# Poisson Limit Theorems for Systems with Product Structure

M. Auer

The author thanks the University of Maryland for their hospitality during this
work. Special thanks go my advisor D. Dolgopyat for suggesting this problem,
and providing helpful comments throughout. Thanks also go to A. Kanigowski,
for supplying his knowledge on many of the examples.

00footnotetext: Key words ergodic theory, hyperbolic dynamical system, limit
distribution, Poisson process, rare events

##### Abstract

We obtain a Poisson Limit for return times to small sets for product systems.
Only one factor is required to be hyperbolic while the second factor is only
required to satisfy polynomial deviation bounds for ergodic sums. In
particular, the second fact can be either elliptic or parabolic. As an
application of our main result, several maps of the form Anosov map $\times$
another map are shown to satisfy a Poisson Limit Theorem at typical points,
some even at all points.
The methods can be extended to certain types of skew products, including
$T,T^{-1}$-maps of high rank.

##  Part I Results

###  1\. Introduction

One of the prominent limit theorems in classical probability theory is the
Poisson Limit Theorem. This is taken advantage of in a variety of
probabilistic models describing waiting times until unlikely events occur; due
to the Poisson Limit Theorem (PLT) such waiting times are usually well
approximated by exponentially distributed variables. It has been a great
discovery that, many deterministic systems satisfy the same kind of limit
theorems for rare events.

Limit distributions of waiting times are most classical for mixing Markov
chains, there one considers returns to small cylinders, for example see
[Pit91]. As remarked there, this result can be immediately generalized to
systems with a Markov partition, the only caveat being that the sets are still
cylinders, so geometrically not the most intuitive class. Nonetheless, waiting
time limits for returns to small balls can be shown in concrete settings; for
example hyperbolic toral automorphisms [DGS04], Gibbs-Markov maps [Zha21],
partially hyperbolic maps [Dol04], some intermittent interval maps [CG93], and
many more. It is sometimes interesting to also ask for explicit rates of
convergence, this can be shown using eg $\alpha$-mixing, see [HSV99], [AS11],
[HV09].
For systems having a spectral gap for the Perron-Frobenius transfer operator
there is a particularly nice method of proving Poisson limits, making use of
the relation of hitting and return time limits obtained in [HSV99]. The
argument has been carried out in [Zwe19], [Zwe16].
Another method is to induce a given system on a small set, such that the
induced system has limits for return times, then one can use methods of
[HWZ14], [BSTV03] to prove the same limits for the original system. This can
be used to show return times limits for certain quadratic maps, or expanding
maps with an indifferent fixed point.
We do not make any claims on completeness of the list of reference given
above, for a more complete picture see [LFF+16].
Some related topics are extreme value laws [FFT11], [FFV17], spatiotemporal
limits [PS20], [Zwe] or Borel-Cantelli like Lemmas [HLSW22], [HNPV13],
[DFL22a].
Similar questions can be asked for flows as well, this topic has not been
studied as thoroughly as the question for maps. As shown in [PY17] for
suspension this reduces to the study for maps.
Poisson Limit Theorem for flows can be reduced to Poisson Limit Theorem for
time 1 map with the target being the set of points which visit $B(x,r)$ in the
next unit of time.

###  2\. Preliminaries

######  Definition 1.

Given a probability-preserving ergodic dynamical system
$(X,\mathcal{A},\mu,T)$ and a measurable set $A\in\mathcal{A}$, we will define
the first return time to $A$ as

$\varphi_{A}(x)=\min(n\geq 1\;|\;T^{n}(x)\in A).$

The first return map shall be denoted by $T_{A}(x)=T^{\varphi_{A}(x)}(x)$, and
the sequence of consecutive returns by

$\Phi_{A}=(\varphi_{A},\varphi_{A}\circ T_{A},\varphi_{A}\circ
T_{A}^{2},...).$

In the following, for some measurable set $A\in\mathcal{A}$ with $\mu(A)>0$,
the measure conditional on $A$ shall be given as $\mu_{A}(B)=\frac{\mu(A\cap
B)}{\mu(A)}$.
The first important result in the study of $\varphi_{A}$, was Kac’s formula,
which calculates the expectation as

$\int_{A}\varphi_{A}\,\text{d}\mu_{A}=\frac{1}{\mu(A)}.$

Hence we will study limits of $\mu(A)\varphi_{A}$ as $\mu(A)\rightarrow 0$.
More explicitly let $(A_{l})_{l\geq 1}$ be a sequence of rare events, that is,
each $A_{l}$ is measurable with $\mu(A_{l})\rightarrow 0$, we want to find
weak limits of the form

$\mu(A_{l})\Phi_{A_{l}}\overset{\mu}{\Rightarrow}\Phi\;\;\;\textit{as
}l\rightarrow\infty,$

or

$\mu(A_{l})\Phi_{A_{l}}\overset{\mu_{A_{l}}}{\Rightarrow}\tilde{\Phi}\;\;\;\textit{as
}l\rightarrow\infty,$

where $\Rightarrow$ denotes convergence in distribution. In the above
situation we shall call $\Phi$ the hitting time limit and $\tilde{\Phi}$ the
return time limit. An important fact is that the hitting and return time
limits are intimately related. In particular an important consequence is the
following.

######  Proposition 2.

([HSV99, Theorem 2.1]) Let $(X,\mathcal{A},\mu,T)$ be an ergodic probability
preserving dynamical system and let $(A_{l})_{l\geq 1}$ be a sequence of rare
events. Then $\Phi\overset{d}{=}\Phi_{Exp}$ is equivalent to
$\tilde{\Phi}\overset{d}{=}\Phi_{Exp}$, where $\Phi_{Exp}$ is an iid process
of standard exponentially distributed random variables.

This suggests that we should expect exponential hitting and return time limits
for geometrically sensible sequences of rare events.
In the following let $X$ be a $C^{r}$ Riemannian manifold with $\dim(X)=d$ and
assume $\mu\ll m_{X}$ the volume on $X$, with continuous density. Most of the
statements can be reformulated to hold for arbitrary invariant $\mu$, but for
the sake of simplicity we shall keep this assumption.

######  Definition 3.

  1. (i)

Let $x^{*}\in X$ and, for $r>0$, denote by $B_{r}(x^{*})$ the geodesic ball of
radius $r$ centred at $x^{*}$. We will say that $T$ satisfies the PLT at
$x^{*}$ if

$\mu(B_{r}(x^{*}))\Phi_{B_{r}(x^{*})}\overset{\mu}{\Rightarrow}\Phi_{Exp}\;\;\;\textit{
as }r\rightarrow 0.$

  2. (ii)

Let

$PLT:=\\{x^{*}\in X\;|\;\textit{$T$ satisfies the PLT at $x^{*}$}\\}.$

If $\mu(PLT)=1$ we will say that $T$ satisfies the PLT almost everywhere, and
if $PLT=X$ we will say that $T$ satisfies the PLT everywhere.

If $T$ is Lipschitz-continuous along the (finite) orbit of a periodic point,
then it does not satisfy the PLT at that point. To see this note first that
the PLT at $x^{*}$ in particular implies via Theorem 2, that for each $N\geq
1$,

$\mu_{B_{r}(x^{*})}(\varphi_{B_{r}(x^{*})}\leq N)\rightarrow 0\;\;\;\textit{
as }r\rightarrow 0.$

Now suppose $x^{*}$ is a point with period $p$, and $|T^{p}(x)-T^{p}(y)|\leq
C|x-y|$ near $x^{*}$ then

$\mu_{B_{r}(x^{*})}(\varphi_{B_{r}(x^{*})}\leq
p)\geq\mu_{B_{r}(x^{*})}(B_{\frac{r}{C}}(x^{*}))=\frac{1}{C^{d}}+o(1).$

In particular chaotic (in the sense of Devaney) transformations can never
satisfy the PLT everywhere, nevertheless we shall construct maps that satisfy
the PLT everywhere. The main goal is to prove that a big class of product
systems satisfies the PLT (almost) everywhere.

###  3\. The PLT for (skew-)products

In this paper we study the PLT for systems that can be written as a product
(or skew product of a special type). Therefore let $Y$ be another Riemannian
$C^{r^{\prime}}$-manifold with $\dim(Y)=d^{\prime}$, and assume
$R:Y\rightarrow Y$ preserves a probability measure $\nu\ll m_{Y}$ with
continuous density. Instead of $T:X\rightarrow X$, consider now some
$T:X\times Y\rightarrow X$. We will prove the PLT for certain systems of the
form $S(x,y)=(T(x,y),R(y))$. The case of direct products can be recovered if
$T(x,y)=T(x)$ is independent of $y$ (which will be the case for most of our
examples). Denote also $T_{y}(x)=T(x,y)$. We will assume that $T_{y}$
preserves a probability measure $\mu$ (independent of $y$). For $n\geq 1$, let
$T^{n}_{y}(x)=T_{R^{n-1}(y)}(T_{R^{n-2}(y)}(...(T_{y}(x))))$. For $y\in Y$ we
introduce analogously the fiberwise return times as

$\displaystyle\varphi_{A,y}^{(1)}(x):=\varphi_{A,y}(x):=\min(j\geq
1\;|\;T_{y}^{j}(x)\in A)$ (1)
$\displaystyle\varphi_{A,y}^{(n+1)}(x):=\min(j\geq\varphi_{A,y}^{(n)}+1\;|\;T_{y}^{j}(x)\in
A)$
$\displaystyle\Phi_{A,y}:=(\varphi_{A,y}^{(1)},\varphi_{A,y}^{(2)},\dots).$

All of the above can be equivalently expressed as the usual returns times,
where we only fix a set in the fiber, e.g

$\varphi_{A,y}(x)=\varphi_{A\times Y}(x,y)=\min(j\geq 1\;|\;S^{j}(x,y)\in
A\times Y),$

but for the sake of the conditions to follow, we think it is best to keep in
mind the definitions in (1) as it emphasises that $y$ should be regarded as
fixed.

We will list here the main assumptions we make in order to prove the PLT.

Here and in the following we will use $\varphi$ to denote first return times
w.r.t different systems, it will however always be clear which system we mean
from the specified set.

  * (MEM)

For almost all $y\in Y$ there are constants $r>0$, $C>1$ and $\gamma>0$ such
that

$\displaystyle\left|\int_{X}\right.$
$\displaystyle\left.\prod_{j=0}^{n-1}f_{j}\circ
T_{y}^{k_{j}}\,\text{d}\mu-\prod_{j=0}^{n-1}\int_{X}f_{j}\,\text{d}\mu\right|$
(2) $\displaystyle\leq Ce^{-\gamma\min_{0\leq j_{1}<j_{2}\leq
n-1}|k_{j_{1}}-k_{j_{2}}|}\prod_{j=0}^{n-1}||f_{j}||_{r},$

for $n\geq 1$, $f_{0},...,f_{n-1}\in C^{r}$ and $0\leq k_{0}\leq...\leq
k_{n-1}$.

  * (EE)

There are $r^{\prime}>0$ and $0<\delta_{1}<1$ such that

$\left|\left|\sum_{j=0}^{n-1}f\circ R^{j}-n\int
f\,\text{d}\nu\right|\right|_{L^{2}(\nu)}\leq
C||f||_{C^{r^{\prime}}}n^{\delta_{1}}\;\;\;\forall f\in
C^{r^{\prime}},\;\forall n\geq 1.$ (3)

  * (LR$(y^{*})$)

There is a $c>0$ such that, for $r>0$ and $\nu$-a.e $y\in B_{r}(y^{*})$, we
have

$\varphi_{B_{r}(y^{*})}(y)\geq c|\log(r)|.$ (4)

  * (SLR$(y^{*})$)

There is a $\psi:(0,\infty)\rightarrow(0,\infty)$ with $|\log(r)|=o(\psi(r))$
as $r\rightarrow 0$ such that, for $r>0$ and $\nu$-a.e $y\in B_{r}(y^{*})$, we
have

$\varphi_{B_{r}(y^{*})}(y)\geq\psi(r).$ (5)

  * (NSR$(x^{*})$)

There is a $\xi:(0,\infty)\rightarrow(0,\infty)$ with $|\log(r)|=o(\xi(r))$ as
$r\rightarrow 0$ such that, for $\nu$-a.e $y\in Y$, we have

$\mu_{B_{r}(x^{*})}(\varphi_{B_{r}(x^{*}),y}\leq\xi(r))\rightarrow
0\;\;\;\textit{ as }r\rightarrow 0.$ (6)

  * (BR$(x^{*},y^{*})$)

One of the following is satisfied

    * –

(SLR$(y^{*})$),

    * –

(NSR$(x^{*})$) AND (LR$(y^{*})$),

    * –

or (NSR$(x^{*})$) AND (LR$(x^{*})$)111With $\varphi_{B_{r}(x^{*}),y}$ instead,
this is, again, required for almost all $y\in Y$..

Colloquially we will also refer to (MEM) as exponential mixing of all orders,
and to (EE) as the Quantitative Ergodic Theorem or effective ergodicity. Both
are standard assumptions and have been studied for many classes of systems.
Conditions (LR), (SLR), and (NSR) all are concerned with the fact that a
points in a small ball $B$ cannot return to $B$ too quickly. Sometimes in
literature the center $x^{*}$ or $y^{*}$ is referred to as a slowly recurrent
point. For technical reason we need to distinguish different versions of slow
recurrence, (SLR) being the strongest.

######  Remark 4.

  1. (i)

In the case $T(x,y)=G_{\tau(y)}(x)$, where $G$ satisfies (a continuous version
of) (MEM)222It is in fact enough to assume exponential mixing. and $\tau$ is
bounded from above and below, the condition (NSR$(x^{*})$) at almost every
$x^{*}$. It was shown in Lemma 4.14 of [DFL22a], albeit for maps instead of
flows, that $G$ satisfies condition (NSR$(x^{*})$) is satisfied333It is shown
that, for every fixed $K>0$,  $\mu_{B_{r}(x^{*})}(\varphi_{B_{r}(x^{*}),y}\leq
K|\log(r)|)\rightarrow 0\;\;\;\textit{ as }r\rightarrow 0.$ But then we also
have  $\mu_{B_{r}(x^{*})}(\varphi_{B_{r}(x^{*}),y}\leq
K(r)|\log(r)|)\rightarrow 0\;\;\;\textit{ as }r\rightarrow 0,$ for some
$K(r)\nearrow\infty$ growing slowly enough. This is a routine argument which
is left to the reader.  at almost every $x^{*}$. Since $\tau$ is bounded, $T$
also has this property.

  2. (ii)

It is shown in [BS01, Lemma 5] that, for a map of positive entropy, condition
(LR) is satisfied at almost every point. This remains true for maps of the
form $T(x,y)=G_{\tau(y)}(x)$, for bounded $\tau$, where $G$ has positive
entropy.

  3. (iii)

Considering the previous remarks it may seem unnecessary to state condition
(SLR). Note however that none of the conditions can be satisfied at periodic
points, and the maps we want to use for $T$ will have plenty periodic points.
$SLR(y^{*})$ will be useful to show the PLT everywhere, if we can choose $R$
without periodic points.

  4. (iv)

In most of the examples we will have

$\left|\left|\sum_{j=0}^{n-1}f\circ R^{j}-n\int
f\,\text{d}\nu\right|\right|_{L^{2}(\nu)}\leq
C||f||_{CH^{r^{\prime}}}n^{\delta_{1}}\;\;\;\forall f\in
H^{r^{\prime}},\;\forall n\geq 1.$

Since $C^{r^{\prime}}\subset H^{r^{\prime}}$ and
$||f||_{H^{r^{\prime}}}\leq||f||_{C^{r^{\prime}}}$ for $f\in C^{r^{\prime}}$,
this implies condition $(EE)$.

######  Theorem 5.

Assume that $S(x,y)=(T(x,y),R(y))$ satisfies conditions (MEM), (EE), and
(BR$(x^{*},y^{*})$) for some points $x^{*}\in X,y^{*}\in Y$. If

$d>\frac{3}{2}\frac{d^{\prime}(r^{\prime}+1)}{1-\delta_{1}}$ (7)

then $S$ satisfies the PLT at $(x^{*},y^{*})$.

######  Corollary 6.

If $T$ preserves a smooth measure and satisfies (MEM), and $R$ satisfies (EE),
then $T\times R$ satisfies the PLT almost everywhere.

If the preserved measure is smooth, then, by [DKRH21], $T$ is Bernoulli, in
particular, it has positive entropy. (NSR$(x^{*})$) and (LR$(x^{*})$) are
satisfied almost everywhere by Remark 4.

For some applications it will be useful to choose $T(x,y)=G_{\tau(y)}(x)$,
where $\int_{Y}\tau\,\text{d}\nu=0$. However in this case $T$ will not satisfy
condition (MEM). Fortunately we can apply similar techniques if ergodic
averages of $\tau$ grow faster than logarithmically. More explicitly denote
$\tau_{n}=\sum_{j=0}^{n-1}\tau\circ R^{j}$, assume there is a
$\zeta:\mathbb{N}\rightarrow(0,\infty)$ with $\log(n)=o(\zeta(n))$ and a
$\kappa>0$ such that

$\nu(|\tau_{n}|<\zeta(n))\leq O(n^{-\kappa}).$ (BA)

######  Theorem 7.

Assume that $S(x,y)=(T(x,y),R(y))$, where $T(x,y)=G_{\tau(y)}(x)$, satisfies
conditions (MEM) with $G$ instead of $T$. Suppose that $R$ satisfies (EE), and
$\tau$ satisfies (BA). Let $x^{*}\in X$, $y^{*}\in Y$. If there is a
$\delta_{2}>0$ such that for small enough $r>0$ we have

$\varphi_{B_{r}(y^{*})}\geq r^{-\delta_{2}}\;\;\;\textit{on }B_{r}(y^{*}),$
(8)

and

$d>\frac{3}{2}\frac{d^{\prime}(r^{\prime}+1)}{1-\delta_{1}}\;\;\;\textit{ and
}\;\;\;\kappa>\frac{d^{\prime}}{\delta_{2}}.$ (9)

then $S$ satisfies the PLT at $(x^{*},y^{*})$.

######  Remark 8.

If in the situation of Theorem 5 resp. 7, in the assumption (EE), we assume a
pointwise estimate instead of the $L^{2}$ bound, we can improve the assumption
(7) resp. (9) on the dimension to

$d>\frac{d^{\prime}(r^{\prime}+1)}{1-\delta_{1}}.$

###  4\. Examples

The definitions of the maps in Examples 9, 12 and Lemma 11 are given in
section 9. For most of the examples we present, the choice of $R$ is more
interesting then the choice of $T$, mostly because (MEM) implies chaotic
behavior and so the PLT in that setting is not surprising. We will thus not
focus too much on $T$ for this section. We only present some examples here,
there are many others one can verify using Theorem 5.

######  Example 9.

Let $T$ be a map satisfying (MEM) on a manifold of sufficiently
high444Sufficient bounds are given in the proofs. dimension then

  1. (i)

if $R$ is a diophantine rotation, then $T\times R$ satisfies the PLT
everywhere;

  2. (ii)

if $R$ is the time $1$ map of a horocycle flow on $\Gamma\backslash
SL_{2}(\mathbb{R})$ where $\Gamma$ is a cocompact lattice, then $T\times R$
satisfies the PLT everywhere;

  3. (iii)

if $R$ is a skew-shift, then $T\times R$ satisfies the PLT everywhere.

######  Example 10.

Note that we so far only gave examples of maps that satisfy the PLT
everywhere. The reason is that, say if $T$ is Anosov, the condition
(BR$(x^{*},y^{*})$) is automatically satisfied for a.e $x^{*}$ and all
$y^{*}$. Due to Remark 4(ii) condition (LR$(x^{*})$) is satisfied almost
everywhere, furthermore, as shown in [DFL22b], $T$ itself satisfies the PLT
almost everywhere, and so in particular (NSR$(x^{*})$) holds almost
everywhere. So in order to apply Theorem 5 on $T\times R$, for some map $R$,
we just have to check (EE), i.e there are $L^{2}$ bounds on equidistribution
in $R$. (EE) can be verified for a big class of maps, in particular it holds
if $R$ satisfies a summable decay of correlation. For example certain
quadratic maps (see [You92]), intermittent interval maps (see [LM13]),…

Theorem 7 can be used to construct $T,T^{-1}$ transformations of zero entropy
that satisfy the PLT. All that remains to do is to construct a $\tau$
satisfying (BA), this can be done with the construction given in [DDKN22].

######  Lemma 11.

Let $R_{\alpha}:\mathbb{T}^{d^{\prime}}\rightarrow\mathbb{T}^{d^{\prime}}$ be
a diophantine rotation, i.e

$\left|\langle k,\alpha\rangle-l\right|>C|k|^{-n}\;\;\;\forall
k\in\mathbb{Z}^{d^{\prime}},k\not=0,l\in\mathbb{Z},$ ((D))

for some $n\geq 1$. For $\frac{n}{2}<\rho<d^{\prime}$ there is a $d\geq 1$ and
a function $\tau\in C^{\rho}(\mathbb{T}^{d^{\prime}},\mathbb{R}^{d})$ such
that $\nu(\tau)=0$, while

$\nu(||\tau_{n}||<\log^{2}(n))=o(n^{-5}).$

Note that in order to apply Theorem 7 we can always make $d$ as big as we
want.

######  Example 12.

Let $R=R_{\alpha}$ be a diophantine rotation with $d^{\prime}=2$ and $n=2$,
$\tau$ be the function from Lemma 11, and let $G$ be the Weyl Chamber flow on
$SL(d,\mathbb{R})/\Gamma$, where $\Gamma$ is a uniform lattice. If $d>10$ then
$S(x,y)=(G_{\tau(y)}(x),R_{\alpha}(y))$ satisfies the PLT everywhere.

###  5\. The delayed PLT

The main tool in the proof will be a PLT, along a subsequence, what we will
call a delayed PLT. This statement in itself is of independent interest, so
let us state a version of this here. Let $\alpha=(\alpha^{(n)})_{n\geq 1}$ be
a sequence of natural numbers, we will refer to $\alpha$ as the delay
sequence, and denote
$\displaystyle\tilde{\alpha}^{(n)}=\sum_{j=1}^{n}\alpha^{(n)}$. For measurable
$A\subset X$ we define the delayed first return time to $A$ along $\alpha$ as

$\displaystyle\varphi_{A,\alpha}^{(1)}(x):=\varphi_{A,\alpha}(x):=\min(j\geq
1\;|\;T^{\tilde{\alpha}^{(j)}}(x)\in A)$ (10)
$\displaystyle\varphi_{A,\alpha}^{(n+1)}(x):=\min(j\geq\varphi_{A,\alpha}^{(n)}\;|\;T^{\tilde{\alpha}^{(j)}}(x)\in
A)$
$\displaystyle\Phi_{A,\alpha}:=(\varphi_{A,\alpha}^{(1)},\varphi_{A,\alpha}^{(2)},...).$

The main example in this paper will be $\alpha^{(n)}=\varphi_{B}^{(n)}$,
however other choices are possible, for example $\alpha^{(n)}=g(n)$, where $g$
is a polynomial, or $\alpha^{(n)}=p_{n}$, where $p_{n}$ is the $n$th prime.
Given a rare sequence $(A_{l})_{l\geq 1}$, we will distinguish between two
cases

  * 1)

the delay sequence is fixed in $l$,

  * 2)

the delay sequence is allowed to vary with $l$.

######  Definition 13.

  1. (i)

Let $x^{*}\in X$ and, for $r>0$, denote by $B_{r}(x^{*})$ the geodesic ball of
radius $r$ centred at $x^{*}$. Let $(\alpha^{(n)})_{n\geq 1}$ be a sequence of
natural numbers. We will say that $T$ satisfies the delayed PLT along $\alpha$
at $x^{*}$ if

$\mu(B_{r}(x^{*}))\Phi_{B_{r}(x^{*}),\alpha}\overset{\mu}{\Rightarrow}\Phi_{Exp}\;\;\;\textit{
as }r\rightarrow 0.$

  2. (ii)

Let $PLT(\alpha)=\\{x^{*}\in X\;|\;T$ satisfies the delayed PLT along $\alpha$
at $x^{*}\\}.$

If $x^{*}\in PLT(\alpha)$ for all sequences of natural numbers $\alpha$, then
we say that $T$ satisfies the delayed PLT at $x^{*}$.

In case 1) the main result is a straightforward modification of Theorems 5 and
7 and hence will not be proven in detail

######  Theorem 14.

Let $\alpha$ be a sequence of positive integers. Assume the conditions of
Theorem 5 or 7 hold replacing (EE) in both cases by

$\left|\left|\sum_{j=0}^{n-1}f\circ R^{\tilde{\alpha}^{(j)}}-n\int
f\,\text{d}\nu\right|\right|_{L^{2}(\nu)}\leq
C||f||_{C^{r^{\prime}}}n^{\delta_{1}}\;\;\;\forall f\in
C^{r^{\prime}},\;\forall n\geq 1.$

Then $S$ satisfies the delayed PLT along $\alpha$ at $(x^{*},y^{*})$.

The proof is analogous to the proof of Theorem 5 resp. 7 (with $\Phi_{B_{l}}$
replaced by $\Phi_{B_{l},\alpha}$ and Lemma 21 replaced by Lemma 33).
Therefore no detailed proof will be given 555Note however that there is no
relation of delayed hitting times to delayed return times as in Proposition
2..

In case 2) the main result is a special case of Proposition 19(III), however
it is worth stating by itself.

######  Theorem 15.

Suppose $T$ satisfies conditions (MEM), (SLR$(x^{*})$) and (NSR$(x^{*})$),
then $T$ satisfies the delayed PLT at $x^{*}$.

##  Part II Proofs

###  6\. PLT along varying subsequences

####  6.1. Approximation

In our work we often need to apply the quantitative ergodicity (EE), and
mixing condition (MEM), for indicator functions, hence we have to approximate
them by functions in $C^{r}$.

######  Definition 16.

Let $M$ be a $C^{r}$ manifold with dimension $dim(M)=d$, $\lambda$ be a
measure on $M$.

$\bullet$ Let $B\subset M$ be a measurable subset. We say that $B$ is
regularly approximable in $C^{r}$ if there is a constant $C>0$ such that, for
$0<\epsilon<\frac{1}{10}\lambda(B)^{\frac{1}{d}}$, there are
$\overline{h},\underline{h}\in C^{r}$ with $\underline{h}\leq
1_{B}\leq\overline{h}$ and for $h\in\\{\overline{h},\underline{h}\\}$

$\displaystyle\lambda(1_{B}\not=h)\leq\lambda(B)^{\frac{d-1}{d}}\epsilon,\;\textit{
while }\;||h||_{C^{r}}\leq C\epsilon^{-r}.$ (11)

$\bullet$ Let $\\{B_{\alpha}\\}$ be measurable subsets of $M$. We say that
$\\{B_{\alpha}\\}$ is regularly approximable in $C^{r}$ if there is a constant
$C>0$ such that each $B_{\alpha}$ is regularly approximable in $C^{r}$ with
the same constant $C$.

######  Lemma 17.

Let $M$, $r$, $d$ and $\lambda$ be as in Definition 16. Assume in addition
that $\lambda$ is absolutely continuous w.r.t volume with bounded density.
Suppose that $\bigcup_{\alpha}B_{\alpha}$ is relatively compact, and there is
an open
$U\supset\overline{\bigcup_{\alpha}B_{\alpha}}\supset\bigcup_{\alpha}B_{\alpha}$
and a $C^{r}$-diffeomorphism $\iota:U\rightarrow V$ for some open set
$V\subset\mathbb{R}^{d}$ such that each $\iota(B_{\alpha})$ is a ball. Then
$\\{B_{\alpha}\\}$ is regularly approximable in $C^{r}$.

###### Proof.

(i) We may assume that $M=\mathbb{R}^{d}$ and $\lambda$ is Lebesgue measure,
otherwise we pick up another constant, which can be absorbed into $C$.
Furthermore we can assume that all $B_{\alpha}$ are balls centered at the
origin. In the following fix $\alpha$ and, dropping the $\alpha$ from our
notation, let $B=B_{t}(0)$. We have $\lambda(B)=C_{1}t^{d}$, where $C_{1}$ is
the volume of the $d$-dimensional unit ball.

(ii) Let $\theta:\mathbb{R}\rightarrow[0,1]$ be a smooth function with
$\theta(x)=1$ if $x<0$, and $\theta(x)=0$ if $x>1$. For $t>\epsilon>0$
consider

$\hat{\theta}(x)=\theta(\epsilon^{-1}(x-t))$

then $\hat{\theta}$ is still smooth and
$||\hat{\theta}||_{C^{r}}=\epsilon^{-r}||\theta||_{C^{r}}$. Consider
$\overline{h}:\mathbb{R}^{d}\rightarrow[0,1]$ given by
$\displaystyle\overline{h}(x)=\hat{\theta}(|x|),$ then

  * •

$\overline{h}$ is smooth, away from the origin because it is the composition
of smooth functions, and near the origin it is constant $1$,

  * •

$\overline{h}(x)=1$ if $|x|<t$ and $\overline{h}(x)=0$ if $|x|>t+\epsilon$,

  * •

and $||\overline{h}||_{C^{r}}\leq C_{3}\epsilon^{-r}$ where
$C_{3}=rC_{2}||\theta||_{C^{r}}$ and $C_{2}$ is the $C^{r}$ norm of the smooth
function $x\mapsto|x|$ on $\\{t\leq|x|\leq t+\epsilon\\}$.

Furthermore we have

$\displaystyle\lambda(\overline{h}\not=1_{B})$
$\displaystyle=\lambda(t\leq|x|\leq t+\epsilon)=C_{1}((t+\epsilon)^{d}-t^{d})$
$\displaystyle\leq C_{1}dt^{d-1}\epsilon\leq
dC_{1}^{\frac{1}{d}}\lambda(B)^{\frac{d-1}{d}}\epsilon,$

all the constants can be absorbed in the constant $C$ from the claim, the
constant only depends or $r,d$.
For $\underline{h}$ repeat the calculations with
$\hat{\theta}=\theta(\epsilon^{-1}(x-(t-\epsilon)))$ instead. ∎

####  6.2. Proof of the PLT along varying subsequences

For our purposes it will not be enough to consider the delayed PLT for a
single rare sequence $(A_{l})_{l\geq 1}$, rather let $K\geq 1$, and
$A_{l}^{(1)},...,A_{l}^{(K)}$ be subsets of $X$ such that $\\{A_{l}^{(k)}\\}$
is regularly approximable in $C^{r}$. Assume that there are
$\omega^{(1)},...,\omega^{(K)}>0$ and $r_{l}\rightarrow 0$ such that

$\mu(A_{l}^{(k)})=\omega^{(k)}r_{l}^{d}+o(r_{l}^{d}).$ (12)

Given $\kappa_{l}^{(n)}\in\\{1,...,K\\}$, for $l,n\geq 1$, define

$\displaystyle\phi_{\kappa_{l},\alpha_{l},y}^{(1)}(x)=\min(n\geq
1\;|\;T_{y}^{\tilde{\alpha}_{l}^{(n)})}(x)\in A_{l}^{(\kappa_{l}^{(n)})}),$
$\displaystyle\phi_{\kappa_{l},\alpha_{l},y}^{(m+1)}(x)=\min(n\geq\phi_{\kappa_{l},\alpha_{l},y}^{(m)}(x)+1\;|\;T_{y}^{\tilde{\alpha}_{l}^{(n)}}(x)\in
A_{l}^{(\kappa_{l}^{(n)})}),$
$\displaystyle\Phi_{\kappa_{l},\alpha_{l},y}=(\phi_{\kappa_{l},\alpha_{l},y}^{(1)},\phi_{\kappa_{l},\alpha_{l},y}^{(2)},...).$

Denote

$p_{l,t}^{(k)}:=\frac{1}{t}\\#\\{j=1,...,t_{l}\;|\;\kappa_{l}^{(j)}=k\\}.$

For now suppose that there are positive constants $\theta^{(k)}>0$ such that,
for all $k=1,...,K$,

$p_{l,t_{l}}^{(k)}\rightarrow\frac{\theta^{(k)}}{\sum_{j=1}^{K}\theta^{(j)}}=:p^{(k)}\;\;\;\textit{as
}l\rightarrow\infty.$ (13)

Later on, when we will prove PLT in product systems, we will choose $\kappa$
and $\theta$ in a specific way666Say we want to prove a PLT for the system
$T\times R$ and sets of the form $B_{r}(x^{*})\times B_{r}(y^{*})$ then we
choose $K=1$ and $\theta>0$ such that $\nu(B_{r}(y^{*}))=\theta
r^{d^{\prime}}+o(r^{d^{\prime}})$., details shall be given in section 7.4, see
in particular Lemma 27.
The main estimate of mixing rates for regularly approximable sets is the
following.

######  Lemma 18.

Let $m\geq 1$, $A^{(1)},...,A^{(k)}\subset X$ be regularly approximable in
$C^{r}$, and $1\leq n_{1}<...<n_{k}$, then

$\left|\mu\left(\bigcap_{i=1}^{k}T^{-n_{i}}A^{(i)}\right)-\prod_{i=1}^{k}\mu(A^{(i)})\right|\leq
K\max_{i=1,...,k}\mu(A^{(i)})^{\frac{d-1}{d}\frac{kr}{kr+1}}e^{-\frac{\gamma
p}{kr+1}},$

where $\displaystyle p=\min_{i=1,...,k-1}|n_{i+1}-n_{i}|$ and the constant
$K>0$ only depends on $k$ and the constant in the definition of regularly
approximable.

###### Proof.

By Lemma 17, for every $\epsilon>0$, there are $h^{(i)}\in C^{r}$ such that
$0\leq h^{(i)}\leq 1_{A^{(i)}}$ and

$\mu(1_{A^{(i)}}\neq
h^{(i)})\leq\mu({A^{(i)}})^{\frac{d-1}{d}}\epsilon,\;\textit{ while
}||h^{(i)}||_{r}<C\epsilon^{-r}.$

We estimate

$\displaystyle\left|\mu\left(\bigcap_{i=1}^{k}T^{-n_{i}}{A^{(i)}}\right)-\prod_{i=1}^{k}\mu(A^{(i)})\right|$
$\displaystyle\leq\left|\mu\left(\bigcap_{i=1}^{k}T^{-n_{i}}{A^{(i)}}\right)-\int_{X}\prod_{i=1}^{k}h^{(i)}\circ
T^{n_{i}}\,\text{d}\mu\right|$
$\displaystyle+\left|\int_{X}\prod_{i=1}^{k}h^{(i)}\circ
T^{-n_{i}}\,\text{d}\mu-\prod_{i=1}^{k}\int_{X}h^{(i)}\circ
T^{n_{i}}\,\text{d}\mu\right|$
$\displaystyle+\left|\prod_{i=1}^{k}\int_{X}h^{(i)}\circ
T^{n_{i}}\,\text{d}\mu-\prod_{i=1}^{k}\mu(A^{(i)})\right|$ $\displaystyle\leq
4k\max_{i=1,...,k}\mu(A^{(i)})^{\frac{d-1}{d}}\epsilon+\tilde{C}e^{-\gamma
p}\epsilon^{-kr}.$

This bound is minimised for

$\epsilon^{*}=\left(\frac{\tilde{C}r}{4k}\max_{i=1,...,k}\mu(A^{(i)})^{-\frac{d-1}{d}}e^{-\gamma
p}\right)^{\frac{1}{kr+1}},$

so

$\left|\mu\left(\bigcap_{i=1}^{k}T^{-n_{i}}A^{(i)}\right)-\prod_{i=1}^{k}\mu(A^{(i)})\right|\leq\hat{C}\max_{i=1,...,k}\mu(A^{(i)})^{\frac{d-1}{d}\left(1-\frac{1}{kr+1}\right)}e^{-p\frac{\gamma}{kr+1}}$
$+\bar{C}\max_{i=1,...,k}\mu(A^{(i)})^{\frac{kr}{kr+1}\frac{d-1}{d}}e^{-\gamma
p\left(1-\frac{kr}{kr+1}\right)}$ $\leq
K\max_{i=1,...,k}\mu(A^{(i)})^{\frac{d-1}{d}\frac{kr}{kr+1}}e^{-\frac{\gamma}{kr+1}p},$

where the constant $K>0$ only depends on the constant in the definition of
regularly approximable. ∎

######  Proposition 19.

Suppose that $T$ satisfies (MEM), $A_{l}^{(1)},...,A_{l}^{(K)}\subset X$, for
$K\geq 1$ $\kappa_{l}$ satisfy (13) for some $p^{(k)}>0$ with
$\displaystyle\sum_{k=1}^{K}p^{(k)}=1$, and let $\alpha_{l}$ be sequences of
natural numbers. Denote $A_{l}=\bigcup_{k=1}^{K}A_{l}^{(k)}$ and suppose that
either

  1. (I)

$\alpha_{l}$ grows faster than $|\log\mu(A_{l})|$ in the sense that

$|\log\mu(A_{l})|=o(\min_{i\geq 2}|\alpha^{(i)}_{l}|),$ (14)

  2. (II)

short returns to $A_{l}$ are rare in the sense that

$\mu_{A_{l}}(\varphi_{A_{l}}\leq a_{l})\rightarrow 0\;\;\;\textit{as
}l\rightarrow\infty,$ (15)

for some sequence $(a_{l})_{l\geq 1}$ with $|\log(\mu(A_{l}))|=o(a_{l})$, and
$\alpha_{l}$ grows at least as fast $|\log\mu(A_{l})|$ in the sense that

$|\log\mu(A_{l})|=O(\min_{i\geq 2}|\alpha^{(i)}_{l}|),$ (16)

  3. (III)

or short returns to $A_{l}$ are rare in the sense that

$\mu_{A_{l}}(\varphi_{A_{l}}\leq a_{l})\rightarrow 0\;\;\;\textit{as
}l\rightarrow\infty,$

for some sequence $(a_{l})_{l\geq 1}$ with $|\log(\mu(A_{l}))|=o(a_{l})$, and
returns are at least logarithmically large, i. e there exists $c>0$ such that

$\varphi_{A_{l}}(x)\geq c\log(\mu(A_{l}))\;\;\;\textit{a.e }x\in A_{l}.$

Then

$\Omega
r_{l}^{d}\Phi_{\kappa_{l},\alpha_{l},y}\overset{\mu}{\Rightarrow}\Phi_{Exp}\;\;\;\textit{as
}l\rightarrow\infty,$

where $\displaystyle\Omega=\sum_{k=1}^{K}\omega^{(k)}p^{(k)}$ and
$\omega^{(k)}$ come from (12).

###### Proof.

(i) Taking a subsequence if necessary, we may assume that there is a
$[0,\infty]$-valued process $\Phi=(\phi^{(1)},\phi^{(2)},...)$ such that

$\Omega
r_{l}^{d}\Phi_{\kappa_{l},\alpha_{l}}\overset{\mu}{\Rightarrow}\Phi\;\;\;\textit{as
}l\rightarrow\infty.$

Denote $\iota:[0,\infty)^{\mathbb{N}}\rightarrow[0,\infty)^{\mathbb{N}}$ the
map

$\iota(x_{1},x_{2},x_{3},...)=(x_{1},x_{1}+x_{2},x_{1}+x_{2}+x_{3},\dots).$

Since $\iota$ is a homeomorphism, showing $\Phi\overset{d}{=}\Phi_{Exp}$ is
equivalent to $\iota(\Phi)\overset{d}{=}\iota(\Phi_{Exp})$, where we use the
obvious extension of $\iota$ to $[0,\infty]^{\mathbb{N}}$. To show the latter,
for $J\geq 1$ and $0<t_{1}<...<t_{J}$, we show

$\mathbb{P}\left(\sum_{j=1}^{k}\phi^{(j)}\leq
t_{k},\;k=1,\dots,J\right)=\mathbb{P}\left(\sum_{j=1}^{k}\phi^{(j)}_{Exp}\leq
t_{k},\;k=1,\dots,J\right).$ (17)

The trick is is to look at the ”dual” object

$S_{\kappa_{l},\alpha_{l}}^{(n)}=\sum_{j=1}^{n}1_{A_{l}^{\kappa_{l}^{(j)}}}\circ
T^{\tilde{\alpha}_{l}^{(j)}}.$

The important relation here is the following

$S_{\kappa,\alpha_{l}}^{(n)}\geq
N\;\iff\;\sum_{i=1}^{N}\varphi_{\kappa,\alpha_{l}}^{(i)}\leq n.$ (18)

Indeed, evaluating both sides at some $x\in X$, the left side says that there
are at least $N$ different times $1\leq j_{1}<...<j_{N}\leq n$ such that
$T^{\tilde{\alpha}_{l}^{(j)}}(x)\in A_{l}^{\kappa_{l}^{(j)}}$. The right hand
side expresses that, if $1\leq j_{1}<...<j_{N}$ are the first $N$ times that
$T^{\tilde{\alpha}_{l}^{(j)}}(x)\in A_{l}^{\kappa_{l}^{(j)}}$, then
$\displaystyle j_{1}+\sum_{i=2}^{N}(j_{i}-j_{i-1})\leq n$. Let $(P_{t})_{t\geq
0}$ be a Poisson process, such that $\Phi_{Exp}$ are the waiting times of
$(P_{t})$, i.e $(P_{t})$ and $\Phi_{Exp}$ are related by

$P_{t}\geq k\iff\sum_{j=1}^{k}\phi^{(j)}_{Exp}\leq t.$

The right side of (17) is equal to

$\mathbb{P}\left(\sum_{j=1}^{k}\phi^{(j)}_{Exp}\leq
t_{k},\;k=1,...,J\right)=\mathbb{P}(P_{t_{k}}\geq k,\;k=1,...,J).$

Due to (18) it is enough to show

$\left(S_{\kappa_{l},\alpha_{l}}^{\left\lfloor\frac{t_{1}}{\Omega
r_{l}^{d}}\right\rfloor},S_{\kappa_{l},\alpha_{l}}^{\left\lfloor\frac{t_{2}}{\Omega
r_{l}^{d}}\right\rfloor},...,S_{\kappa_{l},\alpha_{l}}^{\left\lfloor\frac{t_{d}}{\Omega
r_{l}^{d}}\right\rfloor}\right)\overset{\mu}{\Rightarrow}(P_{t_{1}},...,P_{t_{J}})\;\;\;\textit{as
}l\rightarrow\infty.$

(ii) Taking a further subsequence if necessary, there are $[0,\infty]$-valued
$\tilde{P}_{t_{1}},...,\tilde{P}_{t_{d}}$ such that

$\left(S_{\kappa_{l},\alpha_{l}}^{\left\lfloor\frac{t_{1}}{\Omega
r_{l}^{d}}\right\rfloor},S_{\kappa_{l},\alpha_{l}}^{\left\lfloor\frac{t_{2}}{\Omega
r_{l}^{d}}\right\rfloor},...,S_{\kappa_{l},\alpha_{l}}^{\left\lfloor\frac{t_{J}}{\Omega
r_{l}^{d}}\right\rfloor}\right)\overset{\mu}{\Rightarrow}(\tilde{P}_{t_{1}},...,\tilde{P}_{t_{J}})\;\;\;\textit{as
}l\rightarrow\infty.$

We will show that

  1. (A)

$\tilde{P}_{t_{k}}-\tilde{P}_{t_{k-1}}$ is Poisson distributed with intensity
$t_{k}-t_{k-1}$ for $k=1,...,d$,

  2. (B)

and
$(\tilde{P}_{t_{1}}-\tilde{P}_{t_{0}},\tilde{P}_{t_{2}}-\tilde{P}_{t_{1}},...,\tilde{P}_{t_{J}}-\tilde{P}_{t_{J-1}})$
is an independent vector, where $t_{0}=0$.777This is essentially Watanabe’s
characterisation of Poisson-processes.

Clearly $P_{0}=\tilde{P}_{0}=0$.
For $j=1,\dots,J$ denote
$S_{t_{j},l}=S_{A_{l},\kappa_{l}}^{\left\lfloor\frac{t_{j}}{\Omega
r_{l}^{d}}\right\rfloor}$. Assertions (A) and (B) will follow888Here we apply
the method of moments, see eg [Bil95]. once we show that, for all
$m_{1},...,m_{d}\geq 1$,

$\int_{X}\prod_{j=1}^{J}{{S_{t_{j},l}-S_{t_{j-1},l}}\choose
m_{j}}\,\text{d}\mu=\prod_{j=1}^{J}\frac{(t_{j}-t_{j-1})^{m_{j}}}{m_{j}!}+o(1)\;\;\;\textit{as
}l\rightarrow\infty.$

In the rest of the proof fix $J\geq 1$, $0=t_{0}<t_{1}<\dots<t_{J}$ and
$m_{1},\dots,m_{J}\geq 1$.

(iii) First, for each $j=1,\dots,J$, rewrite

$S_{t_{j},l}-S_{t_{j-1},l}=\sum_{i=\left\lfloor\frac{t_{j-1}}{\Omega
r_{l}^{d}}\right\rfloor+1}^{\left\lfloor\frac{t_{j}}{\Omega
r_{l}^{d}}\right\rfloor}1_{A_{l}^{(\kappa_{l}^{(i)})}}\circ
T^{\tilde{\alpha}_{l}^{(i)}}.$

So

$\prod_{j=1}^{J}{{S_{t_{j},l}-S_{t_{j-1},l}}\choose
m_{j}}=\prod_{j=1}^{d}\sum_{\left\lfloor\frac{t_{j-1}}{\Omega
r_{l}^{d}}\right\rfloor+1\leq
k_{1,j}<...<k_{m_{j},j}\leq\left\lfloor\frac{t_{j}}{\Omega
r_{l}^{d}}\right\rfloor}\prod_{i=1}^{m_{j}}\xi_{i,j}$ (19)

where $\xi_{i,j}=1_{A_{l}^{(\kappa_{l}^{(k_{i,j})})}}\circ
T^{\tilde{\alpha}_{l}^{({k_{i,j})}}}$. To simplify notation we will also
denote $m=(m_{1}+\dots+m_{J})$,
$\displaystyle\omega=\min_{k=1,...,K}\omega^{(k)}$,
$p_{l}=2m\frac{mr+1}{\gamma}\left|\log(\omega r_{l}^{d})\right|$,

$\Delta_{l}:=\\{\textbf{k}=(k_{i,j})_{\begin{subarray}{c}j=1,...,J\\\
i=1,...,m_{j}\end{subarray}}\;\Big{|}\;\left\lfloor\frac{t_{j-1}}{\Omega
r_{l}^{d}}\right\rfloor+1\leq
k_{1,j}\\!\\!<\\!\\!\dots\\!\\!<\\!\\!k_{m_{j},j}\leq\\!\\!\left\lfloor\frac{t_{j}}{\Omega
r_{l}^{d}}\right\rfloor\;\textit{ for }j=1,...,d\\}$

and

$\Delta^{\prime}_{l}:=\\{\textbf{k}\in\Delta_{l}\;|\;\min_{\begin{subarray}{c}j=1,...,J,\;i=1,...,m_{j}\\\
j^{\prime}=1,...,J,\;i^{\prime}=1,...,m_{j^{\prime}},\;(j,i)\not=(j^{\prime},i^{\prime})\end{subarray}}|\tilde{\alpha}_{l}^{k_{i,j}}-\tilde{\alpha}_{l}^{k_{i^{\prime},j^{\prime}}}|\leq
p_{l}\\}$

We will split the sum in (19) into two terms

$\prod_{j=1}^{d}{{S_{t_{j},l}-S_{t_{j-1},l}}\choose m_{j}}=M_{l}+R_{l},$

where

$M_{l}=M_{(t_{j}),l,m}=\sum_{\textbf{k}\in\Delta_{l}\setminus\Delta^{\prime}_{l}}\prod_{\begin{subarray}{c}j=1,...,d\\\
i=1,...,m_{j}\end{subarray}}\xi_{i,j},\quad
R_{l}=R_{(t_{j}),l,m}=\sum_{\textbf{k}\in\Delta^{\prime}_{l}}\prod_{\begin{subarray}{c}j=1,...,d\\\
i=1,...,m_{j}\end{subarray}}\xi_{i,j}.$

We will show that

$\int_{X}M_{l}\,\text{d}\mu\rightarrow\prod_{j=1}^{J}\frac{(t_{j}-t_{j-1})^{m_{j}}}{m_{j}!}\;\textit{
and }\;\int_{X}R_{l}\,\text{d}\mu\rightarrow 0\;\;\;\textit{as
}l\rightarrow\infty.$ (20)

(iv) Let us first treat $M_{l}$. For $l\geq 1$ and $\textbf{k}\in\Delta_{l}$,
by Lemma 18 we have

$\left|\mu\left(\bigcap_{\begin{subarray}{c}j=1,...,d\\\
i=1,...,m_{j}\end{subarray}}T^{-\tilde{\alpha}_{l}^{k_{i,j}}}A_{l}^{(\kappa_{l}^{(k_{i,j})})}\right)-\prod_{j,i}\mu\left(A_{l}^{(\kappa_{l}^{(k_{i,j})})}\right)\right|$
$\leq
K\max_{j,i}\mu\left(A_{l}^{(\kappa_{l}^{(k_{i,j})}}\right)^{\frac{d-1}{d}\frac{mr}{mr+1}}e^{-\frac{\gamma\min_{i,j}\alpha_{l}^{k_{i,j}}}{mr+1}}.$

For $\textbf{k}\in\Delta_{l}\setminus\Delta^{\prime}_{l}$, this yields

$\mu\left(\bigcap_{\begin{subarray}{c}j=1,...,d\\\
i=1,...,m_{j}\end{subarray}}T^{-\tilde{\alpha}_{l}^{k_{i,j}}}A_{l}^{(\kappa_{l}^{(k_{i,j})})}\right)=r_{l}^{md}\prod_{\begin{subarray}{c}j=1,...,d\\\
i=1,...,m_{j}\end{subarray}}\omega_{l}^{(\kappa_{l}^{k_{i,j}})}+o(r_{l}^{md}),$

and the $o$-term does not depend on k. Summing over
$\textbf{k}\in\Delta_{l}\setminus\Delta^{\prime}_{l}$, and using (13), yields

$\displaystyle\int_{X}M_{l}\,\text{d}\mu$
$\displaystyle=\int_{X}\sum_{\textbf{k}\in\Delta_{l}\setminus\Delta^{\prime}_{l}}\prod_{\begin{subarray}{c}j=1,...,d\\\
i=1,...,m_{j}\end{subarray}}1_{A_{l}^{(\kappa_{l}^{(k_{i,j})})}}\circ
T^{\tilde{\alpha}_{l}^{({k_{i,j})}}}\,\text{d}\mu$
$\displaystyle=r_{l}^{md}\sum_{\textbf{k}\in\Delta_{l}\setminus\Delta^{\prime}_{l}}\prod_{\begin{subarray}{c}j=1,...,d\\\
i=1,...,m_{j}\end{subarray}}\omega^{(\kappa_{l}^{k_{i,j}})}+o(1)$
$\displaystyle\overset{*}{=}r_{l}^{md}\prod_{j=1}^{J}\frac{1}{m_{j}!}\left(\sum_{k=1}^{K}\left\lfloor\frac{t_{j}-t_{j-1}}{\Omega
r_{l}^{d}}\right\rfloor p^{(k)}_{l,\left\lfloor\frac{t_{j}-t_{j-1}}{\Omega
r_{l}^{d}}\right\rfloor}\omega^{(k)}\right)^{m_{j}}+o(1)$
$\displaystyle=\Omega^{-m}\prod_{j=1}^{J}\frac{(t_{j}-t_{j-1})^{m_{j}}}{m_{j}!}\left(\sum_{k=1}^{K}p^{(k)}\omega^{(k)}\right)^{m_{j}}+o(1)$
$\displaystyle=\prod_{j=1}^{J}\frac{(t_{j}-t_{j-1})^{m_{j}}}{m_{j}!}+o(1),$

where, in $*$, we consider the unordered sum, i.e the sum over all $(k_{i,j})$
with $\left\lfloor\frac{t_{j-1}}{\Omega r_{l}^{d}}\right\rfloor+1\leq
k_{i,j}\leq\left\lfloor\frac{t_{j}}{\Omega r_{l}^{d}}\right\rfloor$ and
999This condition is analogous to $\Delta_{l}\setminus\Delta^{\prime}_{l}$,
but without requiring $k_{i,j}<k_{i+1,j}$.

$\min_{\begin{subarray}{c}j=1,...,J,\;i=1,...,m_{j}\\\
j^{\prime}=1,...,J,\;i^{\prime}=1,...,m_{j^{\prime}},\;(j,i)\not=(j^{\prime},i^{\prime})\end{subarray}}|\tilde{\alpha}_{l}^{k_{i,j}}-\tilde{\alpha}_{l}^{k_{i^{\prime},j^{\prime}}}|>p_{l}.$

This shows the first assertion of (20).

(v) In order to treat $R_{l}$, first note that under assumption $(I)$ we have
$R_{l}=0$ for big enough $l$. In the following we focus on assumptions $(II)$
and $(III)$. Note that

$\int_{X}R_{l}\,\text{d}\mu=\int_{X}1_{R_{l}\neq
0}R_{l}\,\text{d}\mu\leq\mu(\text{supp}(R_{l}))\|R_{l}\|_{L^{2}}$

and

$\text{supp}(R_{l})\subset\bigcup_{s=1}^{\left\lfloor\frac{t_{J}}{\Omega
r_{l}^{d}}\right\rfloor}T^{-\tilde{\alpha}_{l}^{(j)}}\left(A_{l}\cap\\{\varphi_{A_{l}}\leq
3p_{l}\\}\right)=:U_{l},$

hence

$\mu(\text{supp}(R_{l}))\leq\mu(U_{l})=O(r_{l}^{-d})\mu(A_{l})o(1)=o(1).$

Therefore, in order to show $\int_{X}R_{l}\,\text{d}\mu\rightarrow 0$, it is
enough to show that $(R_{l})_{l\geq 1}$ is bounded in $L^{2}$. Notice that

$R_{l}^{2}\leq\sum_{1\leq k_{1},...,k_{2m}\leq\left\lfloor\frac{t_{J}}{\Omega
r_{l}^{d}}\right\rfloor}\prod_{i=1}^{m}1_{A_{l}^{(\kappa_{l}^{(k_{i})})}}\circ
T^{\tilde{\alpha}_{l}^{(i)}}=S_{t_{J},l}^{2m}.$

We may write

$S_{t_{J},l}^{2m}=\sum_{k=1}^{2m}\genfrac{\\{}{\\}}{0.0pt}{}{2m}{k}{S_{t_{J},l}\choose
k}\leq C_{m}\sum_{k=1}^{2m}{S_{t_{J},l}\choose k}\leq
C_{m}\sum_{k=1}^{2m}(M_{t_{J},l,k}+R_{t_{J},l,k}),$

where $\genfrac{\\{}{\\}}{0.0pt}{}{2m}{k}$ are the Stirling numbers of the
second kind and $\displaystyle
C_{m}=\\!\\!\max_{k=1,...,2m}\genfrac{\\{}{\\}}{0.0pt}{}{2m}{k}$. Now the
previous parts of the proof show that

$\int_{X}M_{t_{J},l,k}\,\text{d}\mu\;\;\;\textit{is bounded as
}l\rightarrow\infty,\textit{ for each fixed }k\geq 1,$

it remains to show that

$\int_{X}R_{t_{J},l,k}\,\text{d}\mu\;\;\;\textit{is bounded as
}l\rightarrow\infty,\textit{ for each fixed }k\geq 1.$

(vi) In order to bound $\int_{X}R_{t_{J},l,m}\,\text{d}\mu$ for fixed $m$ we
first split up $\Delta^{\prime}_{l}$ (we adapt all of the notation above for
$R_{l}=R_{t_{J},l,m}$, $t=t_{J}$) into

$\Delta_{l}^{(j)}:=\left\\{\textbf{k}=(k_{1},...,k_{m})\in\tilde{\Delta}_{l}\;\left|\begin{aligned}
&\exists 1\leq i_{i}<...<i_{m-j}\leq m\textit{ such that }\\\
&|\tilde{\alpha}_{l}^{k_{i+1}}-\tilde{\alpha}_{l}^{k_{i}}|\leq\tilde{p}_{l}\;\forall
i\in\\{1,...,m\\}\setminus\\{i_{1},...,i_{m-j}\\},\\\ &\textit{and
}|\tilde{\alpha}_{l}^{k_{i_{r}+1}}-\tilde{\alpha}_{l}^{k_{i_{r}}}|>\tilde{p}_{l}\;\forall
r=1,...,m-j\end{aligned}\right.\right\\},$

for $j=1,...,m-1$, so that
$\Delta^{\prime}_{l}=\bigcup_{j=1}^{m-1}\Delta_{l}^{(j)}$. Under assumption
$(II)$ there is a constant $c>0$ such that for $\rho=2m\frac{mr+1}{c\gamma}$
and big enough $l$ we have

$|\Delta_{l}^{(j)}|\leq{{m}\choose j}{\left\lfloor\frac{t}{\Omega
r_{l}^{d}}\right\rfloor\choose{{m}-j}}\rho^{j}\leq
m^{m}t^{m}\rho^{m}(\Omega^{-{m}}+1)r_{l}^{d({m}-j)}.$

On the other hand, for $\textbf{k}\in\Delta_{l}^{(j)}$, we can again use Lemma
18 to estimate

$\int_{X}\prod_{s=1}^{m}\xi_{s}\,\text{d}\mu\leq\int_{X}\prod_{s\in\\{1,...,{m}\\}\setminus\\{i_{1},...,i_{j}\\}}\xi_{s}\leq\Omega
r_{l}^{d({m}-j)}+o(r_{l}^{d({m}-j)}),$

where

$\xi_{s}=1_{A_{l}^{\kappa_{l}^{(k_{s})}}}\circ
T^{\tilde{\alpha}_{l}^{(k_{s})}}.$

Summing up we obtain

$\int_{X}R_{l}\,\text{d}\mu\leq
2({m})^{{m}+1}t^{m}\rho^{m}(\Omega^{-{m}+1}+\Omega),$

and we conclude $\int_{X}R_{l}\,\text{d}\mu\rightarrow 0$ under assumption
$(II)$.

(vii) Finally assume $(III)$. For fixed $j=1,...,m-1$, $l\geq 1$ and $1\leq
i_{1}<...<i_{j}\leq\left\lfloor\frac{t}{\Omega r_{l}^{d}}\right\rfloor$ with
$\displaystyle\min_{s=1,...,j-1}|\tilde{\alpha}_{l}^{(i_{s+1})}-\tilde{\alpha}_{l}^{(i_{s})}|\geq
p_{l}$ set

$A_{i_{1},...,i_{j}}=\bigcap_{s=1}^{j}T^{-\tilde{\alpha}_{l}^{(i_{s})}}A_{l}^{\kappa_{l}^{(i_{s})}}.$

By Lemma 18

$\mu(A_{i_{1},...,i_{j}})\leq\omega r_{l}^{dj}+o(r_{l}^{dj}),$

and the $o$ term doesn’t depend on $(i_{1},...,i_{j})$. For $x\in
A_{i_{1},...,i_{j}}$ consider

$\mathfrak{K}_{i_{1},...,i_{j}}(x)=\left\\{\textbf{k}=(k_{1},...,k_{m})\in\Delta_{l}^{j}\;\left|\begin{aligned}
&\exists r_{1},...,r_{j}\;\textit{ such that }k_{r_{s}}=i_{s}\;\forall
s=1,...,j,\\\ &\textit{ and
}\;\prod_{s=1}^{m}\xi_{s}(x)=1\end{aligned}\right\\}\right..$

Then, since $\varphi_{A_{l}}(x)\geq c\log(\mu(A_{l}))$, we have

$|\mathfrak{K}_{i_{1},...,i_{j}}(x)|\leq\left(2\frac{p_{l}}{dc\log(r_{l})}\right)^{m-j}=:\rho^{m-j}.$

At the same time we have

$\text{supp}\left(\prod_{s=1}^{m}\xi_{s}\right)\subset
A_{i_{1},...,i_{j}}\;\;\;\textit{ for
}\textbf{k}\in\mathfrak{K}_{i_{1},...,i_{j}}(x)\;\textit{ for some }x.$

Also every $\textbf{k}\in\tilde{\Delta}_{l}^{(j)}$ is in some
$\mathfrak{K}_{i_{1},...,i_{j}}(x)$, therefore

$\displaystyle\sum_{\textbf{k}\in\tilde{\Delta}_{l}^{(j)}}\int_{X}\prod_{s=1}^{m}\xi_{s}\,\text{d}\mu$
$\displaystyle\leq\sum_{\begin{subarray}{c}1\leq
i_{1}<...<i_{j}\leq\left\lfloor\frac{t}{\Omega r_{l}^{d}}\right\rfloor\\\
\min_{s=1,...,j-1}|\tilde{\alpha}_{l}^{(i_{s+1})}-\tilde{\alpha}_{l}^{(i_{s})}|\geq
p_{l}\end{subarray}}\int_{A_{i_{1},...,i_{j}}}|\mathfrak{K}_{i_{1},...,i_{j}}(x)|\,\text{d}\mu(x)$
$\displaystyle\leq\sum_{\begin{subarray}{c}1\leq
i_{1}<...<i_{j}\leq\left\lfloor\frac{t}{\Omega r_{l}^{d}}\right\rfloor\\\
\min_{s=1,...,j-1}|\tilde{\alpha}_{l}^{(i_{s+1})}-\tilde{\alpha}_{l}^{(i_{s})}|\geq\tilde{p}_{l}\end{subarray}}\int_{A_{i_{1},...,i_{j}}}\rho^{j}\,\text{d}\mu$
$\displaystyle\leq\rho^{j}\left(\omega
r_{l}^{dj}+o(r_{l}^{dj})\right)\left(\left\lfloor\frac{t}{\Omega
r_{l}^{d}}\right\rfloor\right)^{j}$
$\displaystyle\leq\rho^{j}t^{j}\frac{\omega}{\Omega^{j}}+o(1).$

Summing up over $j$ we get

$\int_{X}R_{l}\,\text{d}\mu\leq
m\max(1,\rho^{m})\max(1,t^{m})\frac{\omega}{\max(1,\Omega^{m})}+o(1).$

Following the argument in step (v) this shows
$\int_{X}R_{l}\,\text{d}\mu\rightarrow 0$, hence (20), in case of assumption
(III). This concludes the proof. ∎

######  Remark 20.

(i) Note that assumptions (I), (II), or (III) directly correspond to the three
possible cases of (BR$(x^{*},y^{*})$)–(SLR$(y^{*})$), (NSR$(x^{*})$) AND
(LR$(y^{*})$), or (NSR$(x^{*})$) AND (LR$(x^{*})$) respectively.

(ii) In all of the examples we give in section 4, the $\alpha_{l}$ will
satisfy a condition stronger than (I). In fact, in this set-up, there is a
$\delta_{2}>0$ such that

$\min_{i\geq 2}|\alpha^{(i)}_{l}|\geq\mu(A_{l})^{-\delta_{2}},$

compare also (8)101010The constant $\delta_{2}$ given there is not exactly the
same. In the notation there we have to use $\delta_{2}\frac{d^{\prime}}{d}$..
If this stronger condition is satisfied instead of (I), then we do not need
the full strength of exponential mixing in (MEM). Any superpolynomial rate
will be enough. Details are given in the proof of Theorem 7, but we shall give
a heuristic here.
When using mixing of all order with indicators of the form
$1_{B_{r_{l}}(x^{*})}$, the error term will contain a term coming from the
$C^{r}$ norm in the definition of regularly approximable. In this case, using
(11), this term will be of order $Cr_{l}^{-drm}$. To compensate, say the rate
of mixing is $\psi$, since the gaps $\alpha_{l}$ are large we can multiply
with $\psi(\min_{i\geq 2}|\alpha^{(i)}_{l}|)$. So we want to show

$r_{l}^{-drm}\psi(r_{l}^{-d\delta_{2}})=o(1),$

for all $m\geq 1$, thus $\psi$ should decay superpolynomially.

###  7\. The PLT for rectangles.

####  7.1. The strategy

Let us first outline the strategy of proving Theorem 5.
Say we want to show the PLT for the geodesic balls $(Q_{l})$ converging to
$(x^{*},y^{*})$, then, for every $\epsilon>0$, we find suitable
rectangles111111Using the exponential map, this is a simple exercise in
$\mathbb{R}^{d+d^{\prime}}$. Here we use continuity of the densities of $\mu$
resp. $\nu$! approximating $Q_{l}$ in the sense that
$\bigcup_{k=1}^{K}A_{l}^{(k)}\times B_{l}^{(k)}\subset Q_{l}$ and

$(\mu\times\nu)_{Q_{l}}\left(Q_{l}\setminus\bigcup_{k=1}^{K}A_{l}^{(k)}\times
B_{l}^{(k)}\right)<\epsilon.$

Denote $Q_{l}^{\prime}:=\bigcup_{k=1}^{K}A_{l}^{(k)}\times B_{l}^{(k)}$. We
will use Proposition 19 to show the PLT for $Q^{\prime}_{l}$, i.e

$(\mu\times\nu)(Q^{\prime}_{l})\Phi_{Q^{\prime}_{l}}\rightarrow{(\mu\times\nu)}\Phi_{Exp}\;\;\;\textit{as
}l\rightarrow\infty.$

Letting $\epsilon\rightarrow 0$, the next Lemma will help us conclude that

$(\mu\times\nu)(Q_{l})\Phi_{Q_{l}}\rightarrow{(\mu\times\nu)}\Phi_{Exp}\;\;\;\textit{as
}l\rightarrow\infty.$

The critical approximation here is the following Lemma (see [Zwe]).

######  Lemma 21.

There is a metric $D$ defined on the space of probability measures on
$[0,\infty]^{\mathbb{N}}$ and modelling weak convergence of measures121212 In
the sense that $\lambda_{l}\Rightarrow\lambda$ if and only if
$D(\lambda_{l},\lambda)\rightarrow 0$. such that

$D\left(law_{(\mu\times\nu)_{Q}}((\mu\times\nu)(Q)\Phi_{Q}),law_{(\mu\times\nu)_{Q^{\prime}}}((\mu\times\nu)(Q^{\prime})\Phi_{Q^{\prime}}\right)\leq
7(\mu\times\nu)_{Q}(Q\setminus Q^{\prime}),$

for measurable $Q^{\prime}\subset Q\subset X\times Y$.

We shall prove a more general version of Lemma 21 and define the metric $D$ in
Section 10.

For returns to rectangles, say $A_{l}\times B_{l}$, for fixed $y\in Y$, we can
ignore all the times $j$ where $R^{j}(y)\not\in B_{l}$. Hence denoting
$\alpha_{l}(y)=\Phi_{B_{l}}(y)$, we first show that

$\mu(A_{l})\Phi_{A_{l},\alpha_{l}(y)}\overset{\mu}{\Rightarrow}\Phi_{Exp}\;\;\;\textit{as
}l\rightarrow\infty.$ (21)

To show this, the idea is that, due to assumption (BR), the times
$\tilde{\alpha}_{l}^{(j)}(y)$ are sufficiently far apart to use (MEM), so that
$1_{A_{l}}\circ T_{y}^{\tilde{\alpha}_{l}^{(j)}(y)}$ are ’close to
independent’ and we can use classical arguments. Since (21) is now true for
almost every $y\in Y$, we also have

$\mu(A_{l})\Phi_{A_{l},\Phi_{B_{l}}}\overset{\mu\times\nu}{\Rightarrow}\Phi_{Exp}\;\;\;\textit{as
}l\rightarrow\infty.$ (22)

These are not quite the returns that we wanted to consider. However notice
that, in every step, we skip exactly $\varphi_{B_{l}}$ steps, thus we have the
following relation

$\varphi_{A_{l}\times
B_{l}}(x,y)=\sum_{j=0}^{\varphi_{A_{l},\Phi_{B_{l}}(y)}(x)-1}\varphi_{B_{l}}\circ
R^{j}_{B_{l}}(y).$ (23)

We now use (EE) to control the ergodic sums of $\varphi_{B_{l}}$ and show that

$\mu(A_{l})\nu(B_{l})\Phi_{A_{l}\times
B_{l}}\overset{\mu\times\nu}{\Rightarrow}\Phi_{Exp}\;\;\;\textit{as
}l\rightarrow\infty.$

The same idea works for unions of rectangles
$\displaystyle\bigcup_{k=1}^{K}A_{l}^{(k)}\times B_{l}^{(k)}$.

####  7.2. Quantitative Ergodic Theorem

In section 7.3, it will be convenient to use a pointwise quantitative ergodic
Theorem instead of the $L^{2}$ bound we assume in (EE). We will show first
that such pointwise bounds hold for most points.
More explicitly, we will show that for every sequence $((f_{l})_{l\geq 1})$ of
functions in $L^{2}$ and $\epsilon>0$ there are $L_{y},M_{l}\geq 1$ and a
constant $K>0$ such that

$\left|\sum_{j=1}^{n}f_{l}\circ R^{j}(y)-n\int_{Y}f_{l}d\nu\right|\leq
K||f_{l}||_{C^{r^{\prime}}}n^{\frac{1+2\delta+\epsilon}{3}}\;\;\;\textit{for
a.e }y\in Y,\forall n\geq M_{l},l\geq L_{y}.$

First let us give a small Lemma.

######  Lemma 22.

Let $0<\delta<1$, and $k_{n}=\left\lfloor n^{\delta^{\prime}}\right\rfloor$,
for some $\delta^{\prime}>1$ with $1>\delta^{\prime}(1-\delta)$, then for some
big enough $M\geq 1$ we have

$[M,\infty)\subset\bigcup_{n\geq
1}[k_{n}-k_{n}^{\delta},k_{n}+k_{n}^{\delta}].$

###### Proof.

Since $k_{n}\rightarrow\infty$ as $n\rightarrow\infty$, it is enough to show

$k_{n}+k_{n}^{\delta}\geq k_{n+1}-k_{n+1}^{\delta},$

for big enough $n$. Indeed, using the mean value theorem there is
$\xi_{n}\in[n,n+1]$ such that

$\displaystyle(n+1)^{\delta^{\prime}}-n^{\delta^{\prime}}=\delta^{\prime}\xi_{n}^{\delta^{\prime}-1}\leq\delta^{\prime}(n+1)^{\delta^{\prime}-1}\leq(n+1)^{\delta\delta^{\prime}},$

as soon as $n^{\delta\delta^{\prime}-\delta^{\prime}+1}>\delta^{\prime}$. ∎

######  Proposition 23.

Let $(Y,R,\nu)$ be a probability preserving ergodic dynamical system,
$(f_{l})_{l\geq 1}$ be a sequence of $C^{r}$ functions. Suppose there are
$0<\delta<1$, $C>0$ and $N_{l}\geq 1$ such that

$\left|\left|\sum_{j=1}^{n}f_{l}\circ
R^{j}-n\int_{Y}f_{l}\,\text{d}\nu\right|\right|_{L^{2}}\leq
C||f_{l}||_{r}n^{\delta}\;\;\;\forall n\geq N_{l},\forall l\geq 1.$

Then there are $L_{y},M_{l}\geq 1$ and a constant $\tilde{C}>0$ such that

$\left|\sum_{j=1}^{n}f_{l}\circ
R^{j}(y)-n\int_{Y}f_{l}d\nu\right|\leq\tilde{C}||f_{l}||_{r}n^{\delta^{\prime}}\;\;\;\textit{for
a.e }y\in Y,\forall n\geq M_{l},l\geq L_{y}.$

Furthermore, for any small enough $\epsilon>0$, we can choose

$M_{l}=\max\left(N_{l},\left\lceil
l^{\frac{-\delta_{2}}{(\delta-\delta_{1})\delta_{2}+1}+\epsilon}\right\rceil\right)$

and $\delta^{\prime}=\frac{1+2\delta+\epsilon}{3}$.

###### Proof.

(1) W.l.o.g we may assume $\int_{Y}f_{l}\,\text{d}\nu=0$. For $l\geq 1$,
$n\geq N_{l}$, $K>0$ and $\delta_{1}\in(0,1)$ close to $1$, using the
Chebyshev inequality we have

$\nu\left(\left|\sum_{j=1}^{n}f_{l}\circ
R^{j}\right|>K||f_{l}||_{r}n^{\delta_{1}}\right)\leq\frac{\left|\left|\sum_{j=1}^{n}f_{l}\circ
R^{j}\right|\right|_{L^{2}}^{2}}{(K||f_{l}||_{r}n^{\delta_{1}})^{2}}\leq\frac{C^{2}}{K^{2}}n^{2(\delta-\delta_{1})}.$

Denote $B_{n}^{l,K}=\left\\{\left|\sum_{j=1}^{n}f_{l}\circ
R^{j}\right|>K||f_{l}||_{r}n^{\delta_{1}}\right\\}$. We will use a Borel-
Cantelli argument to show that, for some $K>0$ and for a.e $y$, there are only
finitely many $l\geq 1,n\geq M_{l}$ such that $y\in B_{n}^{l,K}$.

(2) Note that if

$\left|\sum_{j=1}^{n}f_{l}\circ
R^{j}(y)\right|>(2K+1)||f_{l}||_{r}n^{\delta_{1}},$

then

$\left|\sum_{j=1}^{k}f_{l}\circ R^{j}(y)\right|>K||f_{l}||_{r}k^{\delta_{1}},$

for $k\in\mathbb{Z}\cap[n-n^{\delta_{1}},n+n^{\delta_{1}}]$. By Lemma 22, for
any $\delta_{2}>1$ with $1>\delta_{2}(1-\delta_{1})$ we have that there is
some $M\geq 1$, only depending on $\delta_{1}$ and $\delta_{2}$, such that for
each $k\geq M$ there is a $n\geq 1$ such that $B_{k}^{l,K}\subset
B_{k_{n}}^{l,(2K-1)}$, where $k_{n}=\left\lfloor n^{\delta_{2}}\right\rfloor$.
(3) For some small $\tilde{\epsilon}>0$ let

$\tilde{M}_{l}=\left\lceil
l^{\frac{-1}{(\delta-\delta_{1})\delta_{2}+1}+\tilde{\epsilon}}\right\rceil.$

Suppose $2(\delta-\delta_{1})\delta_{2}<-1$ then

$\displaystyle\sum_{l\geq
1}\sum_{n\geq\tilde{M}_{l}}\nu(B_{k_{n}}^{l,(2K+1)})$
$\displaystyle\leq\sum_{l\geq
1}\sum_{n\geq\tilde{M}_{l}}\frac{C^{2}}{(2K+1)^{2}}n^{2(\delta-\delta_{1})\delta_{2}}$
$\displaystyle\leq\frac{C^{2}}{(2K+1)^{2}2(\delta-\delta_{1})\delta_{2}}\sum_{l\geq
1}\tilde{M}_{l}^{2(\delta-\delta_{1})\delta_{2}+1}$ $\displaystyle<\infty.$

By Borel-Cantelli, for a.e $y$ there are only finitely many $l\geq 1$ and
$n\geq\tilde{M}_{l}$ such that $y\in B_{k_{n}}^{l,(2K+1)}$. It follows that,
for large enough $L_{y}\geq 1$

$y\not\in B_{n}^{l,K}\;\;\;\forall l\geq L_{y},n\geq M_{l},$

where

$M_{l}=\max\left(N_{l},\left\lceil
l^{\frac{-\delta_{2}}{(\delta-\delta_{1})\delta_{2}+1}+\epsilon}\right\rceil\right),$

for some small $\epsilon>0$.

(4) Finally for the $\delta_{1},\delta_{2}$ we want

  * •

$\delta_{1}>0$,

  * •

$\delta_{2}>1$,

  * •

$(1-\delta_{1})\delta_{2}<1$,

  * •

and $2(\delta-\delta_{1})\delta_{2}<-1$.

For $\delta_{1}=\frac{1+2\delta+\epsilon}{3}$ and
$\delta_{2}=\frac{3}{2(1-\delta)+2\epsilon}$, these relations are satisfied,
making $\epsilon$ smaller if necessary. ∎

####  7.3. Uniform Estimates

We will show two estimates.

######  Lemma 24.

Suppose for every sequence $((f_{l})_{l\geq 1})$ of functions in $L^{2}$ and
$\epsilon>0$ there are $L_{y},M_{l}\geq 1$ and a constant $K>0$ such that

$\left|\sum_{j=1}^{n}f_{l}\circ R^{j}(y)-n\int_{Y}f_{l}d\nu\right|\leq
K||f_{l}||_{r}n^{\delta}\;\;\;\textit{for a.e }y\in Y,\forall n\geq
M_{l},l\geq L_{y}.$

If $B$ is regularly approximable in $C^{r}$, then

$\left|\sum_{j=1}^{n}1_{B}\circ R^{j}(y)-n\nu(B)\right|\leq
K_{y}n^{1-\frac{1-\delta}{r+1}}\;\;\;\forall n\geq M_{y},\textit{ a.e }y\in
Y.$ (24)

###### Proof.

For each $\epsilon>0$ there are $\underline{h},\overline{h}\in C^{r}$ with
$\underline{h}\leq 1_{B}\leq\overline{h}$ and

$\lambda(1_{B}\not=h)\leq\nu(B)^{\frac{d-1}{d}}\epsilon,\;\textit{ while
}\;||h||_{C^{r}}\leq C\epsilon^{-r},$

for $h\in\\{\overline{h},\underline{h}\\}$. In particular

$\nu(B)-\nu(B)^{\frac{d-1}{d}}\epsilon\leq\int_{Y}\underline{h}\,\text{d}\nu\leq\int_{Y}\overline{h}\,\text{d}\nu\leq\nu(B)+\nu(B)^{\frac{d-1}{d}}\epsilon.$

By assumption, for $h\in\\{\overline{h},\underline{h}\\}$ we have

$\left|\sum_{j=1}^{n}h\circ R^{j}(y)-n\int_{Y}h\,\text{d}\nu\right|\leq
C_{y}||f||_{C^{r}}n^{\delta}\;\;\;\forall n\geq M_{y},\textit{ a.e }y\in Y.$

Therefore

$\displaystyle\sum_{j=1}^{n}1_{B}\circ R^{j}(y)$
$\displaystyle\geq\sum_{j=1}^{n}\underline{h}\circ R^{j}(y)\geq
n\int_{Y}\underline{h}\,\text{d}\nu-C_{y}C\epsilon^{-r}n^{\delta}$
$\displaystyle\geq n\nu(B)-n\nu(B)^{\frac{d-1}{d}}\epsilon-
C_{y}C\epsilon^{-r}n^{\delta}.$

Likewise

$\sum_{j=1}^{n}1_{B}\circ R^{j}(y)\leq
n\nu(B)+n\nu(B)^{\frac{d-1}{d}}\epsilon+C_{y}C\epsilon^{-r}n^{\delta}.$

Hence

$\left|\sum_{j=1}^{n}1_{B}\circ R^{j}(y)-n\nu(B)\right|\leq
n\nu(B)^{\frac{d-1}{d}}\epsilon+C_{y}C\epsilon^{-r}n^{\delta}.$

This bound is minimized for

$\epsilon^{*}=\left(\frac{rC_{y}Cn^{\delta}}{n\nu(B)^{\frac{d-1}{d}}}\right)^{\frac{1}{r+1}},$

so we even have

$\displaystyle\left|\sum_{j=1}^{n}1_{B}\circ R^{j}(y)-n\nu(B)\right|$
$\displaystyle\leq
r^{\frac{1}{r+1}}(C_{y}C)^{\frac{1}{r+1}}n^{1-\frac{1}{r+1}}n^{\frac{\delta}{r+1}}\nu(B)^{\frac{d-1}{d}\left(1-\frac{1}{r+1}\right)}$
$\displaystyle\;\;\;\;\;+n^{\frac{r}{r+1}}n^{\delta\left(1-\frac{r}{r+1}\right)}\left(\frac{rC_{y}C}{\nu(B)^{\frac{d-1}{d}}}\right)^{-\frac{r}{r+1}}$
$\displaystyle\leq K_{y}n^{1-\frac{1-\delta}{r+1}},$

where the constant $K_{y}>0$ only depends on $r$, $C_{y}$ and $C$. ∎

######  Lemma 25.

Let $(B_{l})_{l\geq 1}$ be a sequence of rare events, and assume that

$\left|\sum_{j=1}^{n}1_{B_{l}}\circ R^{j}_{B_{l}}(y)-n\nu(B_{l})\right|\leq
C_{l,y}n^{1-\frac{1-\delta}{r+1}}\;\;\;\textit{for a.e }y\in Y,\forall n\geq
M_{y}M_{l},l\geq L_{y},$

for some constant $\delta>0$, $M_{y}\geq 1$ only depending on $y$, $M_{l}\geq
1$ depending only on $l$, and $C_{l,y}>0$ depending on $l$ and $y$, and
$L_{y}\geq 1$ depending only on $y$. Then

$\left|\frac{\nu(B_{l})}{sN_{l}(y)}\sum_{j=0}^{\lceil
sN_{l}(y)\rceil-1}\varphi_{B_{l}}\circ R^{j}_{B_{l}}(y)-1\right|\leq
3s^{-\frac{1-\delta}{r+1}}+\frac{2}{s}.$

where

$N_{l}(y)=\max\left(M_{l},\left\lceil\frac{1}{2\nu(B_{l})}\right\rceil,\min\left(n\geq
1\;|\;n^{-\frac{1-\delta}{r+1}}\leq\frac{1}{2C_{l,y}}\nu(B_{l})\right)\right).$

###### Proof.

Let $y\in Y$ be as in the statement. To simplify the notation denote
$\displaystyle g(n)=n^{1-\frac{1-\delta}{r+1}}$. For $l\geq L_{y}$ and $n\geq
M_{y}M_{l}$ we have

$n\nu(B_{l})-C_{l,y}g(n)\leq\sum_{j=1}^{n}1_{B_{l}}\circ R^{j}_{B_{l}}(y)\leq
n\nu(B_{l})+C_{l,y}g(n).$

and thus

$\sum_{j=0}^{\lfloor n\nu(B_{l})-C_{l,y}g(n)\rfloor-1}\varphi_{B_{l}}\circ
R^{j}_{B_{l}}(y)\leq n\leq\sum_{j=0}^{\lceil
n\nu(B_{l})+C_{l,y}g(n)\rceil-1}\varphi_{B_{l}}\circ R^{j}_{B_{l}}(y).$

Rewrite the summation limits above as $\displaystyle\left\lfloor
n\left(\nu(B_{l})-\frac{C_{l,y}g(n)}{n}\right)\right\rfloor-1$ and
$\displaystyle\left\lceil
n\left(\nu(B_{l})+\frac{C_{l,y}g(n)}{n}\right)\right\rceil-1$ respectively.
For $N_{l}(y)$ as in the statement and $s\geq M_{y}$ we have

$\displaystyle\sum_{j=0}^{\lceil sN_{l}(y)\rceil-1}\varphi_{B_{l}}\circ
R^{j}_{B_{l}}(y)\leq\frac{sN_{l}(y)}{\nu(B_{l})-\frac{g(\lceil
sN_{l}(y)\rceil)}{\lceil sN_{l}(y)\rceil}}+1.$

Thus

$\displaystyle\frac{\nu(B_{l})}{sN_{l}(y)}\sum_{j=0}^{\lceil
sN_{l}(y)\rceil-1}\varphi_{B_{l}}\circ
R^{j}_{B_{l}}(y)\leq\frac{\nu(B_{l})}{\nu(B_{l})-\frac{g(\lceil
sN_{l}(y)\rceil)}{\lceil sN_{l}(y)\rceil}}+\frac{2}{s}$ $\displaystyle\leq
1+\frac{\frac{g(\lceil sN_{l}(y)\rceil)}{\lceil
sN_{l}(y)\rceil}}{\nu(B_{l})-\frac{g(\lceil sN_{l}(y)\rceil)}{\lceil
sN_{l}(y)\rceil}}+\frac{2}{s}\leq 1+2\frac{g(\lceil
s\rceil)}{s+1}+\frac{2}{s}.$

Analogously $\displaystyle\frac{\nu(B_{l})}{sN_{l}(y)}\sum_{j=0}^{\lceil
sN_{l}(y)\rceil-1}\varphi_{B_{l}}\circ R^{j}_{B_{l}}(y)\geq 1-2\frac{g(\lceil
s\rceil)}{s+1}-\frac{2}{s}.$ ∎

We shall also use the following.

######  Proposition 26.

Suppose $R$ satisfies condition (EE), and let $(B_{l})_{l\geq 1}$ be a
sequence of rare events such that $\\{B_{l}\\}$ is regularly approximable.
Then for every $\epsilon>0$ and for almost every $y\in Y$

$\sup_{l\geq 1}\left|\frac{\nu(B_{l})}{sN_{l}(y)}\sum_{j=1}^{\lceil
sN_{l}(y)\rceil}\varphi_{B_{l}}^{(j)}(y)-1\right|\rightarrow 0\;\;\;\textit{as
}s\rightarrow\infty,\;\textit{a.e }y\in Y,$

where

$N_{l}(y)=\left\lceil\left(\frac{K_{y}}{\nu(B_{l})}\right)^{\frac{3}{2}\frac{r+1}{1-\delta_{1}}+\epsilon}\right\rceil,$

for some $K_{y}>0$ depending only on $y$ and $\epsilon$.

###### Proof.

Take $y\in Y$ as in the conclusion of Proposition 23, then

$\left|\sum_{j=1}^{n}1_{B}\circ R^{j}(y)-n\nu(B)\right|\leq
K_{y}n^{1-\frac{1}{r+1}(1-\delta_{1})}\;\;\;\forall n\geq M_{y},\textit{ a.e
}y\in Y.$

Now apply Lemma 25. ∎

####  7.4. PLT scaled by returns to $\\{B_{l}\\}$

For this section let $K\geq 1$, and $A_{l}^{(1)},...,A_{l}^{(K)}$ resp.
$B_{l}^{(1)},...,B_{l}^{(K)}$ be subsets of $X$ resp. $Y$ regularly
approximable in $C^{r}$ resp. $C^{r^{\prime}}$. Suppose that there are
$r_{l}\searrow 0$, and positive constants $\omega^{(k)},\theta^{(k)}>0$ such
that

$\mu(A_{l}^{(k)})=\omega^{(k)}r_{l}^{d}+o(r_{l}^{d}),\;\textit{ and
}\;\nu(B_{l}^{(k)})=\theta^{(k)}r_{l}^{d^{\prime}}+o(r_{l}^{d^{\prime}})\;\;\;\forall
l\geq 1,k=1,...,K,$

and, for each $l$, $B_{l}^{(1)},...,B_{l}^{(K)}$ are disjoint. Denote
$\displaystyle B_{l}=\bigcup_{k=1}^{K}B_{l}^{(k)}$, which, by disjointness of
$B_{l}^{(1)},...,B_{l}^{(K)}$, is also regularly approximable in
$C^{r^{\prime}}$, and $\alpha_{l}=\Phi_{B_{l}}$. Consider

$\kappa_{l}^{(n)}(y)=k\;\;\;\textit{ if }\;\;R_{B_{l}}^{n}(y)\in B_{l}^{(k)},$

which is well-defined as the $B_{l}^{(k)}$ are disjoint, and

$\displaystyle\phi_{\kappa_{l},\alpha_{l},y}^{(1)}(x)=\min\left(n\geq
1\;|\;T_{y}^{\tilde{\alpha}_{l}^{(n)})}(x)\in
A_{l}^{(\kappa_{l}^{(n)})}\right),$
$\displaystyle\phi_{\kappa_{l},\alpha_{l},y}^{(m+1)}(x)=\min\left(n\geq\phi_{\kappa_{l},\alpha_{l},y}^{(m)}(x)+1\;|\;T_{y}^{\tilde{\alpha}_{l}^{(n)}}(x)\in
A_{l}^{(\kappa_{l}^{(n)})}\right),$
$\displaystyle\Phi_{\kappa_{l},\alpha_{l},y}=(\phi_{\kappa_{l},\alpha_{l},y}^{(1)},\phi_{\kappa_{l},\alpha_{l},y}^{(2)},...).$

Following the steps outlined in §7.1 we will first show

$\Omega
r_{l}^{d^{\prime}}\Phi_{\kappa_{l},\alpha_{l}}\overset{\mu}{\Rightarrow}\Phi_{Exp}\;\;\;\textit{as
}l\rightarrow\infty,$

for
$\Omega=\frac{\sum_{k=1}^{K}\omega^{(k)}\theta^{(k)}}{\sum_{k=1}^{K}\theta^{(k)}}$,
and then use the relation

$\varphi_{\bigcup_{k=1}^{K}A_{l}^{(k)}\times
B_{l}^{(k)}}(x,y)=\sum_{j=0}^{\varphi_{\kappa_{l},\Phi_{B_{l}}(y)}(x)-1}\varphi_{B_{l}}\circ
R^{j}_{B_{l}}(y).$ (25)

to obtain

$(\mu\times\nu)\left(\bigcup_{k=1}^{K}A_{l}^{(k)}\times
B_{l}^{(k)}\right)\Phi_{\bigcup_{k=1}^{K}A_{l}^{(k)}\times
B_{l}^{(k)}}\overset{\mu\times\nu}{\Rightarrow}\Phi_{Exp}\;\;\;\textit{as
}l\rightarrow\infty.$ (26)

Denote

$p_{l,t}^{(k)}(y):=\frac{1}{t}\\#\\{j=1,...,t_{l}\;|\;\kappa_{l}^{(j)}(y)=k\\}=\frac{1}{t}\sum_{j=1}^{t}1_{B_{l}^{(k)}}(R_{B_{l}}^{j}(y)).$

We first show that, for each $k$, a.e $y\in Y$, and $t_{l}=O(r_{l}^{-d})$, we
have

$p_{l,t_{l}}^{(k)}(y)\rightarrow\frac{\theta^{(k)}}{\sum_{j=1}^{K}\theta^{(j)}}=:p^{(k)}\;\;\;\textit{as
}l\rightarrow\infty.$ (27)

######  Lemma 27.

Suppose $R$ satisfies (EE) and

$d>\frac{3}{2}\frac{d^{\prime}(r^{\prime}+1)}{1-\delta_{1}}$

Then $\kappa_{l}$ satisfies (27).

###### Proof.

Let $\epsilon>0$ be such that
$d>\frac{d^{\prime}(r^{\prime}+1)}{1-\frac{1+2\delta_{1}+\epsilon}{3}}$ and
denote $\delta^{\prime}=\frac{1+2\delta_{1}+\epsilon}{3}$. (i) Using Lemma 24,
and disjointness, for each $B_{l}^{(k)}$, we get a function
$\tilde{g}(n)=O\left(n^{1-\frac{1}{r+1}\left(1-\delta^{\prime}\right)}\right)$
such that

$\left|\sum_{j=1}^{n}1_{B_{l}^{(k)}}\circ
R^{j}(y)-n\nu(B_{l}^{(k)})\right|\leq\tilde{g}(n),\quad\textit{
and}\quad\left|\sum_{j=1}^{n}1_{B_{l}}\circ
R^{j}(y)-n\nu(B_{l})\right|\leq\tilde{g}(n)$

for all $n\geq M_{y}$. Lemma 25 yields

$\left|\sum_{i=0}^{n-1}\varphi_{B_{l}}\circ
R_{B_{l}}^{i}(y)-\frac{n}{\nu(B_{l})}\right|\leq
3\frac{\tilde{g}(n)}{\nu(B_{l})^{2}}+2,$

for $n\geq\tilde{N}(y)$, where

$\tilde{N}_{l}(y)=\max\left(M_{y},\min\left(n\geq
1\;|\;\frac{\tilde{g}(n^{\prime})}{n^{\prime}}\leq\frac{1}{2}\nu(B_{l})\;\forall
n^{\prime}\geq n\right)\right).$

(ii) Rewrite

$p_{l,n}^{(k)}=\frac{1}{n}\sum_{i=1}^{\sum_{j=1}^{n}\varphi_{B_{l}}\circ
R_{B_{l}}^{j}(y)}1_{B_{l}^{(k)}}\circ R^{i}.$

For $\tilde{\alpha}_{l}^{(n)}=\sum_{j=1}^{n}\varphi_{B_{l}}\circ
R_{B_{l}}^{j}(y)$ we have

$\displaystyle np_{l,n}^{(k)}(y)$
$\displaystyle=\sum_{j=1}^{\tilde{\alpha}_{l}^{(n)}(y)}1_{B_{l}^{(k)}}\circ
R^{j}(y)\leq\sum_{j=1}^{\left\lceil\frac{n}{\nu(B_{l})}+2\frac{\tilde{g}(n)}{\nu(B_{l})^{2}}+2\right\rceil}1_{B_{l}^{(k)}}\circ
R^{j}(y)$
$\displaystyle\leq\left(\frac{n}{\nu(B_{l})}+2\frac{\tilde{g}(n)}{\nu(B_{l})^{2}}+3\right)\nu(B_{l}^{(k)})+\tilde{g}\left(\frac{n}{\nu(B_{l})}+2\frac{\tilde{g}(n)}{\nu(B_{l})^{2}}+3\right)$
$\displaystyle\leq
n\left(\frac{\nu(B_{l}^{(k)})}{\nu(B_{l})}+2\frac{\tilde{g}(n)\nu(B_{l}^{(k)})}{n\nu(B_{l})^{2}}\right)+3\nu(B_{l}^{(k)})+\tilde{g}\left(6\frac{n}{\nu(B_{l})}\right)$

for
$n\geq\max\left(\tilde{N}_{l}(y),\left\lceil\nu(B_{l})^{-1}\right\rceil\right):=\hat{N}_{l}(y)$.
So we have

$p_{l,n}^{(k)}\leq\frac{\nu(B_{l}^{(k)})}{\nu(B_{l})}+\frac{\tilde{g}(n)}{n}r_{l}^{-d^{\prime}}O_{l}(1)+r_{l}^{d^{\prime}}O_{l}(1)+\frac{\tilde{g}(nr_{l}^{-d^{\prime}}O_{l}(1))}{n},$

where the subscripts in $O_{l}$ are used to indicate that the error terms only
depend on $l$. The upper bound is similar. So

$\displaystyle\left|p_{l,n}^{(k)}-\frac{\theta^{(k)}}{\sum_{j=1}^{K}\theta^{(j)}}\right|=o_{l}(1)+\frac{\tilde{g}(n)}{n}r_{l}^{-d^{\prime}}O_{l}(1)+r_{l}^{d^{\prime}}O_{l}(1)+\frac{\tilde{g}(nr_{l}^{-d^{\prime}}O_{l}(1))}{n}$
$\displaystyle=o_{l}(1)+O_{n}\left(n^{-\frac{1}{r^{\prime}+1}\left(1-\delta_{1}\right)}\right)r_{l}^{-d^{\prime}}O_{l}(1)+O_{n}\left(n^{-\frac{1}{r^{\prime}+1}\left(1-\delta_{1}\right)}\right)O_{l}\left(r_{l}^{d^{\prime}-\frac{d^{\prime}}{r+1}\left(1-\delta_{1}\right)}\right)$
$\displaystyle=o_{l}(1)+O_{n}\left(n^{-\frac{1}{r^{\prime}+1}\left(1-\delta_{1}\right)}\right)r_{l}^{-d^{\prime}}O_{l}(1)=o_{l}(1),$

for $n\geq\overline{N}_{l}(y)$, where

$\overline{N}_{l}(y)=\max\left(\hat{N}_{l}(y),\left\lceil
r_{l}^{-\frac{d^{\prime}(r^{\prime}+1)}{1-\delta_{1}}-\epsilon}\right\rceil\right),$

for some $\epsilon>0$. Choosing $\epsilon>0$ so small that

$\frac{d^{\prime}(r^{\prime}+1)}{1-\delta_{1}}+\epsilon<d$

yields $\overline{N}_{l}(y)=o\left(r_{l}^{-d}\right)$, thus proving the Lemma.
∎

####  7.5. Adding in the gaps

Now all that’s left to do is to add back in the gaps. We use (22) to conclude

$(\mu\times\nu)\left(\bigcup_{k=1}^{K}A_{l}^{(k)}\times
B_{l}^{(k)}\right)\Phi_{\bigcup_{k=1}^{K}A_{l}^{(k)}\times
B_{l}^{(k)}}\overset{\mu\times\nu}{\Rightarrow}\Phi_{Exp},\;\;\;\textit{as
}l\rightarrow\infty.$

To do this we make use of the relation

$\varphi_{\bigcup_{k=1}^{K}A_{l}^{(k)}\times
B_{l}^{(k)}}=\sum_{j=0}^{\varphi_{\kappa_{l},\alpha_{l},y}(x)-1}\varphi_{B_{l}}\circ
R^{j}_{B_{l}}(y).$

Let us formulate an abstract result which will allow us to sum up the gaps.
For completeness let us first argue the following probability theory fact.

######  Lemma 28.

Let $(\Omega,\mathbb{P})$ be a probability space, and
$E_{l}:\Omega\rightarrow[0,\infty)$ non-negative real random variables, such
that there are positive random variables $\mu_{l}:\Omega\rightarrow(0,1)$ with

$\mu_{l}E_{l}\overset{\mathbb{P}}{\Rightarrow}E\;\;\;\textit{as
}l\rightarrow\infty,$

for some non-negative random variable $E$ with $\mathbb{P}(E=0)=0$. Then for
any $N_{l}:\Omega\rightarrow[0,\infty)$ with

$\mu_{l}N_{l}\rightarrow 0\;\;\;\textit{as }l\rightarrow\infty\;\textit{
pointwise a.e}$

we have

$\mathbb{P}(E_{l}\leq N_{l})\rightarrow 0\;\;\;\textit{as
}l\rightarrow\infty.$

###### Proof.

Let $\epsilon>0$, there is a $\delta>0$ such that
$\mathbb{P}(E\leq\delta)<\epsilon$, and the distribution function of $E$ is
continuous at $\delta$. By Jegorow’s Theorem, there is a measurable
$K\subset\Omega$ with $\mathbb{P}(K^{c})<\epsilon$ and

$\mu_{l}N_{l}\rightarrow 0\;\;\;\textit{as }l\rightarrow\infty\;\textit{
uniformly on }K.$

Choose $\tilde{L}\geq 1$ so big that $N_{l}\leq\delta\mu_{l}^{-1}$ on $K$ for
$l\geq\tilde{L}$. Now choose $L\geq\tilde{L}$ so big that

$\mathbb{P}(\mu_{l}E_{l}\leq\delta)\leq 2\epsilon\;\;\;\forall l\geq L,$

it follows that

$\mathbb{P}(E_{l}\leq
N_{l})\leq\mathbb{P}(\mu_{l}E_{l}\leq\delta)+\mathbb{P}(K^{c})\leq 3\epsilon,$

for $l\geq L$. ∎

######  Proposition 29.

Let $(\Omega,\mathbb{P})$ be a probability space, and
$E_{l}:\Omega\rightarrow\mathbb{N}$ be positive integer valued observables.
Assume there are positive real numbers $q_{l}\searrow 0$, and a
$[0,\infty)$-valued random variable $E$ with $\mathbb{P}(E=0)=0$ such that

$q_{l}E_{l}\overset{\mathbb{P}}{\Rightarrow}E\;\;\;\textit{as
}l\rightarrow\infty,$

Let $\beta^{(l)}_{j}:\Omega\rightarrow[0,\infty)$ be non-negative random
variables, and assume there are $N_{l}:\Omega\rightarrow(0,\infty)$ with
$q_{l}N_{l}\rightarrow 0$ as $l\rightarrow\infty$ a.e and positive random
variables $b^{(l)}:\Omega\rightarrow(0,\infty)$ such that

$\sup_{l\geq 1}\left|\frac{1}{sN_{l}b^{(l)}}\sum_{j=1}^{\lceil
sN_{l}\rceil}\beta_{j}^{(l)}(\omega)-1\right|\rightarrow 0\;\;\;\textit{as
}s\rightarrow\infty,\;\textit{ a.e. }.$ (UC)

Then
$\displaystyle\frac{q_{l}}{b^{(l)}}\sum_{j=1}^{E_{l}}\beta_{j}^{(l)}\overset{\mathbb{P}}{\Rightarrow}E\;\;\;\textit{as
}l\rightarrow\infty.$

######  Remark 30.

In our context we use $(\Omega,\mathbb{P})=(X\times Y,\mu\times\nu)$,
$R_{l}=\phi_{A_{l},\Phi_{B_{l}}}$, $\beta_{j}^{(l)}=\varphi_{B_{l}}\circ
R_{B_{l}}^{j}$, $q_{l}=\mu(A_{l})$ and $b^{(l)}=\frac{1}{\nu(B_{l})}$. The
existence of $N_{l}$ is the content of Proposition 26.

###### Proof.

(i) Let $F$ be the distribution function of $E$ and $C=\\{t\;|\;F\textit{ is
continuous at }t\\}$ the set of its continuities. Let $t\in C$, and
$\epsilon>0$ such that $\frac{t}{1+\epsilon},\frac{t}{1-\epsilon}\in C$.
(ii) By Jegorow’s Theorem, for $l\geq 1$, there is a measurable set
$K\subset\Omega$ with $\mathbb{P}(K^{c})<\epsilon$

$\sup_{l\geq 1}\sup_{\omega\in
K}\left|\frac{1}{sN_{l}b^{(l)}}\sum_{j=1}^{\lceil
sN_{l}\rceil}\beta_{j}^{(l)}(\omega)-1\right|\rightarrow 0\;\;\;\textit{as
}s\rightarrow\infty.$

(iii) Choose $S>0$ so big that

$(1-\epsilon)b^{(l)}\leq\frac{1}{sN_{l}}\sum_{j=1}^{\lceil
sN_{l}\rceil}\beta_{j}^{(l)}\leq(1+\epsilon)b^{(l)}\;\;\;\textit{on
}K,\;\forall s\geq S,l\geq 1.$

Then

$(1-\epsilon)q_{l}E_{l}\leq\frac{q_{l}}{b^{(l)}}\sum_{j=1}^{E_{l}}\beta_{j}^{(l)}(\omega)\leq(1+\epsilon)q_{l}E_{l}\;\;\;\textit{
on }K\cap\\{E_{l}\geq SN_{l}\\},\;\forall l\geq 1.$

We get

$\displaystyle\mathbb{P}\left(\frac{q_{l}}{b^{(l)}}\sum_{j=1}^{E_{l}}\beta_{j}^{(l)}\leq
t\right)-\mathbb{P}(E_{l}\leq SN_{l})-\mathbb{P}(K^{c})$
$\displaystyle\leq\mathbb{P}\left(K\cap\\{E_{l}\geq
SN_{l}\\}\cap\left\\{\frac{q_{l}}{b^{(l)}}\sum_{j=1}^{E_{l}}\beta_{j}^{(l)}\leq
t\right\\}\right)\leq\mathbb{P}\left(q_{l}E_{l}\leq\frac{t}{1-\epsilon}\right).$

Likewise

$\displaystyle\mathbb{P}\left(\frac{q_{l}}{b^{(l)}}\sum_{j=1}^{E_{l}}\beta_{j}^{(l)}\leq
t\right)$ $\displaystyle\geq\mathbb{P}\left(K\cap\\{E_{l}\geq
SN_{l}\\}\cap\left\\{q_{l}E_{l}\leq\frac{t}{1+\epsilon}\right\\}\right)$
$\displaystyle\geq\mathbb{P}\left(q_{l}E_{l}\leq\frac{t}{1+\epsilon}\right)-\mathbb{P}(E_{l}\leq
SN_{l})-\mathbb{P}(K^{c}).$

Taking $\displaystyle\limsup_{l\rightarrow\infty}$ resp.
$\displaystyle\liminf_{l\rightarrow\infty}$ in (7.5) and using Lemma 28 we
obtain

$\displaystyle
F\left(\frac{t}{1+\epsilon}\right)-\epsilon\leq\liminf_{l\rightarrow\infty}\mathbb{P}\left(\frac{q_{l}}{b^{(l)}}\sum_{j=1}^{E_{l}}\beta_{j}^{(l)}\leq
t\right)$
$\displaystyle\leq\limsup_{l\rightarrow\infty}\mathbb{P}\left(\frac{q_{l}}{b^{(l)}}\sum_{j=1}^{E_{l}}\beta_{j}^{(l)}\leq
t\right)\leq F\left(\frac{t}{1-\epsilon}\right)+\epsilon.$

Since $C\subset(0,\infty)$ is dense, we can let $\epsilon\searrow 0$ while
$\frac{t}{1+\epsilon},\frac{t}{1-\epsilon}\in C$, this yields
$\displaystyle\mathbb{P}\left(\frac{q_{l}}{b^{(l)}}\sum_{j=1}^{E_{l}}\beta_{j}^{(l)}\leq
t\right)\rightarrow F(t)\;\;\;\textit{as }l\rightarrow\infty.$ ∎

######  Remark 31.

  1. (a)

We can extend this statement to sequences in the following manner: under the
assumptions of the proposition, let $E^{k}_{l}:\Omega\rightarrow\mathbb{N}$ be
such that

$\mathbb{P}(E^{1}_{l},E^{2}_{l},...)\overset{\mu}{\Rightarrow}(E^{1},E^{2},...)\;\;\;\textit{as
}l\rightarrow\infty,$

for some $E^{k}:\Omega\rightarrow[0,\infty)$ with $\mathbb{P}(E^{k}=0)=0$.
Then

$\frac{q_{l}}{b^{(l)}}\left(\sum_{j=1}^{E^{1}_{l}}\beta_{j}^{(l)},\sum_{j=1}^{E^{2}_{l}}\beta_{j}^{(l)},...\right)\overset{\mathbb{P}}{\Rightarrow}(E^{1},E^{2},...)\;\;\;\textit{as
}l\rightarrow\infty.$

The proof of this statement is almost the same as for the proposition,
therefore we won’t repeat it.

  2. (b)

The probability measure $\mathbb{P}$ can be replaced by a sequence
$(\mathbb{P}_{l})_{l\geq 1}$ by also replacing $(UC)$ with

$\displaystyle\forall\epsilon>0\textit{ there is are}$ $\displaystyle\textit{
measurable sets }K_{l}\subset\Omega\textit{ with
}\limsup_{l\rightarrow\infty}\mathbb{P}_{l}(K_{l}^{c})<\epsilon\;\textit{ such
that}$ $\displaystyle\sup_{l\geq 1}\sup_{\omega\in
K_{l}}\left|\frac{1}{sN_{l}b^{(l)}}\sum_{j=1}^{\lceil
sN_{l}\rceil}\beta_{j}^{(l)}(\omega)-1\right|\rightarrow 0\;\;\;\textit{as
}s\rightarrow\infty.$

###### Proof of Theorem 5.

(i) Let $r_{l}\searrow 0$, and denote by $Q_{l}=B_{r_{l}}(x^{*},y^{*})$ the
geodesic ball of radius $r_{l}$ centred at $(x^{*},y*)$, and let $\epsilon>0$.
Wlog $r_{1}$ is small enough that the exponential map at $(x^{*},y^{*})$ is a
diffeomorphism from the ball of radius $2r_{1}$ in $\mathbb{R}^{d+d^{\prime}}$
onto $B_{2r_{1}}(x^{*},y^{*})$. It is now easy to construct, for some $K\geq
1$, sets $A_{l}^{(1)},...,A_{l}^{(K)}$ and $B_{l}^{(1)},...,B_{l}^{(K)}$ as in
section 6. Let $\omega^{(k)},\theta^{(k)}>0$ be such that131313Choose
$x^{*},y^{*}$ such that the densities of $\mu,\nu$ are positive at the
respective points.

$\mu(A_{l}^{(k)})=\omega^{(k)}r_{l}^{d}+o(r_{l}^{d}),\;\textit{ and
}\;\nu(B_{l}^{(k)})=\theta^{(k)}r_{l}^{d^{\prime}}+o(r_{l}^{d^{\prime}})\;\;\;\forall
l\geq 1,k=1,...,K,$

and set
$\Omega=\frac{\sum_{k=1}^{K}\omega^{(k)}\theta^{(k)}}{\sum_{k=1}^{K}\theta^{(k)}}$.

Due to Lemma 17 all those sets can be chosen to be regularly approximable,
such that141414Here we again use the continuity of the density.
$(\mu\times\nu)_{Q_{l}}\left(Q_{l}\setminus\bigcup_{k=1}^{K}A_{l}^{(k)}\times
B_{l}^{(k)}\right)<\epsilon$ for all $l\geq 1$. Let $\Lambda>0$ be such that
$(\mu\times\nu)(Q_{l})=\Lambda r_{l}^{d+d^{\prime}}+o(r_{l}^{d+d^{\prime}})$,
then

$\left|\Lambda-\Omega\sum_{k=1}^{K}\theta^{(k)}\right|<\epsilon$

(ii) For $y$ as in assumption (MEM), consider
$\alpha_{l}(y)=\Phi_{\bigcup_{k=1}^{K}B_{l}^{(k)}}(y)$, and

$\kappa_{l}^{(n)}(y)=k\;\;\;\textit{ if
}\;\;R_{\bigcup_{j=1}^{K}B_{l}^{(j)}}^{n}(y)\in B_{l}^{(k)}.$

By disjointness $\kappa_{l}(y)$ is well-defined. By Lemma 27, $\kappa_{l}(y)$
satisfies $(\dagger)$, with $a_{l}=r_{l}^{d}$ and
$p^{(k)}=\frac{\theta^{(k)}}{\sum_{s=1}^{K}\theta^{(s)}}$. We can use
Proposition 19 to obtain

$\Omega
r_{l}^{d}\Phi_{\kappa_{l}(y),\alpha_{l}(y)}\overset{\mu}{\Rightarrow}\Phi_{Exp}\;\;\;\textit{as
}l\rightarrow\infty,\textit{ a.e }y.$

Since this is true for a.e $y\in Y$, we also have

$\Omega
r_{l}^{d}\Phi_{\kappa_{l},\alpha_{l}}\overset{\mu\times\nu}{\Rightarrow}\Phi_{Exp}\;\;\;\textit{as
}l\rightarrow\infty.$

(iii) Using Lemma 24 for $B_{l}^{(k)}$ for each $k=1,...,K$, and summing over
$k$, we get

$\left|\sum_{j=1}^{n}1_{B_{l}}\circ R^{j}(y)-n\nu(B_{l})\right|\leq
K_{y}\tilde{g}(n)\;\;\;\forall n\geq N,\textit{ a.e }y\in Y,$

for some
$\tilde{g}(n)=o\left(n^{1-\frac{1}{r^{\prime}+1}(1-\delta_{1})}\right)$ not
depending on $l$ and $y$, and $K_{y}>0$ only depending on $y$. By Lemma 25,
$\alpha_{l}$ satisfies $(UC)$ with

$b^{(l)}=\frac{1}{\nu(B_{l})}=\frac{1}{\sum_{k=1}^{K}\theta^{(k)}}r_{l}^{-d^{\prime}}+o(r_{l}^{-d^{\prime}})$

and

$N_{l}=\max\left(\left\lceil\frac{1}{2\nu(B_{l})}\right\rceil,\min\left(n\geq
1\;|\;C\left(n^{\delta_{1}-1}\right)^{\frac{1}{r^{\prime}+1}}\leq\frac{1}{2}\nu(B_{l})\right)\right),$

for some constant $C>0$. In particular,
$N_{l}=O\left(r_{l}^{-d^{\prime}\frac{r^{\prime}+1}{1-\delta_{1}}}\right)=o(a_{l})$.
Note that, for $Q_{l}^{\prime}=\bigcup_{k=1}^{K}A_{l}^{(k)}\times B_{l}^{(k)}$
we have

$\displaystyle\Phi_{Q_{l}^{\prime}}=(\sum_{j=0}^{\varphi_{\kappa_{l},\alpha_{l}}^{(1)}-1}\varphi_{B_{l}}\circ
R^{j}_{B_{l}},\sum_{j=\varphi_{\kappa_{l},\alpha_{l}}^{(1)}}^{\varphi_{\kappa_{l},\alpha_{l}}^{(2)}-1}\varphi_{B_{l}}\circ
R^{j}_{B_{l}},...).$

We can apply Proposition 29

$\Omega\left(\sum_{k=1}^{K}\theta^{(k)}\right)r_{l}^{d+d^{\prime}}\Phi_{Q_{l}^{\prime}}\overset{\mu\times\nu}{\Rightarrow}\Phi_{Exp}\;\;\;\textit{as
}l\rightarrow\infty.$

(iv) By Proposition 2 also

$\Omega\left(\sum_{k=1}^{K}\theta^{(k)}\right)r_{l}^{d+d^{\prime}}(\mu\times\nu)(Q_{l}^{\prime})\Phi_{Q_{l}^{\prime}}\overset{{\mu\times\nu}_{Q_{l}^{\prime}}}{\Rightarrow}\Phi_{Exp}\;\;\;\textit{as
}l\rightarrow\infty.$

At the same time, taking a subsequence if necessary, there are
$[0,\infty]$-valued processes $\Phi$ and $\tilde{\Phi}$ such that

$\displaystyle(\mu\times\nu)(Q_{l})\Phi_{Q_{l}}\overset{\mu\times\nu}{\Rightarrow}\Phi\;\;\;\textit{as
}l\rightarrow\infty,\;\textit{ and}$
$\displaystyle(\mu\times\nu)(Q_{l})\Phi_{Q_{l}}\overset{(\mu\times\nu)_{Q_{l}}}{\Rightarrow}\tilde{\Phi}\;\;\;\textit{as
}l\rightarrow\infty.$

Hence $D\left(\Phi_{Exp},\tilde{\Phi}\right)\leq\epsilon,$ where $D$ is given
by Lemma 21. Since this is true for every $\epsilon>0$ we have
$\tilde{\Phi}\overset{d}{=}\Phi_{Exp}$, and by Corollary 2 also
$\Phi\overset{d}{=}\Phi_{Exp}$. ∎

###  8\. The skewing time

Here we will prove Theorem 7, to do this we will verify that the map
$T(x,y)=G_{\tau(y)}(x)$ satisfies superpolynomial mixing of all orders, i.e
condition (SPM) of Remark 20(iii).

######  Lemma 32.

In the setting of Theorem 7, suppose that

$\nu(|\tau_{n}|<\zeta(n))=O(n^{-\kappa})\;\;\;\textit{as }n\rightarrow\infty,$

for some $\zeta:\mathbb{N}\rightarrow(0,\infty)$ and
$\kappa>\frac{d^{\prime}}{\delta_{2}}$, and $\sum_{l\geq
1}r_{l}^{\frac{1}{2}(d^{\prime}-\delta_{2}\kappa)}<\infty$. Then, for each
$t>0$, there is a set $\mathfrak{G}_{t}$ with $\nu(\mathfrak{G}_{t})=1$ such
that, for $y\in\mathfrak{G}_{t}$, there are $L_{y,t}>0$ and sets
$\mathcal{B}_{l,y,t}\subset\\{1,...,\left\lceil\frac{t}{\mu(A_{l})}\right\rceil\\}$
with $|\mathcal{B}_{l,y,t}|=o(\mu(A_{l})^{-1})$ such that

$|\tau_{\tilde{\alpha}_{l}^{(n)}}(y)-\tau_{\tilde{\alpha}_{l}^{(m)}}(y)|\geq\zeta\left(\tilde{\alpha}_{l}^{(n)}(y)-\tilde{\alpha}_{l}^{(m)}(y)\right)$
$\forall l\geq L_{t,y},1\leq
n<m\leq\left\lceil\frac{t}{\mu(A_{l})}\right\rceil,n\not\in\mathcal{B}_{l,y,t}$

where $\alpha_{l}=\Phi_{B_{l}}$.

###### Proof.

Fix $t>0$, to keep notation simple we assume
$\mu(A_{l})=r_{l}^{d}+o(r_{l}^{d})$ and
$\nu(B_{l})=r_{l}^{d^{\prime}}+o(r_{l}^{d^{\prime}})$, otherwise there is an
extra constant in the estimates below.

(i) We call $\displaystyle
n\in\left\\{1,...,\left\lceil\frac{t}{\mu(A_{l})}\right\rceil\right\\}$ a
$(l,y)$-bad return if there is a $m>n$ such that

$|\tau_{\tilde{\alpha}_{l}^{(n)}}(y)-\tau_{\tilde{\alpha}_{l}^{(m)}}(y)|<\zeta(\tilde{\alpha}_{l}^{(n)}(y)-\tilde{\alpha}_{l}^{(m)}(y)),$

denote $\mathcal{B}_{l,y}=\\{n\geq 1\;|\;n\textit{ is a }(l,y)-\textit{bad
return}\\}$. Let $0<\epsilon_{1}$ We call $y\in Y$ an $l$-bad point if
$|\mathcal{B}_{l,y}|<r_{l}^{-d+\epsilon_{1}}$.

(ii) Using Lemma 25 and Jegorov’s theorem, for $\epsilon_{2}>0$, we can find a
measurable $G=G_{\epsilon_{2}}\subset Y$ with $\nu(G^{c})<\epsilon_{2}$ and an
$\tilde{L}\geq 1$ depending on $G$ such that

$\tilde{\alpha}_{l}^{(\left\lfloor\frac{t}{\mu(A_{l}}\right\rfloor)}\leq
2tr_{l}^{-(d+d^{\prime})}\;\;\;\forall y\in G,l\geq\tilde{L}.$

(iii) For $l\geq 1$ denote
$G_{l}=\\{\tilde{\alpha}_{l}^{\left\lceil\frac{t}{\mu(A_{l})}\right\rceil}\leq
2tr_{l}^{-(d+d^{\prime})}\\}$, we have

$\nu(y\in G_{l}\;|\;y\textit{ is
}l\textit{-bad})\leq\frac{\int_{G_{l}}|\mathcal{B}_{l,y}|\,\text{d}\nu}{r_{l}^{-d+\epsilon}}\leq
r_{l}^{d-\epsilon}\sum_{n=1}^{\left\lceil\frac{t}{\mu(A_{l})}\right\rceil}\nu(y\;|\;n\textit{
is }(l,y)\textit{-bad})$ $\leq r_{l}^{d-\epsilon}\sum_{j=1}^{\lceil
2tr_{l}^{-(d+d^{\prime})}}\nu\left(T^{-j}\left(\exists i\geq
1\;|\;|\tau_{\tilde{\alpha}_{l}^{(i)}}|<\zeta\left(\tilde{\alpha}_{l}^{(i)}\right)\right)\right)$
$\leq r_{l}^{d-\epsilon}\sum_{j=1}^{\lceil
2tr_{l}^{-(d+d^{\prime})}}\nu(\exists i\geq
r_{l}^{-\delta_{2}}\;|\;|\tau_{i}|<\zeta(i))$ $\leq 2Ktr_{l}^{d-\epsilon-
d-d^{\prime}+\delta_{2}\kappa}\leq 2Ktr_{l}^{\delta_{2}\kappa-d^{\prime}},$

for some constant $K>0$, for small enough $\epsilon$ this is summable. An
application of the Borel-Cantelli Lemma yields that for almost every $y\in Y$;
for big enough $l$, either $y\not\in G_{l}$ or

$|\tau_{\tilde{\alpha}_{l}^{(n)}}(y)-\tau_{\tilde{\alpha}_{l}^{(m)}}(y)|\geq\zeta(\tilde{\alpha}_{l}^{(n)}(y)-\tilde{\alpha}_{l}^{(m)}(y))\;\;\;\forall
1\leq
n<m\leq\left\lceil\frac{t}{\mu(A_{l})}\right\rceil,n\not\in\mathcal{B}_{l,y,}.$

At the same time, by Proposition 26 we have $G_{l}\nearrow Y$. Thus a.e $y$ is
in $G_{l}$ for big enough $l$, and the conclusion follows. ∎

###### Proof of Theorem 7.

In order to keep notation simple we will only show the PLT for regularly
approximable rectangles, this can be easily extended to geodesic balls, by
following the same arguments as in the proof of Theorem 5.

(i) For a.e $y$ and $0=t_{0}<t_{1}<...<t_{J}$ choose
$L_{y}=L_{y,t_{1}+...+t_{J}}$ and
$\mathcal{B}_{l,y}=\mathcal{B}_{l,y,t_{1}+...+t_{J}}$ as in Lemma 32. For such
a $y$ and $l\geq L_{y}$ (in the following we suppress $y$ from the notation)
consider

$S_{t_{j},l}=\sum_{i=1}^{\left\lceil\frac{t_{j}}{\mu(A_{l})}\right\rceil}1_{A_{l}}\circ
T^{\tilde{\alpha}_{l}^{(i)}}=S^{\prime}_{t_{j},l}+S_{t_{j},l}^{\prime\prime},$

where

$S^{\prime}_{t_{j},l}=\sum_{\begin{subarray}{c}i=1,...,\left\lceil\frac{t_{j}}{\mu(A_{l})}\right\rceil\\\
i\not\in\mathcal{B}_{l}\end{subarray}}1_{A_{l}}\circ
T^{\tilde{\alpha}_{l}^{(j)}}.$

As in Proposition 19, the first goal is to show

$\left(S_{t_{1},l}-S_{t_{0},l},...,S_{t_{J},l}-S_{t_{J-1},l}\right)\overset{\mu}{\Rightarrow}\left(P_{t_{1}-t_{0}},...,P_{t_{J}-t_{J-1}}\right)\;\;\;\textit{as
}l\rightarrow\infty,$

where $(P_{t})$ is a standard Poisson process. Since
$||S_{t_{j},l}^{\prime\prime}||_{L^{1}}\rightarrow 0$ for all $j=1,...,J$, it
is equivalent to show

$\left(S^{\prime}_{t_{1},l}-S^{\prime}_{t_{0},l},...,S^{\prime}_{t_{J},l}-S^{\prime}_{t_{J-1},l}\right)\overset{\mu}{\Rightarrow}\left(P_{t_{1}-t_{0}},...,P_{t_{J}-t_{J-1}}\right)\;\;\;\textit{as
}l\rightarrow\infty.$

For $m_{1},...,m_{J}\geq 1$ it will be enough to show

$\int_{X}\prod_{j=1}^{J}{{S^{\prime}_{t_{j},l}-S^{\prime}_{t_{j-1},l}}\choose
m_{j}}\,\text{d}\mu=\prod_{j=1}^{J}\frac{(t_{j}-t_{j-1})^{m_{j}}}{m_{j}!}.$
(28)

We have

$\prod_{j=1}^{J}{{S^{\prime}_{t_{j},l}-S^{\prime}_{t_{j-1},l}}\choose
m_{j}}=\sum_{\begin{subarray}{c}\left\lceil\frac{t_{j-1}}{\mu(A_{l})}\right\rceil+1\leq
k_{j,1}<...<k_{j,m_{j}}\leq\left\lceil\frac{t_{j}}{\mu(A_{l})}\right\rceil\\\
k_{i,j}\not\in\mathcal{B}_{l}\textit{ for
}j=1,...,J,\;i=1,...,m_{j}\end{subarray}}\prod_{i,j}1_{A_{l}}\circ
T^{\tilde{\alpha}_{l}^{(k_{i,j})}}.$ (29)

(ii) Due to assumption (MEM) for $G$, and Lemma 32, we have

$\left|\int_{X}\prod_{j=1}^{m}f_{j}\circ
T^{\tilde{\alpha}_{l}^{(n_{j})}}\,\text{d}\mu-\prod_{j=1}^{m}\int_{X}f_{j}\,\text{d}\mu\right|\leq
C_{y}\psi(\min_{j\not=j^{\prime}}|\tilde{\alpha}_{l}^{(n_{j})}-\tilde{\alpha}_{l}^{(n_{j^{\prime}})}|)\prod_{j=1}^{m}||f_{j}||_{C^{r}},$

where $\psi(x)=e^{-\gamma\zeta(x)}$ and $\zeta$ is as in assumption (BA), for
$f_{1},...,f_{m}\in C^{r}$ and $1\leq n_{1}\leq...\leq
n_{m}\leq\left\lceil\frac{t1+...+t_{J}}{r_{l}^{d}}\right\rceil$ with
$n_{i}\not\in\mathcal{B}_{l}$. Due to (8) and assumption (BA) we have

$\psi(\min_{j\not=j^{\prime}}|\tilde{\alpha}_{l}^{(n_{j})}-\tilde{\alpha}_{l}^{(n_{j^{\prime}})}|)=O(r_{l}^{w_{l}}),$

for some $w_{l}>0$ with $w_{l}\rightarrow\infty$ as $l\rightarrow\infty$.
Approximating $1_{A_{l}}$ by functions in $C^{r}$ it is
straightforward151515The calculation is analogous to Lemma 18. to show that

$\left|\int_{X}\prod_{i,j}1_{A_{l}}\circ
T^{\tilde{\alpha}_{l}^{(k_{i,j})}}\,\text{d}\mu-\mu(A_{l})^{m_{1}+...+m_{J}}\right|=o(\mu(A_{l})^{m_{1}+...+m_{J}}),$

for $k_{i,j}$ as in (29). The sum in (29) has

$\mu(A_{l})^{-(m_{1}+...+m_{J})}\prod_{j=1}^{J}\frac{(t_{j}-t_{j-1})^{m}_{j}}{m_{j}!}+o(\mu(A_{l})^{-(m_{1}+...+m_{J})})$

many terms, so (28) follows. ∎

###  9\. Examples

Here we verify conditions (EE) and (BR) for the examples listed in Section 4.

####  9.1. Diophantine rotations

Let $\alpha\in\left((0,1)\setminus\mathbb{Q}\right)^{d^{\prime}}$ satisfy a
diophantine condition, i.e there are $C>0$ and $n\geq 1$ such that

$\left|\langle k,\alpha\rangle-l\right|>C|k|^{-n}\;\;\;\forall
k\in\mathbb{Z}^{d^{\prime}},k\not=0,l\in\mathbb{Z},$ (D)

and $R=R_{\alpha}:x\mapsto x+\alpha\;(mod\;1)$, for
$x\in\mathbb{T}^{d^{\prime}}$, the rotation by $\alpha$. In fact almost all
$\alpha$ satisfy $(D)$ for some $n>d^{\prime}$ (this is a consequence of a
higher dimensional version of Khinchin’s Theorem, see eg [BRV16]).
Note that $(D)$ implies that there is a constant $C^{\prime}>0$. such that

$\left|1-e^{2\pi i\langle k,\alpha\rangle}\right|\geq
C^{\prime}|k|^{-n}\;\;\;\forall k\in\mathbb{Z}^{d^{\prime}}\setminus\\{0\\}.$

Property $(BR)$ follows directly from $(D)$. Due to the self-symmetry of
$R_{\alpha}$ it is enough to consider returns of $x=0$ to a rectangle
$(-r,r)^{d^{\prime}}$, but

$R_{\alpha}^{m}(0)\in(-r,r)^{d^{\prime}}\iff|m\alpha-k|<r\;\;\;\textit{for
some }k\in\mathbb{Z}^{d^{\prime}}.$

Then, by $(D)$, $Cm^{-n}<r$, equivalently $m>(C^{-1}r)^{-\frac{1}{n}}$.
To show effective equidistribution $(EE)$, we solve the homological equation.
Let $f\in H^{n}(\mathbb{T}^{d^{\prime}})$ with $\int f=0$. Then

$f(x)=\sum_{k\in\mathbb{Z}^{d^{\prime}}}a_{k}e^{2\pi i\langle k,x\rangle},$

where
$\displaystyle\sum_{k\in\mathbb{Z}^{d^{\prime}}}|a_{k}|^{2}\sum_{j_{1}+...+j_{d^{\prime}}=n}\prod_{i=1}^{d^{\prime}}|k_{i}|^{2j_{i}}<\infty$
and $a_{0}=0$. To solve $f\\!\\!=\\!\\!g-g\circ R_{\alpha}$ we write

$g(x)=\sum_{k\in\mathbb{Z}^{d^{\prime}}}b_{k}e^{2\pi i\langle k,x\rangle}.$

By comparing coefficients, this is satisfied for $b_{k}=\frac{a_{k}}{1-e^{2\pi
i\langle k,\alpha\rangle}}$ for $k\not=0$ and $b_{0}=0$. We have

$\sum_{k\in\mathbb{Z}^{d^{\prime}}}|b_{k}|^{2}\leq(C^{\prime})^{2}\sum_{k\in\mathbb{Z}^{d^{\prime}}}|a_{k}|^{2}|k|^{2n}\leq(C^{\prime})^{2}\sum_{k\in\mathbb{Z}}|a_{k}|^{2}\sum_{j_{1}+...+j_{d^{\prime}}=n}\prod_{i=1}^{d^{\prime}}|k_{i}|^{2j_{i}}<\infty.$

In particular $||g||_{L^{2}}\leq C^{\prime}||f||_{H^{n}}$.
Thus for every161616If $\int h\not=0$ consider $f=h-\int h$. $h\in
H^{n}(\mathbb{T}^{d^{\prime}})$ we have

$\left|\left|\sum_{j=1}^{J}h\circ
R_{\alpha}^{j}-J\int_{\mathbb{T}}h\,\text{d}\lambda\right|\right|_{L^{2}}\leq
C^{\prime\prime}||h||_{H^{n}}.$

Due to Remark 4(iii), condition (EE) is satisfied with $r=n$ and
$\delta_{1}=0$.

In this particular case (7) becomes $\displaystyle
d>\frac{3}{2}d^{\prime}(n+1).$

####  9.2. Horocycle flows

Consider the classical horocycle flow $h_{t}$ on (compact) homogeneous space
$\Gamma\backslash PSL(2,\mathbb{R})$ generated by

$h_{t}=\begin{bmatrix}1&t\\\ 0&1\end{bmatrix}.$

For fixed $t>0$ we will consider the time $t$ map $R=h_{t}$.

Condition (BR) follows from the relation $h_{e^{2s}t}=g_{s}\circ h_{t}\circ
g_{-s}$, where $g_{s}$ is the geodesic flow

$g_{s}=\begin{bmatrix}e^{s}&0\\\ 0&e^{-s}\end{bmatrix}.$

Indeed, we will show that there is a constant $c>0$ such that for small enough
$r>0$, $0<|s|<cr^{-1}$, and $x,y\in\Gamma/PSL(2,\mathbb{R})$ with
$d(x,y)<\frac{c}{2}r^{\frac{1}{2}}$ we have

$d(h_{s}x,y)\geq\frac{c}{2}r^{\frac{1}{2}}.$

By the triangle inequality it is enough to show $d(h_{s}x,x)\geq
cr^{\frac{1}{2}}$. By compactness choose $c=\inf_{x}d(h_{1}x,x)>0$ (by
Hedlund’s Theorem there are no periodic orbits). Let $t=\frac{\log(|s|)}{2}$,
then

$d(h_{s}x,x)=d(g_{t}h_{sgn(s)}g_{-t}x,g_{t}g_{-t}x)\geq
e^{-|t|}d(h_{sgn(s)}g_{-t}x,g_{-t}x)\geq cr^{\frac{1}{2}}.$

For small enough $r$, $g_{t}$ contracts distances at most by a factor of
$e^{-|t|}$. Renaming $r=\frac{c}{2}r^{\frac{1}{2}}$ gives (BR).

In order to show effective ergodicity (EE), we combine [FFT16, Corollary 2.8]
and [FF03, Theorem 1.2 and Corollary 1.3] to conclude that

$\left|\left|\sum_{j=0}^{n-1}f\circ R^{j}-n\int
f\,\text{d}\nu\right|\right|_{L^{2}(\nu)}\leq
C||f||_{C^{12}}\;\;n^{\alpha+\epsilon}\;\;\;\forall f\in C^{12},\;\forall
n\geq 1,$

for some171717Denoting $\mu_{0}>0$ the smallest positive eigenvalue of the
Casimir operator, we have
$\displaystyle\alpha=\max\left(\frac{1+\sqrt{\max(0,1-4\mu_{0})}}{2},\frac{5}{6}\right)$.
$\alpha\in(0,1)$ and all $\epsilon>0$. Then Theorem 5 applies with

$d>\frac{3}{2}\times\frac{2\times 7}{1-\alpha}=\frac{21}{1-\alpha}.$

####  9.3. Skew shifts

Let $\alpha\in(0,1)\setminus\mathbb{Q}$ satisfy the diophantine condition
$(D)$ for some $n\geq 2$ and $R:\mathbb{T}^{2}\rightarrow\mathbb{T}^{2}$ be
given by

$R(x,y)=(x+\alpha,y+x).$

Since $R$ has a Diophantine rotation as a factor (BR) is satisfied by §9.1.

For $k=(k_{1},k_{2})\in\mathbb{Z}^{2}$ denote $e_{k}(x)=e^{2\pi i\langle
k,x\rangle}$. Note that

$\langle e_{k},e_{k^{\prime}}\circ
R^{j}\rangle_{L^{2}(\mathbb{T}^{2})}=\delta^{(k_{1},k_{2})}_{(k^{\prime}_{1}+jk^{\prime}_{2},k^{\prime}_{2})}.$

For $f\in H^{2}(\mathbb{T})$ we can write
$f=\sum_{k\in\mathbb{Z}^{2}}a_{k}e_{k}$. If $a_{(k_{1},0)}\equiv 0$ (in
particular $\int f=0$) then

$\sum_{j\geq 1}\left|\langle f,f\circ R^{j}\rangle_{L^{2}}\right|=\sum_{j\geq
1}\left|\sum_{k\in\mathbb{Z}^{2}}a_{(k_{1},k_{2})}\overline{a_{(k_{1}+jk_{2},k_{2})}}\right|\\!\leq\\!\left(\sum_{k\in\mathbb{Z}^{2}}|a_{(k_{1},k_{2})}|\right)^{2}\\!\\!\\!\leq
C||f||_{H^{2}}^{2},$

where $C>0$ does not depend on $f$. From this we obtain

$\displaystyle\left|\left|\sum_{j=1}^{J}f\circ
R^{j}\right|\right|_{L^{2}(\mathbb{T}^{2})}^{2}$
$\displaystyle=\sum_{j=1}^{J}\sum_{j^{\prime}=1}^{J}\langle f\circ
R^{j^{\prime}},f\circ R^{j}\rangle_{L^{2}(\mathbb{T}^{2})}$
$\displaystyle\leq\sum_{j=0}^{J-1}(J-j)\langle f,f\circ
R^{j}\rangle_{L^{2}(\mathbb{T}^{2})}\leq CJ||f||_{H^{2}(\mathbb{T}^{2})}^{2}.$

For general $f\in H^{n}(\mathbb{T}^{2})$ with $\int f=0$, again write
$f=\sum_{k\in\mathbb{Z}^{2}}a_{k}e_{k}$ and set

$\displaystyle
f_{1}=\sum_{k\in\mathbb{Z}^{2},k_{2}\not=0}a_{k}e_{k}\;\;\textit{ and
}\;\;f_{2}=\sum_{k\in\mathbb{Z}^{2},k_{2}=0}a_{k}e_{k}.$

Applying the above, and the analysis for diophantine rotations, we find

$\left|\left|\sum_{j=1}^{J}f\circ
R^{j}\right|\right|_{L^{2}(\mathbb{T}^{2})}^{2}\leq(CJ+C^{\prime})||f||_{H^{n}(\mathbb{T}^{2})}^{2}.$

Thus the condition in Proposition 23 is satisfied with $\delta=\frac{1}{2}$.
So we can apply Theorem 5 with $d>3(n+1).$

####  9.4. Example 12

Recall the definition of the Weyl Chamber flow on $\Gamma\backslash
SL(d,\mathbb{R})$. Let $d\geq 3$, and $\Gamma$ be a uniform lattice. Denote by
$D_{+}$ the subgroup of diagonal elements of $SL(d,\mathbb{R})$ with positive
entries. It is easy to see that $D_{+}$ is isomorphic to $\mathbb{R}^{d-1}$.
$D_{+}$ acts on $\Gamma\backslash Sl(d,\mathbb{R})$ by right translation,
giving us a $\mathbb{R}^{d-1}$-action. By [BEG20] the action $G$ satisfies (a
$\mathbb{R}^{d-1}$ version of) (MEM).

The diophantine rotation $R_{\alpha}$ satisfies (EE) and (BR) by §9.1. Hence
we can apply Theorem 7.

###  10\. Robustness of delayed returns

Lastly we mention the proof for the delayed PLT. All off the above proofs can
be done using $\Phi_{B_{l},\alpha}$ instead of $\Phi_{B_{l}}$, this shows
(with the notation from the proof of Theorem 5)

$(\mu\times\nu)\left(\bigcup_{k=1}^{K}A_{l}^{(k)}\times
B_{l}^{(k)}\right)\Phi_{\bigcup_{k=1}^{K}A_{l}^{(k)}\times
B_{l}^{(k)},\alpha}\overset{\mu}{\Rightarrow}\Phi_{Exp}.$

To conclude, we only need a version of the approximation Lemma 21 for delayed
return times.

Let $(M,d_{M})$ be a compact metric space, let $(\tilde{\vartheta}_{n})_{n\geq
1}$ be a sequence of Lipschitz functions on $M$ dense in $C(M)$, and denote
$\vartheta_{n}=\frac{\tilde{\vartheta}_{n}}{||\tilde{\vartheta}_{n}||_{Lip}}$.
The metric

$D_{M}(\lambda,\lambda^{\prime})=\sum_{n\geq
1}2^{-n}\left|\int_{[0,\infty]}\vartheta_{n}\,\text{d}\lambda-\int_{M}\vartheta_{n}\lambda\right|,$

for probability measures $\lambda$ and $\lambda^{\prime}$, models
distributional convergence181818In the sense that
$\lambda_{n}\Rightarrow\lambda$ iff $D_{M}(\lambda,\lambda_{n})\rightarrow
0$..

######  Lemma 33.

Let $(X,\mu,T)$ be a probability-preserving dynamical system, $\alpha$ be a
sequence of natural numbers, $(A_{l})_{l\geq 1}$ be a sequence of rare events,
and $\Phi=(\phi^{(1)},\phi^{(2)},...)$ be a random process in $[0,\infty)$.
Assume that, for each $\delta>0$, there is a sequence of rare events
$(A^{(\delta)}_{l})_{l\geq 1}$ with $A_{l}^{(\delta)}\subset A_{l}$ and
$\mu_{A_{l}}(A_{l}\setminus A_{l}^{(\delta)})<\delta$ such that

$\mu(A_{l}^{(\delta)})\Phi_{A_{l}^{(\delta)},\alpha}\overset{\mu}{\Rightarrow}\Phi\;\;\;;\textit{as
}l\rightarrow\infty,\forall\delta>0.$

Then

$\mu(A_{l})\Phi_{A_{l},\alpha}\overset{\mu}{\Rightarrow}\Phi\;\;\;\textit{as
}l\rightarrow\infty.$

###### Proof.

Taking a subsequence if necessary, we may assume that there is a
$[0,\infty]$-valued random process $\Phi^{\prime}$ with

$\mu(A_{l})\Phi_{A_{l},\alpha}\overset{\mu}{\Rightarrow}\Phi^{\prime}\;\;\;\textit{as
}l\rightarrow\infty.$

For $s,t\in[0,\infty]$ denote $d_{[0,\infty]}(s,t)=|e^{-s}-e^{-t}|$, where by
convention $e^{-\infty}=0$, this makes $([0,\infty],d_{[0,\infty]}$ a compact
metric space. Then $([0,\infty]^{\mathbb{N}},d_{[0,\infty]^{\mathbb{N}}})$ is
a compact metric space with $diam([0,\infty]^{\mathbb{N}})=1$, where

$d_{[0,\infty]^{\mathbb{N}}}((s_{j}),(t_{j}))=\sum_{j\geq
1}2^{-j}d_{[0,\infty]}(s_{j},t_{j}).$

We claim that for every $\epsilon>0$ there exist $\delta_{0}>0$ and an $L\geq
1$ such that

$D_{[0,\infty]^{\mathbb{N}}}\left(law_{\mu}(\mu(A_{l}^{(\delta)})\Phi_{A_{l}^{(\delta)},\alpha}),law_{\mu}(\mu(A_{l})\Phi_{A_{l},\alpha})\right)<5\epsilon\;\;\;\forall
l\geq L.$ (30)

Then taking $l\rightarrow\infty$ shows
$D_{[0,\infty]^{\mathbb{N}}}(\Phi,\Phi^{\prime})<\epsilon$ and the conclusion
follows by $\epsilon\rightarrow 0$.

Let $1>\epsilon>0$. First note that191919For $k\geq 1$ and $s,t\in[0,\infty]$
we have $d_{[0,\infty]}(ks,kt)\geq d_{[0,\infty]}(s,t)\leq|s-t|$.

$D_{[0,\infty]^{\mathbb{N}}}\left(law_{\mu}(\mu(A_{l}^{(\delta)})\Phi_{A_{l}^{(\delta)},\alpha}),law_{\mu}(\mu(A_{l})\Phi_{A_{l}^{(\delta)},\alpha})\right)<\delta,$

so it is enough to show that there exist $\epsilon>\delta_{0}>0$ and an $L\geq
1$ such that

$D_{[0,\infty]^{\mathbb{N}}}\left(law_{\mu}(\mu(A_{l})\Phi_{A_{l}^{(\delta)},\alpha}),law_{\mu}(\mu(A_{l})\Phi_{A_{l},\alpha})\right)<4\epsilon\;\;\;\forall
l\geq L.$ (31)

Denote
$\Phi_{A_{l},\alpha}=(\varphi_{A_{l},\alpha}^{(1)},\varphi_{A_{l},\alpha}^{(2)},...)$
and
$\Phi_{A_{l}^{(\delta)},\alpha}=(\varphi_{A_{l}^{(\delta)},\alpha}^{(1)},\varphi_{A_{l}^{(\delta)},\alpha}^{(2)},...)$.
Now choose $J\geq 1$ so big that $\sum_{j\geq J}2^{-j}<\epsilon$, and $T>0$
such that

$\mathbb{P}\left(\sum_{j=1}^{J}\phi^{(j)}>T\right)<\epsilon.$

For $\delta=\min\left(\frac{\epsilon}{2},\frac{\epsilon}{2T}\right)$ choose
$L\geq 1$ so big that

$\mu\left(\sum_{j=1}^{J}\mu(A_{l}^{(\delta)})\varphi_{A_{l}^{(\delta)},\alpha}^{(j)}>T\right)<2\epsilon\;\;\;\forall
l\geq L.$

Since
$\displaystyle\sum_{j=1}^{J}\varphi_{A_{l}^{(\delta)},\alpha}^{(j)}>\sum_{j=1}^{J}\varphi_{A_{l},\alpha}^{(j)}$
and $\mu(A_{l}^{(\delta)})>(1-\delta)\mu(A_{l})>\frac{1}{2}\mu(A_{l})$, in
particular

$\mu\left(\sum_{j=1}^{J}\mu(A_{l})\varphi_{A_{l},\alpha}^{(j)}>2T\right)<2\epsilon\;\;\;\forall
l\geq L.$

Now, for $j=1,...,j$, we have

$\displaystyle\mu$
$\displaystyle\left(\varphi_{A_{l},\alpha}^{(j)}\not=\varphi_{A_{l}^{(\delta)},\alpha}^{(j))}\right)\leq\mu\left(\varphi_{A_{l},\alpha}^{(j)}\not=\varphi_{A_{l}^{(\delta)},\alpha}^{(J))}\right)$
$\displaystyle\leq\mu\left(\bigcup_{i=1}^{\left\lfloor\frac{2T}{\mu(A_{l})}\right\rfloor}T^{-\tilde{\alpha}^{(i)}}(A_{l}\setminus
A_{l}^{(\delta)})\right)+\mu\left(\sum_{j=1}^{J}\mu(A_{l})\varphi_{A_{l},\alpha}^{(j)}>2T\right)$
$\displaystyle\leq\frac{2T}{\mu(A_{l})}\mu(A_{l})\delta+2\epsilon\leq
3\epsilon.$

Thus

$\displaystyle
D_{[0,\infty]^{\mathbb{N}}}\left(law_{\mu}(\mu(A_{l})\Phi_{A_{l}^{(\delta)},\alpha}),law_{\mu}(\mu(A_{l})\Phi_{A_{l},\alpha})\right)$
$\displaystyle=\sum_{j\geq
1}2^{-j}\int_{X}d_{[0,\infty]}(\mu(A_{l})\varphi_{A_{l},\alpha}^{(j)},\mu(A_{l})\varphi_{A_{l}^{(\delta)},\alpha}^{(j))})\,\text{d}\mu\leq\sum_{j=1}^{J}2^{-j}3\epsilon+\epsilon\leq
4\epsilon$

proving (31). ∎

### References

  * [AS11] M. Abadi and B. Saussol.  Hitting and returning to rare events for all alpha-mixing processes.  Stochastic Process. Appl., 121:314–323, 2011.
  * [BEG20] M. Björklund, M. Einsiedler, and A. Gorodnik.  Quantitative multiple mixing.  J. Eur. Math. Soc. (JEMS), 22(5):1475–1529, 2020.
  * [Bil95] P. Billingsley.  Probability and measure.  John Wiley $\&$ Sons, 1995.
  * [BRV16] V. Beresnevich, F. Ramírez, and S. Velani.  Metric Diophantine approximation: aspects of recent work.  In Dynamics and analytic number theory, volume 437 of London Math. Soc. Lecture Note Ser., pages 1–95. Cambridge Univ. Press, Cambridge, 2016.
  * [BS01] L. Barreira and B. Saussol.  Hausdorff dimension of measures via Poincaré recurrence.  Comm. Math. Phys., 219(2):443–463, 2001.
  * [BSTV03] H. Bruin, B. Saussol, S. Troubetzkoy, and S. Vaienti.  Return time statistics via inducing.  Ergodic Theory Dyn. Syst., 23:991–1013, 2003.
  * [CG93] P. Collet and A. Galves.  Statistics of close visits to the indifferent fixed point of an interval map.  Journal of Statistical Physics, 1993.
  * [DDKN22] D. Dolgopyat, C. Dong, A. Kanigowski, and P. Nándori.  Flexibility of statistical properties for smooth systems satisfying the central limit theorem.  Invent. Math., 230(1):31–120, 2022.
  * [DFL22a] D. Dolgopyat, B.. Fayad, and S. Liu.  Multiple Borel-Cantelli lemma in dynamics and multilog law for recurrence.  J. Mod. Dyn., 18:209–289, 2022.
  * [DFL22b] D. Dolgopyat, B. Fayad, and S. Liu.  Multiple Borel-Cantelli lemma in dynamics and multilog law for recurrence.  J. Mod. Dyn., 18:209–289, 2022.
  * [DGS04] M. Denker, M. Gordin, and A. Sharova.  A poisson limit theorem for toral automorphisms.  Illinois journal of mathematics, 48:1–20, 2004.
  * [DKRH21] D. Dolgopyat, A. Kanigowski, and F. Rodriguez-Hertz.  Exponential mixing implies bernoulli, 2021.
  * [Dol04] D. Dolgopyat.  Limit theorems for partially hyperbolic systems.  Transactions of AMS, 356:1637–1689, 2004.
  * [FF03] L. Flaminio and G. Forni.  Invariant distributions and time averages for horocycle flows.  Duke Math. J., 119 (3):465–526, 2003.
  * [FFT11] A. C. M. Freitas, J. M. Freitas, and M. Todd.  Extreme value laws in dynamical systems for non-smooth observations.  J. Stat. Phys., 142(1):108–126, 2011.
  * [FFT16] L. Flaminio, G. Forni, and J. Tanis.  Effective equidistribution of twisted horocycle flows and horocycle maps.  Geom. Funct. Anal., 26(5):1359–1448, 2016.
  * [FFV17] A. C. M. Freitas, J. M. Freitas, and S. Vaienti.  Extreme value laws for non stationary processes generated by sequential and random dynamical systems.  Ann. Inst. Henri Poincaré Probab. Stat., 53(3):1341–1370, 2017\.
  * [HLSW22] M. Hussain, B. Li, D. Simmons, and B. Wang.  Dynamical Borel-Cantelli lemma for recurrence theory.  Ergodic Theory Dynam. Systems, 42(6):1994–2008, 2022.
  * [HNPV13] N. Haydn, M. Nicol, T. Persson, and S. Vaienti.  A note on Borel-Cantelli lemmas for non-uniformly hyperbolic dynamical systems.  Ergodic Theory Dynam. Systems, 33(2):475–498, 2013.
  * [HSV99] M. Hirata, B. Saussol, and S. Vaienti.  Statistics of return times: a general framework and new applications.  Comm. Math. Phys., 206(1):33–55, 1999.
  * [HV09] N. Haydn and S. Vaienti.  The compound Poisson distribution and return times in dynamical systems.  Probab. Theory Related Fields, 144(3-4):517–542, 2009.
  * [HWZ14] N. Haydn, N. Winterberg, and R. Zweimüller.  Return-time statistics, hitting-time statistics and inducing.  In Ergodic theory, open dynamics, and coherent structures, volume 70 of Springer Proc. Math. Stat., pages 217–227. Springer, New York, 2014.
  * [LFF+16] V. Lucarini, D. Faranda, A. C. M. Freitas, J. M. Freitas, M. Holland, T. Kuna, M. Nicol, M. Todd, and S. Vaienti.  Extremes and recurrence in dynamical systems.  Pure and Applied Mathematics (Hoboken). John Wiley & Sons, Inc., Hoboken, NJ, 2016.
  * [LM13] S. Luzzatto and I. Melbourne.  Statistical properties and decay of correlations for interval maps with critical points and singularities.  Comm. Math. Phys., 320(1):21–35, 2013.
  * [Pit91] B. Pitskel.  Poisson limit law for markov chains.  Ergodic Theory and Dynamical Systems, 11(3):501–513, 1991.
  * [PS20] F. Pène and B. Saussol.  Spatio-temporal Poisson processes for visits to small sets.  Israel J. Math., 240(2):625–665, 2020.
  * [PY17] M. J. Pacifico and F. Yang.  Hitting times distribution and extreme value laws for semi-flows.  Discrete Contin. Dyn. Syst., 37(11):5861–5881, 2017.
  * [You92] L.S. Young.  Decay of correlations for certain quadratic maps.  Commun. Math. Phys., 146:123–138, 1992.
  * [Zha21] X. Zhang.  A Poisson limit theorem for Gibbs-Markov maps.  Dyn. Syst., 36(1):88–103, 2021.
  * [Zwe] R. Zweimüller.  Hitting-times and positions in rare events.  arXiv:1810.10381v3[math.DS].  preprint.
  * [Zwe16] R. Zweimüller.  The general asymptotic return-time process.  Isr. J. Math., 212:1–36, 2016.
  * [Zwe19] R. Zweimüller.  Hitting-time limits for some exceptional rare events of ergodic maps.  Stochastic Process. Appl., 129(5):1556–1567, 2019.