{}_{E,N}}(x)-1)\sim d(R_{{}_{E,N}}(x))\sim G(N-w(N,E,x))\,.$

###### Proof.

Here we use the notions of slowly and regularly varying functions from
Appendix B. By a remark in [AN03], condition (2.18) implies that the sequence
$(a(n))$ defined in (4.12) is a regularly varying function with exponent $1$,
and the same is true for its inverse $d(n)$ (see for example [BGT87, Prop.
1.5.14]). For such a function we have that $d(b_{n})\sim d(b_{n}-1)$ for all
diverging sequences of positive numbers $(b_{n})$. This shows that for
$\mu$-a.e. $x\in X$ we have $d(R_{{}_{E,N}}(x)-1)\sim d(R_{{}_{E,N}}(x))$.

Moreover, from (4.8) and again from $d$ being regularly varying with index 1,
we have $d(R_{{}_{E,N}}(x))\sim d(\alpha(N-w(N,E,x)))$, where $\alpha(n)$ is
given in (2.11). Using now the theory of the _de Bruijn conjugate_ of a slowly
varying function (see [BGT87, Thm. 1.5.13]), letting $a(n)=n/\ell(n)$ with

$\ell(n):=\sum_{k=0}^{n}\,\mu(A_{\geq\Gamma(k)})\,,$

by assumption (ii)-(e) we know that $\ell$ is super-slowly varying at infinity
with rate function itself (see Definition B.5), and by [BGT87, Cor. 2.3.4] it
follows that its de Bruijn conjugate $\ell^{\\#}$ satisfies $\ell^{\\#}(n)\sim
1/\ell(n)$. Hence

$d(n)\sim n\,\ell(n)$

and by (2.19)

$d(\alpha(n))\sim\alpha(n)\,\ell\Big{(}\alpha(n)\Big{)}\sim G(n)$

as $n\to\infty$. It follows that

$d\left(\alpha(N-w(N,E,x))\right)\sim G\left(N-w(N,E,x)\right)$

and the lemma is proved. ∎

######  Remark 4.6.

Here we show that under condition (2.18) also the sequence $(\alpha(n))$
defined in (2.11) is a regularly varying function with exponent $1$. Indeed by
the remark in [AN03], condition (2.18) implies that
$\gamma(t)=\sum_{k=1}^{\infty}\mu\left(A_{k}\right)\,\min\left\\{k\,L(k),t\right\\}$
is normalised slowly varying. (Indeed, the remark only states slow variation,
but proves actually normalised slow variation.) Since $L(t)$ is normalised
slowly varying and $nL(n)$ is non-decreasing, we can show that

$\displaystyle\gamma(t\,L(t))=\sum_{k=1}^{\infty}\mu\left(A_{k}\right)\,\min\left\\{k\,L(k),t\,L(t)\right\\}=\sum_{k=1}^{\lfloor
t\rfloor}\mu(A_{>k})\,\Big{(}k\,L(k)-(k-1)L(k-1)\Big{)}$

is normalised slowly varying as well. Referring to (B.2), we may write
$\gamma(t)=\exp(\int_{\kappa}^{t}(\eta(x)/x)\mathrm{d}x)$ with $\eta(x)$
tending to zero and
$L(t)=\exp(\int_{\kappa}^{t}(\varepsilon(x)/x)\mathrm{d}x)$ with
$\varepsilon(x)$ tending to zero, by extending the domain of definition of $L$
by linear interpolation. Then we have for $t\notin\mathbb{N}$

(4.20)
$\frac{\mathrm{d}}{\mathrm{d}t}\gamma(t\,L(t))=\frac{\gamma(t\,L(t))\,\eta(t\,L(t))\,\left(1+\varepsilon(t)\right)}{t}$

and since $\eta(t\,L(t))\,\left(1+\varepsilon(t)\right)$ tends to zero,
$\gamma(t\,L(t))$ is normalised slowly varying (see eq. (1.3.4) and subsequent
comments in [BGT87]).

In the next steps we will compare $\gamma(t\,L(t))$ with the wandering rate
$w_{n}(E):=\sum_{k=1}^{n-1}\mu(A_{>k})$. Letting
$\widetilde{L}(k):=1/(k\,L(k)-(k-1)L(k-1))$, we have

$\displaystyle w_{n}(E)=$
$\displaystyle\sum_{k=1}^{n-1}\,\mu(A_{>k})\,(k\,L(k)-(k-1)L(k-1))\,\frac{1}{k\,L(k)-(k-1)L(k-1)}=$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{n-1}\,\Big{(}\gamma(k\,L(k))-\gamma((k-1)L(k-1))\Big{)}\,\widetilde{L}(k)=\sum_{k=1}^{n-1}\,\int_{k-1}^{k}\,\widetilde{L}(k)\,\left(\frac{\mathrm{d}}{\mathrm{d}t}\gamma(t\,L(t))\right)\,\mathrm{d}t\,.$

Since $\widetilde{L}(k)$ is asymptotic to $1/L(k)$ it is slowly varying, then
by (4.20) and [BGT87, Prop. 1.5.9a] also $w_{n}(E)$ is slowly varying. Thus,
$\alpha(n)$ is a regularly varying function with exponent $1$.

######  Lemma 4.7.

The limit in (4.18) holds for the sequence $(U_{j})$.

###### Proof.

For the sequence $(U_{j})$, by (4.13) and (4.19) we first write

$S_{U_{j}}f(x)=\sum_{n=1}^{j}\,(f^{E}\circ T^{n-1}_{{}_{E}})(x)\sim
G(U_{j}-w(U_{j},E,x))+\max_{1\leq k\leq j}\,G\Big{(}(\varphi_{{}_{E}}\circ
T^{k-1}_{{}_{E}})(x)\Big{)}\,.$

By (4.7)

$w(U_{j},E,x)=\max_{1\leq k\leq j}\,(\varphi_{{}_{E}}\circ
T^{k-1}_{{}_{E}})(x)$

and since $G(n)$ is non-decreasing it follows

$S_{U_{j}}f(x)\sim G(U_{j}-w(U_{j},E,x))+G(w(U_{j},E,x))\sim G(U_{j})$

by Lemma B.1. ∎

######  Lemma 4.8.

The limit in (4.18) holds for a sequence $(N_{j})$ such that
$N_{j}\in{\mathcal{I}}_{j}$ for all $j$.

###### Proof.

By (4.15) and (4.7), for all $j$ it holds

$w(N_{j},E,x)=\max_{1\leq k\leq R_{{}_{E,N_{j}}}(x)-1}\,(\varphi_{{}_{E}}\circ
T^{k-1}_{{}_{E}})(x)\,.$

Hence by (4.14) and (4.19) we have, since $G(n)$ is non-decreasing,

$\displaystyle S_{N_{j}}f(x)\sim$
$\displaystyle\,G(N_{j}-w(N_{j},E,x))+G(w(N_{j},E,x))+$
$\displaystyle+G\Big{(}\Big{(}\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N_{j}}}(x)-1}\Big{)}(x)\Big{)}-G\Big{(}\tau_{{}_{E,x}}(R_{{}_{E,N_{j}}}(x))-N+1\Big{)}\,.$

Moreover, since $(f^{E}\circ T_{{}_{E}}^{n-1})$ satisfies the assumptions of
Lemma 2.2 with $W=1$, we apply (2.10) to obtain

$G\Big{(}\Big{(}\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N_{j}}}(x)-1}\Big{)}(x)\Big{)}=o(d(R_{{}_{E,N_{j}}}(x)))\,,$

because by (4.15)

$G\Big{(}\Big{(}\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N_{j}}}(x)-1}\Big{)}(x)\Big{)}\leq G\Big{(}\max_{1\leq
k\leq R_{{}_{E,N_{j}}}(x)-1}\,(\varphi_{{}_{E}}\circ
T^{k-1}_{{}_{E}})(x)\Big{)}$

so that $G\Big{(}\Big{(}\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N_{j}}}(x)-1}\Big{)}(x)\Big{)}$ is the 2-nd maximum in

$\left\\{G\Big{(}(\varphi_{{}_{E}}\circ
T^{k-1}_{{}_{E}})(x)\Big{)}\,:\,k=1,\dots,R_{{}_{E,N_{j}}}(x)\right\\}\,.$

Since $G(n)$ is non-negative, by using (4.19) and Lemma B.1 the statement is
proved. ∎

######  Lemma 4.9.

The limit in (4.18) holds for a subsequence $(N_{j})$ such that
$N_{j}\in{\mathcal{J}}_{j}^{\eta}$ for all $j$.

###### Proof.

Looking at (4.14), we first consider the last terms on the right-hand side. By
the definition of $(U_{j})$ and (2.13)

(4.21) $\displaystyle\Big{(}\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N_{j}}}(x)-1}\Big{)}(x)=\Big{(}\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,U_{j}}}(x)-1}\Big{)}(x)=(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{j})(x)=U_{j+1}-U_{j}\,,$
$\displaystyle\tau_{{}_{E,x}}(R_{{}_{E,N_{j}}}(x))-N_{j}=\tau_{{}_{E,x}}(R_{{}_{E,U_{j}}}(x))-N_{j}=\tau_{{}_{E,x}}(j+1)-N_{j}=U_{j+1}-N_{j}$

and by (4.16), the sequence $U_{j+1}-U_{j}$ is diverging. Moreover writing

$(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{j})(x)=\Big{(}U_{j+1}-N_{j}+1\Big{)}+\Big{(}N_{j}-U_{j}-1\Big{)}$

and using that $N_{j}\in{\mathcal{J}}_{j}^{\eta}$, letting
$a_{j}:=U_{j+1}-N_{j}+1$ and $b_{j}:=N_{j}-U_{j}-1$, we have that $b_{j}$ is
diverging by (4.17). Then by Lemma B.1 we have

(4.22) $G\Big{(}\Big{(}\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N_{j}}}(x)-1}\Big{)}(x)\Big{)}\sim
G\Big{(}U_{j+1}-N_{j}+1\Big{)}+G\Big{(}N_{j}-U_{j}-1\Big{)}.$

Moreover, we use (4.22) and the fact that $G(n)=nL(n)$ with $L$ slowly varying
to prove that

(4.23) $G\Big{(}\Big{(}\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N_{j}}}(x)-1}\Big{)}(x)\Big{)}-G\Big{(}U_{j+1}-N_{j}+1\Big{)}\sim
G\Big{(}N_{j}-U_{j}-1\Big{)}\,.$

In fact, by (4.22) for all $\epsilon>0$ there exists $\bar{j}\in\mathbb{N}$
such that for all $j\geq\bar{j}$ we have

$1-\epsilon\leq\frac{G\Big{(}\Big{(}\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N_{j}}}(x)-1}\Big{)}(x)\Big{)}}{G\Big{(}U_{j+1}-N_{j}+1\Big{)}+G\Big{(}N_{j}-U_{j}-1\Big{)}}\leq
1+\epsilon.$

Since $N_{j}-U_{j}\geq 1$ and $\eta\in(0,\frac{1}{2})$, for
$N_{j}\in{\mathcal{J}}_{j}^{\eta}$ we have

$U_{j+1}-N_{j}+1\leq\frac{1}{2}(N_{j}-U_{j})+1\leq N_{j}-U_{j}-1\,,$

hence $G(U_{j+1}-N_{j}+1)\leq G(N_{j}-U_{j}-1)$ because $G$ is non-decreasing.
It follows that for $j\geq\bar{j}$

$1-2\epsilon\leq\frac{G\Big{(}\Big{(}\varphi_{{}_{E}}\circ
T_{{}_{E}}^{R_{{}_{E,N_{j}}}(x)-1}\Big{)}(x)\Big{)}-G\Big{(}U_{j+1}-N_{j}+1\Big{)}}{G\Big{(}N_{j}-U_{j}-1\Big{)}}\leq
1+2\epsilon\,.$

By the arbitrariness of $\epsilon$ we have proved (4.23).

Using (4.13) and Lemma 4.7 for the first two terms on the right-hand side of
(4.14), and (4.23) for the last difference of (4.14), we have

$S_{N_{j}}f(x)\sim G(U_{j})+G(N_{j}-U_{j}-1)\,.$

Finally Lemma B.1 and the fact that $N_{j}-U_{j}$ is diverging imply
$S_{N_{j}}f(x)\sim G(N_{j})$ and the lemma is proved. ∎

######  Lemma 4.10.

The limit in (4.18) holds for a subsequence $(N_{j})$ such that
$N_{j}\in{\mathcal{J}}_{j}\setminus{\mathcal{J}}_{j}^{\eta}$ for all $j$.

###### Proof.

As in the proof of Lemma 4.9 we need to study the last two terms on the right-
hand side of (4.14), for which we recall (4.21). For the first two terms,
Lemma 4.7 shows that they are asymptotic to $G(U_{j})$.

To estimate the third and fourth term of (4.14) we aim to apply Lemma B.2, for
which we use that $U_{j+1}-U_{j}$ tends to infinity. Moreover we now need to
consider two cases. Let us introduce the sets

${\mathcal{L}}^{0}_{j}:=\left\\{N\in{\mathcal{J}}_{j}\,:\,\lim_{j\to\infty}\,\frac{U_{j+1}-U_{j}}{U_{j}}=0\right\\}$

and

${\mathcal{L}}_{j}:=\left\\{N\in{\mathcal{J}}_{j}\,:\,\exists\,\ell\in(0,\infty]\,\text{
s.t.}\,\lim_{j\to\infty}\,\frac{U_{j+1}-U_{j}}{U_{j}}=\ell\right\\}\,.$

It is clear that for all sequences $(N_{j})$ there exists a subsequence in
${\mathcal{L}}^{0}_{j}$ or in ${\mathcal{L}}_{j}$. Thus we restrict ourselves
to the cases $N_{j}\in{\mathcal{L}}^{0}_{j}$ for all $j$, or
$N_{j}\in{\mathcal{L}}_{j}$ for all $j$.

First let
$N_{j}\in({\mathcal{J}}_{j}\setminus{\mathcal{J}}_{j}^{\eta})\cap{\mathcal{L}}^{0}_{j}$
for all $j$. Then $U_{j}\sim U_{j+1}$ and $N_{j}\in(U_{j},U_{j+1})$ imply
$G(U_{j})\sim G(U_{j+1})\sim G(N_{j})$ and Lemma 4.7 and the fact that
$S_{U_{j}}f(x)<S_{N_{j}}f(x)<S_{U_{j+1}}f(x)$ imply that the limit (4.18)
follows for the sequences in ${\mathcal{L}}^{0}_{j}$.

Let now
$N_{j}\in({\mathcal{J}}_{j}\setminus{\mathcal{J}}_{j}^{\eta})\cap{\mathcal{L}}_{j}$
for all $j$ and let $\ell\in(0,\infty)$ be the limit of
$(U_{j+1}-U_{j})/U_{j}$. Notice that in this case we also have
$\lim_{j}(U_{j+1}-N_{j})=\infty$. A similar argument works for the case
$\ell=\infty$. Letting $a_{j}:=U_{j+1}-U_{j}$ and $b_{j}:=U_{j+1}-N_{j}+1$, we
apply Lemma B.2 to write

(4.24)
$G(U_{j+1}-U_{j})-G(U_{j+1}-N_{j}+1)\sim(N_{j}-U_{j}-1)\,L(U_{j+1}-U_{j})\,.$

To study the term $L(U_{j+1}-U_{j})$, first by Lemma 2.6 for $\mu$-a.e. $x\in
E$ we have for $j$ large enough

(4.25) $U_{j+1}-U_{j}=\max_{1\leq k\leq
R_{{}_{E,U_{j}}}(x)}\,(\varphi_{{}_{E}}\circ T_{{}_{E}}^{k-1})(x)\leq
U_{j}\,\xi(U_{j})\,.$

Then we want to use assumption (ii)-(c) to obtain the asymptotic behaviour of
$L(U_{j+1}-U_{j})$.

Since
$N_{j}\in({\mathcal{J}}_{j}\setminus{\mathcal{J}}_{j}^{\eta})\cap{\mathcal{L}}_{j}$,
for all $\epsilon>0$ there exists $j$ big enough such that
$U_{j+1}-U_{j}\geq(\ell-\epsilon)U_{j}$. From (4.25) it follows that
$\tilde{\xi}(U_{j}):=(U_{j+1}-U_{j})/U_{j}$ satisfies the conditions of
assumption (ii)-(c), hence $L(U_{j+1}-U_{j})\sim L(U_{j})$. From (4.24) and
Lemma 4.7 we have

(4.26) $S_{N_{j}}f(x)\sim G(U_{j})+(N_{j}-U_{j}-1)\,L(U_{j})\,.$

At this point, if

$\liminf_{j\to\infty}\,\frac{N_{j}-U_{j}}{U_{j}}=0\,,$

for all $\epsilon>0$ and for $j$ big enough we have, up to the choice of a
subsequence, $0<N_{j}-U_{j}<\epsilon U_{j}$, and thus

$G(U_{j})+(N_{j}-U_{j}-1)\,L(U_{j})\leq
G(U_{j})+\epsilon\,U_{j}\,L(U_{j})\leq(1+\epsilon)\,G(N_{j})\,,$

and in the other direction

$G(U_{j})\geq
G\Big{(}\frac{1}{1+\epsilon}\,N_{j}\Big{)}\apprge\frac{1}{1+\epsilon}\,G(N_{j})\,.$

Hence by the arbitrariness of $\epsilon$, we find $S_{N_{j}}f(x)\sim G(N_{j})$
in (4.26).

Instead, if

$\liminf_{j\to\infty}\,\frac{N_{j}-U_{j}}{U_{j}}=\bar{\ell}>0\,,$

then for all $\epsilon>0$ and for $j$ big enough we have
$N_{j}-U_{j}\geq(\bar{\ell}-\epsilon)U_{j}$. Hence letting
$\tilde{\xi}(U_{j}):=(N_{j}-U_{j})/U_{j}$, we can apply assumption (ii)-(c)
since

$\bar{\ell}-\epsilon\leq\tilde{\xi}(U_{j})\leq\frac{U_{j+1}-U_{j}}{U_{j}}\leq\xi(U_{j})\,.$

Thus it follows that $L(N_{j}-U_{j})\sim L(U_{j})$, and in (4.26) we find

$S_{N_{j}}f(x)\sim G(U_{j})+(N_{j}-U_{j}-1)\,L(N_{j}-U_{j})\sim
G(U_{j})+G(N_{j}-U_{j})\sim G(N_{j})$

by applying Lemma B.1 and the fact that $(U_{j})$ and $(N_{j}-U_{j})$ are
diverging sequences.

This concludes the proof of the lemma. ∎

From Lemmas 4.7 \- 4.10 the limit in (4.18) holds and the theorem is proved
for functions $f$ constant on $E_{n}$.

In the general case, for a function $f:X\to\mathbb{R}_{\geq 0}$ satisfying
assumption (ii) we obtain that

(4.27) $N\,L_{1}(N)\sim S_{N}g_{1}(x)\leq S_{N}f(x)\leq S_{N}g_{2}(x)\sim
N\,L_{2}(N)$

for $\mu$-a.e. $x$, since we can apply Lemmas 4.7 \- 4.10 to $g_{1},g_{2}$.
Moreover, for $\mu$-a.e. $x\in A_{n}$ we have $g_{1}^{E}(n)\leq f^{E}(x)\leq
g_{2}^{E}(n)$ for all $n\in\mathbb{N}$. Since $g_{1}^{E}(n)\sim g_{2}^{E}(n)$
by assumption (ii)-(b), the function $G(n)$ which is asymptotically equivalent
to $f^{E}$ is well-defined and satisfies $G(n)\sim g_{i}^{E}(n)=nL_{i}(n)$ for
$i=1,2$. Thus by (4.27) we obtain $S_{N}f(x)\sim G(N)$ for $\mu$-a.e. $x$ and
the theorem is proved. ∎

##  5\. Proofs of the statements from Section 3

Proof of Proposition 3.4. In the proof of [BDM21, Theorem 3.1] it is shown
that $(E,S_{{}_{E}})$ is a fibred system with respect to the level sets of the
return time function $\varphi_{{}_{E}}$, hence it is $\psi$-mixing. It remains
to show that $\sum_{n\geq 1}\,\psi(n)/n<\infty$.

Using (3.3) we need to study the sequences $\sigma(k)$ and $\gamma(k)$.
Condition (h4) is satisfied with $N=1$ and $U_{1}=E$, so that $\gamma(k)\equiv
0$. Moreover it is proved in [BDM21] (see the proof of Proposition 3.9 and the
proof that (h3) holds for $(E,S_{{}_{E}})$), that there exists a constant
$C>0$ such that $\sigma(k)\leq C\,d(k)$, where $d(k)$ is defined by

$d(k)=\sum_{j=0}^{2}\left(\left|\frac{f_{k+j+2}}{f_{k+6}}-\frac{f_{k+j+1}}{f_{k+5}}\right|+\left|\frac{f_{k+j+2}}{f_{k+6}}-\frac{f_{k+j}}{f_{k+4}}\right|\right)$

being $(f_{k})$ the sequence recursively defined as

$f_{0}=0,\,f_{1}=1,\,f_{2}=0,\quad f_{k+3}=f_{k+2}+f_{k},\,\,\forall\,k\geq
0\,.$

Then given the distinct roots $\lambda,\mu,\bar{\mu}$ of the polynomial
$p(t)=t^{3}-t^{2}-1$, with $\lambda>1$ and
$|\mu|^{2}=|\bar{\mu}|^{2}=\lambda^{-1}<1$, there exist constants
$c_{1},c_{2},c_{3}\in\mathbb{C}$ such that

$f_{k}=c_{1}\,\lambda^{k}+c_{2}\,\mu^{k}+c_{3}\,\bar{\mu}^{k}\,,\quad\forall\,k\geq
0\,.$

It follows that $d(k)=O(\lambda^{-k})$, so that $\sigma(k)=O(\lambda^{-k})$,
and finally $\sum_{n\geq 1}\,\psi(n)/n<\infty$. ∎

Proof of Proposition 3.6. We show that the assumptions of Theorem 2.7 are
satisfied with $G(N)\sim N$ as $N\to\infty$. It is clear that assumptions
(ii)-(a), (b) and (c) are satisfied by (c1). In addition, since
$\Gamma_{i}(k)\sim k$ we can argue as in the proof of Lemma 4.3 to show that
(c2) implies (ii)-(d).

Thus it remains to show that (ii)-(e) is satisfied. First, in this case we
have $\alpha(n)\sim n/\ell(n)$, hence (2.19) becomes
$\ell(N/\ell(N))\sim\ell(N)$. Finally, using that $\ell(n)$ is an increasing
sequence which diverges, for all $\delta\in[0,1]$ we have the inequalities

$\ell\Big{(}n\,\ell^{\delta}(n)\Big{)}\geq\ell(n)$

and

$\ell\left(\frac{n\,\ell^{\delta}(n)}{\ell(n\,\ell^{\delta}(n))}\right)\leq\ell\left(\frac{n\,\ell(n)}{\ell(n)}\right)=\ell(n)\,.$

Thus by (c3)

$\ell(n)\leq\ell\Big{(}n\,\ell^{\delta}(n)\Big{)}\sim\ell\left(\frac{n\,\ell^{\delta}(n)}{\ell(n\,\ell^{\delta}(n))}\right)\leq\ell(n)$

and we have shown that

(5.1) $\lim_{N\to\infty}\,\frac{\ell(n\,\ell^{\delta}(n))}{\ell(n)}=1$

for all $\delta\in[0,1]$. This can be easily made uniform in $\delta$, proving
that $\ell$ is super-slowly varying at infinity with rate function itself (see
Definition B.5). However we need this property of $\ell$ only in the proof of
Lemma 4.5 to apply [BGT87, Cor. 2.3.4], for which we only need (5.1) with
$\delta=1$.

Thus all the assumptions of Theorem 2.7 are satisfied with $G(N)\sim
g_{i}^{E}(N)\sim N$. ∎

Proof of Proposition 3.9. All assumptions of Theorem 2.7 except for (ii)-(e)
are satisfied, hence we can make use of all the results leading to the proof
of Theorem 2.7 except for Lemma 4.5.

We will only show the divergence result of this proposition for the
subsequence $(U_{j})$ fulfiling $U_{j}=\tau_{{}_{E,x}}(j)$ for $j\geq 1$. By
(2.14) and (4.13) we have for this subsequence

(5.2) $\sum_{n=1}^{U_{j}}\,(f\circ T^{n-1})(x)\sim
d(R_{{}_{E,U_{j}}}(x)-1)+\max_{1\leq k\leq R_{{}_{E,U_{j}}}(x)-1}\,(f^{E}\circ
T^{k-1}_{{}_{E}})(x).$

To estimate the first summand we obtain by the same argumentation as in Lemma
4.5 that $d(R_{{}_{E,U_{j}}}(x))\sim d(\alpha(U_{j}-w(U_{j},E,x)))$. We may
use again $\mu(A_{\geq\Gamma_{n}})\sim\log n/n$ with which we calculate
$a(n)\sim 2\,n/\log^{2}(n)$ with $a$ as in (4.12). Hence, we obtain for its
asymptotic inverse function $d(n)\sim n\,\log^{2}(n)/2$. Using then
$\alpha(n)\sim n/\log n$ from above and (4.8) implies

(5.3)
$d(R_{{}_{E,U_{j}}}(x)-1)\sim\frac{1}{2}\,(U_{j}-w(U_{j},E,x))\,\log(U_{j}-w(U_{j},E,x)).$

On the other hand, for the second summand of (5.2) we obtain from

$(f^{E}\circ T_{{}_{E}}^{k-1})(x)\sim(\varphi_{{}_{E}}\circ
T_{{}_{E}}^{k-1})(x)\,\log(\varphi_{{}_{E}}\circ T_{{}_{E}}^{k-1})(x)\,,$

with $\sim$ in the meaning that $(f^{E}\circ T_{{}_{E}}^{k-1})(x)$ large, and
(4.7) that

(5.4) $\max_{1\leq k\leq R_{{}_{E,U_{j}}}(x)-1}\,(f^{E}\circ
T^{k-1}_{{}_{E}})(x)\sim w(U_{j},E,x)\,\log w(U_{j},E,x).$

From Theorem 2.4 and the fact that $\alpha$ is regularly varying with index
$1$ we can conclude that there exist $u_{1}^{\prime}<u_{2}^{\prime}$ such that
for $\mu$-a.e. $x\in X$ we have

$\liminf_{j}w(U_{j},E,x)/U_{j}<u_{1}^{\prime}<u_{2}^{\prime}<\limsup_{j}w(U_{j},E,x)/U_{j}\,.$

Indeed if the limit of $w(U_{j},E,x)/U_{j}$ existed, the result of Theorem 2.4
would contradict [A97, Thm. 2.4.2]. Hence, (5.2), (5.3) and (5.4) together
imply the statement of Proposition 3.9. ∎

##  Appendix A Proof of Lemma 2.2

Lemma 2.2 is an equivalent formulation to [AN03, Theorem 1.1]. Instead of the
condition in (2.9) this theorem asks for the following conditions to hold:
$J_{r}\coloneqq\sum_{n=1}^{\infty}\epsilon(n)^{r}/n<\infty$ with
$\epsilon(n)=n\,\left(\log^{+}L\right)^{\prime}(n)$ and
$L(t)=\mathbb{E}\left(\min\\{t,Y_{1}\\}\right)$. We have that

$\displaystyle L(t)$
$\displaystyle=\int_{0}^{t}x\mathrm{d}F(x)+t\,\left(1-F(t)\right)=t\,F(t)-\int_{0}^{t}F(x)\mathrm{d}x+t\,\left(1-F(t)\right)=\int_{0}^{t}(1-F(x))\mathrm{d}x.$

Furthermore,

$\displaystyle\left(\log^{+}L\right)^{\prime}(t)=\frac{L^{\prime}(t)}{L(t)}=\frac{1-F(t)}{\int_{0}^{t}(1-F(x))\mathrm{d}x}\quad\text{
and
}\quad\epsilon(t)=\frac{t\,\left(1-F(t)\right)}{\int_{0}^{t}(1-F(x))\mathrm{d}x}.$

Thus, the condition
$J_{r}\coloneqq\sum_{n=1}^{\infty}\epsilon(n)^{r}/n<\infty$ and (2.9) are
equivalent.

Moreover, in [AN03] the norming sequence $(d(n))$ is set to be the inverse of
$a(t)=t/L(t)=t/\int_{0}^{t}(1-F(x))\mathrm{d}x$. Since by the remark before
Theorem 1.1 in [AN03], $L$ is slowly varying (see Appendix B),
$\lim_{t\to\infty}t/L(t)=\infty$. Moreover, one can easily verify that
$a^{\prime}(t)>0$ on an interval $[K,\infty)$ and thus one can consider $a(t)$
as an invertible function.

Equation (2.10) follows from Theorem 1.1.(ii) and its following remark in
[AN03]. ∎

##  Appendix B Slowly varying functions

In this section we collect the results on slowly varying functions used in the
proof of Theorem 2.7. We recall that a function
$L:\mathbb{R}_{>0}\to\mathbb{R}$ is called _regularly varying (at infinity)
with index $\gamma\in\mathbb{R}$_ if for all $r>0$ it holds

$\lim_{x\to\infty}\,\frac{L(rx)}{L(x)}=r^{\gamma}\,.$

If $\gamma=0$, $L$ is called a _slowly varying (at infinity)_. We refer to
[BGT87] for more details. By Karamata’s representation theorem (see for
example [BGT87, Thm. 1.3.1]) we have that each slowly varying function $L$ can
be written as

(B.1) $\displaystyle
L(x)=c(x)\,\exp\left(\int_{\kappa}^{x}\frac{\eta(t)}{t}\mathrm{d}t\right),$

where $c(x)$ tends to a constant $C$ as $x\to\infty$, $\kappa\geq 0$ and
$\eta(x)$ tends to zero as $x\to\infty$, and a function $L$ is called
_normalized slowly varying_ if it can be written as

(B.2) $\displaystyle
L(x)=C\,\exp\left(\int_{\kappa}^{x}\frac{\eta(t)}{t}\mathrm{d}t\right).$

It is immediately clear that for each slowly varying function $L$ there exists
a normalised slowly varying function $\widetilde{L}$ such that
$L(x)\sim\widetilde{L}(x)$.

In the proof of our results we need _Potter’s bound_ , see for example [BGT87,
Thm. 1.5.6]. If $L$ is a slowly varying function at infinity, then for all
constants $\delta>0$ and $A>1$ there exists $C=C\left(\delta,A\right)$ such
that for all $x,y\geq C$ we have

(B.3) $\frac{L\left(x\right)}{L\left(y\right)}\leq
A\,\max\left\\{\left(\frac{x}{y}\right)^{\delta}\,,\left(\frac{y}{x}\right)^{\delta}\right\\}.$

######  Lemma B.1.

If $(a_{n})$ and $(b_{n})$ are two sequences of non-negative reals, at least
one of them diverging, and $L$ is non-negative and slowly varying such that
$(n\,L(n))$ is non-decreasing, then

$a_{n}\,L(a_{n})+b_{n}\,L(b_{n})\sim(a_{n}+b_{n})\,L(a_{n}+b_{n})\,.$

###### Proof.

We begin writing

$\displaystyle
a_{n}L(a_{n})+b_{n}L(b_{n})=\left(a_{n}+b_{n}\right)L(a_{n})-b_{n}\,\left(L(a_{n})-L(b_{n})\right).$

Without restriction of generality we assume that $a_{n}>b_{n}$ for all $n$.
(Otherwise we could just define two new sequences
$c_{n}:=\max\\{a_{n},b_{n}\\}$ and $d_{n}:=\min\\{a_{n},b_{n}\\}$ and continue
with these sequences.) By Potter’s bound (B.3) we have for all $\epsilon>0$
that there exists $N\in\mathbb{N}$ such that for $n\geq N$

$L(a_{n}+b_{n})\leq\left(1+\epsilon\right)L(a_{n})\,\left(\frac{a_{n}+b_{n}}{a_{n}}\right)^{\epsilon}\leq(1+\epsilon)\,2^{\epsilon}\,L(a_{n})$

and

$L(a_{n}+b_{n})\geq\left(1-\epsilon\right)L(a_{n})\,\left(\frac{a_{n}}{a_{n}+b_{n}}\right)^{\epsilon}\geq(1-\epsilon)\,2^{-\epsilon}\,L(a_{n})\,,$

implying $L(a_{n}+b_{n})\sim L(a_{n})$ and thus
$\left(a_{n}+b_{n}\right)L(a_{n})\sim\left(a_{n}+b_{n}\right)L(a_{n}+b_{n})$.

Next, we have a closer look at $b_{n}\,\left(L(a_{n})-L(b_{n})\right)$
distinguishing two cases. For a given $\epsilon>0$ we first look at the
subsequence $n_{k}$ fulfilling $b_{n_{k}}\leq\epsilon\,a_{n_{k}}$ for which we
have

$\displaystyle
b_{n_{k}}\,\left(L(a_{n_{k}})-L(b_{n_{k}})\right)<\epsilon\,\left(a_{n_{k}}+b_{n_{k}}\right)L(a_{n_{k}})\sim\epsilon\,\left(a_{n_{k}}+b_{n_{k}}\right)L\left(a_{n_{k}}+b_{n_{k}}\right)\,.$

On the other hand, $n\,L(n)$ being non-decreasing and non-negative we have

$\displaystyle b_{n_{k}}\,\left(L(a_{n_{k}})-L(b_{n_{k}})\right)$
$\displaystyle\geq-
b_{n_{k}}\,L\left(b_{n_{k}}\right)\geq-\epsilon\,\left(a_{n_{k}}+b_{n_{k}}\right)\,L\left(\epsilon\,\left(a_{n_{k}}+b_{n_{k}}\right)\right)$
$\displaystyle\sim-\epsilon\,\left(a_{n_{k}}+b_{n_{k}}\right)\,L\left(a_{n_{k}}+b_{n_{k}}\right).$

Next, we look at the subsequence fulfilling $b_{n_{k}}>\epsilon\,a_{n_{k}}$.
Applying again (B.3), we have that for $k$ big enough

$L(a_{n_{k}}+b_{n_{k}})\leq(1+\epsilon)\,L(b_{n_{k}})\,\left(\frac{a_{n_{k}}+b_{n_{k}}}{b_{n_{k}}}\right)^{\epsilon}\leq\frac{(1+\epsilon)^{1+\epsilon}}{\epsilon^{\epsilon}}\,L(b_{n_{k}})$

and

$L(a_{n_{k}}+b_{n_{k}})\geq\frac{1}{1+\epsilon}\,L(b_{n_{k}})\,\left(\frac{b_{n_{k}}}{a_{n_{k}}+b_{n_{k}}}\right)^{\epsilon}\geq\frac{\epsilon^{\epsilon}}{(1+\epsilon)^{1+\epsilon}}\,L(b_{n_{k}})\,.$

Since $L(a_{n_{k}}+b_{n_{k}})\sim L(a_{n_{k}})$ we also have that for $k$ big
enough

$\frac{1}{1+\epsilon}\leq\frac{L(a_{n_{k}}+b_{n_{k}})}{L(a_{n_{k}})}\leq
1+\epsilon\,,$

therefore there is a constant $c>0$ such that choosing $k$ big enough we have

$\displaystyle b_{n_{k}}\,\left(L(a_{n_{k}})-L(b_{n_{k}})\right)$
$\displaystyle\leq(a_{n_{k}}+b_{n_{k}})\left((1+\epsilon)L(a_{n_{k}}+b_{n_{k}})-\frac{\epsilon^{\epsilon}}{(1+\epsilon)^{1+\epsilon}}\,L(a_{n_{k}}+b_{n_{k}})\right)$
$\displaystyle\leq
c\,\epsilon^{1/2}\,(a_{n_{k}}+b_{n_{k}})\,L(a_{n_{k}}+b_{n_{k}})$

for $\epsilon\in(0,1)$, and similarly

$\displaystyle b_{n_{k}}\,\left(L(a_{n_{k}})-L(b_{n_{k}})\right)$
$\displaystyle\geq
b_{n_{k}}\left(\frac{1}{1+\epsilon}\,L(a_{n_{k}}+b_{n_{k}})-\frac{(1+\epsilon)^{1+\epsilon}}{\epsilon^{\epsilon}}\,L(a_{n_{k}}+b_{n_{k}})\right)$
$\displaystyle\geq-c\,\epsilon^{1/2}\,(a_{n_{k}}+b_{n_{k}})\,L(a_{n_{k}}+b_{n_{k}})$

for $\epsilon\in(0,1)$.

Since $\epsilon$ was chosen arbitrarily, we have that

$b_{n_{k}}\,\left(L(a_{n_{k}})-L(b_{n_{k}})\right)=o\left((a_{n_{k}}+b_{n_{k}})\,L(a_{n_{k}}+b_{n_{k}})\right)\,,$

and the lemma is proved. ∎

######  Lemma B.2.

Let $(a_{n})$ and $(b_{n})$ be two sequences of non-negative reals with
$a_{n}>b_{n}$ and $(a_{n})$ tending to infinity. Further let $L$ be a non-
negative normalized slowly varying function. Then

(B.4) $\displaystyle a_{n}L(a_{n})-b_{n}L(b_{n})\sim(a_{n}-b_{n})L(a_{n})\,.$

###### Proof.

First we look at a subsequence $n_{j}$ for which $b_{n_{j}}$ is bounded, in
this case (B.4) immediately holds. Hence, we may assume without loss of
generality that also $b_{n}$ tends to infinity. We have

$\displaystyle a_{n}L(a_{n})-b_{n}L(b_{n})$
$\displaystyle=\left(a_{n}-b_{n}\right)L(a_{n})+b_{n}\,\left(L(a_{n})-L(b_{n})\right)$
$\displaystyle=b_{n}\left(\frac{a_{n}}{b_{n}}-1\right)L(a_{n})+b_{n}\,\left(1-\frac{L(b_{n})}{L(a_{n})}\right)\,L(a_{n}).$

Hence, we have to show that

$\displaystyle
1-\frac{L(b_{n})}{L(a_{n})}=o\left(\frac{a_{n}}{b_{n}}-1\right).$

To do so, using (B.2) we note that

$\displaystyle\frac{L(a_{n})}{L(b_{n})}$
$\displaystyle=\exp\left(\int_{b_{n}}^{a_{n}}\frac{\eta(t)}{t}\mathrm{d}t\right)\leq\exp\left(\sup_{y\in[b_{n},a_{n}]}\eta(y)\int_{b_{n}}^{a_{n}}\frac{1}{t}\mathrm{d}t\right)$

and similarly

$\displaystyle\frac{L(a_{n})}{L(b_{n})}$
$\displaystyle=\exp\left(\int_{b_{n}}^{a_{n}}\frac{\eta(t)}{t}\mathrm{d}t\right)\geq\exp\left(\inf_{y\in[b_{n},a_{n}]}\eta(y)\int_{b_{n}}^{a_{n}}\frac{1}{t}\mathrm{d}t\right).$

Since $(b_{n})$ tends to infinity and $\eta$ tends to zero we can conclude
that for all $\epsilon>0$ there exists $N\in\mathbb{N}$ such that for all
$n\geq N$ we have

$\displaystyle\left(\frac{a_{n}}{b_{n}}\right)^{-\epsilon}$
$\displaystyle\leq\frac{L(a_{n})}{L(b_{n})}\leq\left(\frac{a_{n}}{b_{n}}\right)^{\epsilon}.$

Hence, for all $\epsilon>0$ there exists $N\in\mathbb{N}$ such that for all
$n\geq N$

$\displaystyle\frac{\left|1-\frac{L(b_{n})}{L(a_{n})}\right|}{\left|\frac{a_{n}}{b_{n}}-1\right|}$
$\displaystyle\leq\frac{\left(\frac{a_{n}}{b_{n}}\right)^{\epsilon}-1}{\frac{a_{n}}{b_{n}}-1}.$

Since the function $g\colon(1,\infty)\to\mathbb{R}$ given by
$(x^{\epsilon}-1)/(x-1)$ is monotonically decreasing and tends to $\epsilon$
for $x\to 1^{+}$, it follows that for all $\epsilon>0$ there exists
$N\in\mathbb{N}$ such that for all $n\geq N$

$\displaystyle\left|1-\frac{L(b_{n})}{L(a_{n})}\right|\leq\epsilon\,\left|\frac{a_{n}}{b_{n}}-1\right|\,.$

Since $\epsilon$ was arbitrary, the statement is proven. ∎

######  Lemma B.3.

Let $L$ be a slowly varying function at infinity. Then there exist two
normalised slowly varying functions $L^{-}$ and $L^{+}$ such that
$L^{-}(n)\leq L(n)\leq L^{+}(n)$ for all $n\in\mathbb{N}$ and $L^{-}(n)\sim
L(n)\sim L^{+}(n)$ as $n\to\infty$.

###### Proof.

Let $L$ be represented as in (B.1) for fixed $c(x)$, $\kappa$ and $\eta(x)$.
Our approach is to find two normalised slowly varying functions $c^{-},c^{+}$
fulfilling $c^{-}\leq c\leq c^{+}$ and $c^{-}(n)\sim c(n)\sim c^{+}(n)$. Then
we may set

$L^{\pm}(n)=c^{\pm}(n)\,\exp\left(\int_{\kappa}^{n}\eta(t)/t\mathrm{d}t\right)\,.$

Since a product of two normalised slowly varying functions is still normalised
slowly varying, the functions $L^{\pm}$ fulfil all the required properties. We
will in the following only give the construction of $c^{-}$ as the
construction of $c^{+}$ follows analogously.

We notice that a function $\ell$ is normalised slowly varying if it fulfils
$\ell^{\prime}(x)=o\left(\ell(x)/x\right)$ almost everywhere. This follows
immediately by taking the derivative of (B.2), but see also [BGT87, p. 15]. We
construct a function $c^{-}$ on a line $[\gamma_{1},\infty)$ which is
continuous and piecewise differentiable.

In order to define $c^{-}$ let $\Gamma\coloneqq\left\\{n\in\mathbb{N}\colon
c(n)<c(k)\text{ for all }k>n\right\\}$ and let $(\gamma_{n})$ be the ordered
sequence of elements in $\Gamma$, that is we have
$\gamma_{1}<\gamma_{2}<\ldots$ and $c(\gamma_{1})<c(\gamma_{2})<\ldots$. The
set $\Gamma$ contains infinitely many elements unless
$\min\left\\{c(n),C\right\\}=C$ for sufficiently large $n$ which is a trivial
case in which we might simply set $c^{-}(n)=\min\left\\{c(n),C\right\\}$. In
this case $(c^{-})^{\prime}(x)=0$ for $x$ sufficiently large and thus
$(c^{-})^{\prime}(x)\leq(x\log x)^{-1}$ holds immediately.

As we are only interested in the limit behaviour, it is sufficient to define
$c^{-}$ at $\mathbb{R}_{\geq\gamma_{1}}$ and we set
$c^{-}(\gamma_{i})=c(\gamma_{i})$, for all $i\in\mathbb{N}$. For
$x\in(\gamma_{i},\gamma_{i}+1)$ we set

$\displaystyle
c^{-}(x):=\max\left\\{(\log\,x)^{1/2}+c(\gamma_{i})-(\log\,\gamma_{i})^{1/2},c(\gamma_{i+1})\right\\}.$

The function $c^{-}$ is then either piecewise constant and on those parts its
derivative equals zero, or has derivative equal to
$1/2(\log\,x)^{-1/2}\,x^{-1}=o(x^{-1})$. This is sufficient since in our case,
$c^{-}$ is bounded by $C$. Thus, at those points at which $c^{-}$ is
differentiable, it fulfils
$\left(c^{-}\right)^{\prime}(x)=o\left(c^{-}(x)/x\right)$ and there are only
countably many points on which it is not differentiable. On the other hand,
since $(\log\,x)^{1/2}$ tends to infinity, $c^{-}$ tends to $C$. ∎

######  Lemma B.4.

Let $h$ be a slowly varying function with $h(t)\geq 1+h_{0}$ for all $t$ and
for some $h_{0}>0$, and let $L$ be a slowly varying function written as in
(B.1) with $\eta(t)=o(1/\log\xi(t))$ as $t\to\infty$. Then for all functions
$\tilde{h}$ which satisfy $c\leq\tilde{h}(n)\leq h(n)$ for some constant $c>0$
and all $n$, it holds $L(n\,\tilde{h}(n))\sim L(n)$ as $n\to\infty$.

The proof of the lemma follows by applying results in the theory of functions
with regular variations for which we first need to introduce some definitions.

######  Definition B.5.

Given a function $h:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$, a function
$\ell:\mathbb{R}_{>0}\to\mathbb{R}$ is called _super-slowly varying at
infinity with rate function_ $h$ if

$\lim_{x\to\infty}\frac{\ell\left(x\,h(x)^{\delta}\right)}{\ell(x)}=1\text{
uniformly in }0\leq\delta\leq 1.$

######  Definition B.6.

A function $\phi:\mathbb{R}_{>0}\to\mathbb{R}$ is called _self-controlled_ if
there exist positive constants $\gamma,\Gamma,T$ such that
$\gamma\phi(t)\leq\phi(t+\delta\phi(t))\leq\Gamma\phi(t)$, for all $t\geq T$
and $\delta\in[0,1]$.

######  Lemma B.7 ([BGT87, Thm. 3.12.5]).

Let $h:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ be such that
$\phi(t):=\log(h(e^{t}))$ is self-controlled and there exists $T>0$ such that
$1/\phi$ is locally integrable over $[T,\infty)$. Then
$L:\mathbb{R}_{>0}\to\mathbb{R}$ is super-slowly varying with rate function
$h$ if and only if $L$ can be written as in (B.1) with $\eta(t)=o\left(1/\log
h(t)\right)$ as $t\to\infty$.

_Proof of Lemma B.4._ Let us use Lemma B.7. Since $h$ is a slowly varying
function and $h(t)\geq 1+h_{0}$ for all $t$, the function
$\phi(t):=\log(h(e^{t}))$ is such that $1/\phi$ is locally integrable.
Moreover we notice that
$\phi(t+\delta\phi(t))=\log(h(e^{t}\,h(e^{t})^{\delta}))$ and by using
Potter’s bound (B.3) for $h$ we obtain for all $\epsilon>0$ that there exists
$T>0$ such that for $t>T$ we have

$\frac{h(e^{t}\,h(e^{t})^{\delta})}{h(e^{t})}\leq(1+\epsilon)\,\max\left\\{h(e^{t})^{\delta\epsilon}\,,\,h(e^{t})^{-\delta\epsilon}\right\\}\,.$

Since $h(t)\geq 1+h_{0}$, we have $\phi(t)\geq\log(1+h_{0})$, hence for all
$\epsilon\in(0,h_{0})$

$\phi(t+\delta\phi(t))\leq\log(1+\epsilon)+(1+\delta\epsilon)\,\phi(t)\leq(2+\delta\epsilon)\,\phi(t)\,.$

Analogously, we obtain
$\phi(t+\delta\phi(t))\geq\left(\left(1-\delta\epsilon\right)+\log(1-\epsilon)/\log(1+h_{0})\right)\phi(t)$.
If we choose $\epsilon$ sufficiently small, then the factor before $\phi(t)$
is bounded away from zero which implies that $\phi$ is self-controlled.

Then by Lemma B.7, we obtain that if $L$ is a slowly varying function written
as in (B.1) with $\eta(t)=o(1/\log h(t))$ as $t\to\infty$, it is super-slowly
varying with rate function $h$. Let now $\tilde{h}$ be a function satisfying
$c\leq\tilde{h}(n)\leq h(n)$ for all $n$ and for a constant $c>0$. Up to
dividing by $c$ and using that $\eta(t)=o(1/\log(h(t)/c))$ as $t\to\infty$, we
can set $c=1$. Then we can write $\tilde{h}(n)=h(n)^{\delta(n)}$ where
$\delta(n)\in[0,1]$ for all $n$. By Definition B.5 it follows

$\left|\frac{L(n\,\tilde{h}(n))}{L(n)}-1\right|=\left|\frac{L(n\,h(n)^{\delta(n)})}{L(n)}-1\right|\leq\sup_{\delta\in[0,1]}\left|\frac{L(n\,h(n)^{\delta})}{L(n)}-1\right|\to
0\quad\text{as $n\to\infty$.}$

and the lemma is proved. ∎

## References

  * [A97] J. Aaronson, “An introduction to infinite ergodic theory”, Mathematical Surveys and Monographs 50, American Mathematical Society, Providence, RI, 1997.
  * [AN03] J. Aaronson, H. Nakada, _Trimmed sums for non-negative, mixing stationary processes_ , Stochastic Process. Appl. 104 (2003), no. 2, 173–192.
  * [AN05] J. Aaronson, H. Nakada, _On the mixing coefficients of piecewise monotonic maps_ , Israel J. Math. 148 (2005), 1–10.
  * [BGT87] N. H. Bingham, C. M. Goldie, J. L. Teugels, “Regular variation”, Encyclopedia of Mathematics and its Applications 27, Cambridge University Press, Cambridge, 1987.
  * [BDM21] C. Bonanno, A. Del Vigna, S. Munday, _A slow triangle map with a segment of indifferent fixed points and a complete tree of rational pairs_ , Monatsh. Math. 194 (2021), no. 1, 1–40.
  * [BGL18] C. Bonanno, P. Giulietti, M. Lenci, _Infinite mixing for one-dimensional maps with an indifferent fixed point_ , Nonlinearity 31 (2018), no. 11, 5180–5213.
  * [BL21] C. Bonanno, M. Lenci, _Pomeau-Manneville maps are global-local mixing_ , Discrete Contin. Dyn. Syst. 41 (2021), no. 3, 1051–1069.
  * [B05] R. C. Bradley, _Basic properties of strong mixing conditions. A survey and some open questions_ , Probab. Surv. 2 (2005), 107–144.
  * [DV86] H. G. Diamond, J. D. Vaaler, _Estimates for partial sums of continued fraction partial quotients_ , Pacific J. Math. 122 (1986), no. 1, 73–82.
  * [GHPZ] S. Galatolo, M. Holland, T. Persson, Y. Zhang, _Anomalous time-scaling of extreme events in infinite systems and Birkhoff sums of infinite observables_ , Discrete Contin. Dyn. Syst. 41 (2021), no. 4, 1799–1841.
  * [G01] T. Garrity, _On periodic sequences for algebraic numbers_ , J. Number Theory 88 (2001), no. 1, 86–103.
  * [H93] E. Haeusler, _A nonstandard law of the iterated logarithm for trimmed sums_ , Ann. Probab. 21 (1993), no. 2, 831–860.
  * [HM87] E. Haeusler, D. M. Mason, _Laws of the iterated logarithm for sums of the middle portion of the sample_ , Math. Proc. Cambridge Philos. Soc. 101 (1987), no. 2, 301–312.
  * [H14] A. Haynes, _Quantitative ergodic theorems for weakly integrable functions_ , Ergodic Theory Dynam. Systems 34 (2014), no. 2, 534–542.
  * [KS19a] M. Kesseböhmer, T. I. Schindler, _Strong laws of large numbers for intermediately trimmed sums of i.i.d. random variables with infinite mean_ , J. Theoret. Probab. 32 (2019), no. 2, 702–720.
  * [KS19b] M. Kesseböhmer, T. I. Schindler, _Strong laws of large numbers for intermediately trimmed Birkhoff sums of observables with infinite mean_ , Stochastic Process. Appl. 129 (2019), no. 10, 4163–4207. Corrigendum in Stochastic Process. Appl. 130 (2020), no. 11, 7019.
  * [KS20a] M. Kesseböhmer, T. I. Schindler, _Intermediately trimmed strong laws for Birkhoff sums on subshifts of finite type_ , Dyn. Syst. 35 (2020), no. 2, 275–305.
  * [KS20b] M. Kesseböhmer, T. I. Schindler, _Mean convergence for intermediately trimmed Birkhoff sums of observables with regularly varying tails_ , Nonlinearity 33 (2020), no. 10, 5543–5566.
  * [KM92] H. Kesten, R. A. Maller, _Ratios of trimmed sums and order statistics_ , Ann. Probab. 20 (1992), no. 4, 1805–1842.
  * [LM18] M. Lenci, S. Munday, _Pointwise convergence of Birkhoff averages for global observables_ , Chaos 28 (2018), no. 8, 083111, 16 pp.
  * [M06] F. Móricz, _On the harmonic averages of numerical sequences_ , Arch. Math. (Basel) 86 (2006), no. 4, 375–384.
  * [NN03] H. Nakada, R. Natsui, _On the metrical theory of continued fraction mixing fibred systems and its application to Jacobi-Perron algorithm_ , Monatsh. Math. 138 (2003), no. 4, 267–288.
  * [S18] T. I. Schindler, _Trimmed sums for observables on the doubling map_ , arXiv:1810.03223 [math.DS]
  * [S95] F. Schweiger, “Ergodic theory of fibred systems and metric number theory”, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
  * [T83] M. Thaler, _Transformations on $[0,1]$ with infinite invariant measures_, Israel J. Math. 46 (1983), no. 1-2, 67–96.
  * [Z00] R. Zweimüller, _Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points_ , Ergodic Theory Dynam. Systems 20 (2000), no. 5, 1519–1549.
  * [Z09] R. Zweimüller, “Surrey notes on infinite ergodic theory”,
http://mat.univie.ac.at/%7Ezweimueller/MyPub/SurreyNotes.pdf