Birkhoff sums of infinite observables and anomalous time-scaling of extreme events in infinite systems

# Birkhoff sums of infinite observables and anomalous time-scaling of extreme events in infinite systems

## Abstract.

We establish quantitative results for the statistical behaviour of infinite systems. We consider two kinds of infinite system:

• a conservative dynamical system preserving a -finite measure such that ;

• the case where is a probability measure but we consider the statistical behaviour of an observable which is non-integrable: .

In the first part of this work we study the behaviour of Birkhoff sums of systems of the kind ii). For certain weakly chaotic systems, we show that these sums can be strongly oscillating. However, if the system has superpolynomial decay of correlations or has a Markov structure, then we show this oscillation cannot happen. In this case we prove asymptotic relations between the behaviour of , the local dimension of , and on the growth of Birkhoff sums (as time tends to infinity). We then establish several important consequences which apply to infinite systems of the kind i). This includes showing anomalous scalings in extreme event limit laws, or entrance time statistics. We apply our findings to non-uniformly hyperbolic systems preserving an infinite measure, establishing anomalous scalings in the case of logarithm laws of entrance times, dynamical Borel–Cantelli lemmas, almost sure growth rates of extremes, and dynamical run length functions.

###### Key words and phrases:
Conservative dynamics; infinite invariant measure; Birkhoff sum; Logatithm law; hitting time; extreme values; Borel–Cantelli; intermittent system; Run lenght
###### 2010 Mathematics Subject Classification:
37A40; 37A50; 37A25; 60G70; 37D25.
Acknowledgements. S. Galatolo thanks GNAMPA-INdAM for partial support during this research. M. Holland acknowledges the support by the EPSRC: EP/P034489/1, and the support from the mathematics departments at the University of Pisa, and the University of Houston. T. Persson thanks institut Mittag-Leffler and the program “Fractal Geometry and Dynamics”, during which parts of this paper was written. Y. Zhang acknowledges the support by the NSFC: 11701200 and 11871262, and the support from the mathematics departments at the University of Warwick, the University of Exeter, the University of Pisa and the Centre Physique Théorique, Marseille.

## 1. Introduction

### 1.1. Infinite Observables

We consider a dynamical system preserving a probability measure , together with an observable function . Let us consider the case where the observable is non-integrable, i.e. , and the Birkhoff sum

 Sn(x):=n−1∑k=0ϕ(fk(x)).

The pointwise ergodic theorem implies that grows to infinity faster than any linear increasing speed, for almost each . For these systems, Aaronson [2, Theorem 2.3.2] has shown that for any sequence , if then either

 limsupn→∞Sn(x)b(n)=∞a.e.orliminfn→∞Sn(x)b(n)=0a% .e. (1)

Thus, for these kind of systems a kind of pointwise ergodic theorem cannot hold for the asymptotic behaviour of the ratio . It is then natural to investigate the speed of growth of such Birkhoff sums quantitatively, and study their asymptotic behaviour from a coarser point of view. We approach this problem in the first part of the paper. In the second part of the paper we consider applications of these studies to understand several quantitative ergodic features of systems preserving an infinite measure. We set up a general framework and give examples of application to a family of intermittent, non uniformly hyperbolic maps, finding a kind of anomalous time-scaling for serveral quantitative statistical properties of the dynamics related to extreme events and hitting times. The understanding of the asymptotic behaviour of Birkhoff sums of infinite observables also has other important applications. We mention as an example the works [46, 47] where this is used to estimate the speed of mixing on area preserving flows on surfaces.

To obtain estimates from above on the behaviour of , the following general result is useful.

###### Proposition 1.1 (Aaronson [1, Proposition 2.3.1]).

If is increasing, and

 ∫a(ϕ(x))dμ(x)<∞,

then for -a.e.

 limn→∞a(Sn)n=0.
###### Remark 1.2.

Consider the case where for some and , and denote by the local dimension of at . From Proposition 1.1, if we let for some we get . This implies that for each and -almost all , we have eventually (as )

 Sn(x)≤nkdμ(~x)+ε. (2)

As it will be shown in Proposition 2.9 and in Section 8, there are systems for which the asymptotic behaviour of is strongly oscillating, or far from the estimate given in (2).

Thus, establishing convergence (or finding the typical growth rate) of is in general non-trivial, and suitable assumptions are needed on the system to get a definite asymptotic behaviour for . Lower bound estimates on the growth rates of have been given in [11, 21] under assumptions related to hitting time statistics and recurrence. These assumptions include having a logarithm law for the hitting time or a dynamical Borel–Cantelli property for certain shrinking target sets. We now review these connections in greater detail.

#### Known relations between Birkhoff sums of infinite observables, hitting times and Borel–Cantelli properties

Our first main result, Theorem 2.5 establishes almost sure bounds on the growth rate of under mild assumptions on the system and the non integrable observable function . We include the case where the system has super-polynomial decay of correlations and allow the obserbable to be quite general. We only impose a regularity assumption on the level sets . In particular our results allow for the fact these sets might not be homeomorphic to balls (in a given Riemannian metric), e.g.  might be a tube or another regular set.1

Let us now briefly discuss the hitting time scaling behaviour indicators considered in [21] and their relation with . Let be a ball with centre and radius . We define the first hitting (or entrance) time of the orbit of to by

 τr(x,~x):=min{n∈N:n>0, fn(x)∈B(~x,r)}.

Then define the hitting time indicators as

 ¯¯¯¯¯H(x,~x):=limsupr→0log(τr(x,~x))−log(r),H––(x,~x):=liminfr→0log(τr(x,~x))−log(r).

To help in understanding the sense of these definitions, we remark that according to the definitions, scales like . If observables of the form are considered then relations between and the behaviour of Birkhoff sums of infinite observables are proved in [21]. Among these, it is shown that for each , eventually (),

 Sn(x)≥nk¯¯¯¯H(x,~x)−ε

holds -a.e. We recall that and have been estimated in many systems (see e.g. [22, 24, 26, 27, 28, 29, 37] and references therein) and are related to the local dimension of the invariant measure in strongly chaotic systems, while in weakly chaotic or non chaotic ones they also have relations with the arithmetical properties of the system. In particular it is proved that in fastly mixing systems

 ¯¯¯¯¯H(x,~x)=dμ(~x)

holds for -a.e.  (see Proposition 3.2 for a precise statement) hence implying for -almost every , the lower estimate

 Sn(x)≥nkdμ(~x)−ε

holds for the observable , and for large (compare with 2).

In the recent paper [11], it is supposed that the system has absolutely continuous invariant measure, on a space of dimension and to satisfy a strong Borel–Cantelli assumption.2 Under this assumption, it is shown that for each and -almost every , for the observables of the kind with , we have eventually ()

 nkD−ε≤Sn(x)≤nkD(logn)kD+ε.

Other similar results are given in the case the system is exponentially mixing and the invariant measure has density in , or in particular cases of intermittent maps.

#### Growth of Birkhoff sums and extremes.

Given a measure preserving system consider the maximum process

 Mn(x):=max0≤k≤n−1ϕ(fk(x)), (3)

where is an observable function. In the case where on all of , it is clear that . Hence can provide a lower bound for . In [15] it is proved that if a process is generated by i.i.d. random variables, and then (-almost surely). Conversely, in the case of infinite observables the behaviour of gives good lower bounds in many interesting systems, approaching the general upper bound given in Proposition 1.1. This is indeed the strategy used to get lower bounds to in [11, 21] and in the present paper to get Theorem 2.5.

In the classical probabilistic literature the statistical properties of such are of interest to those working in extreme value theory, [15, 20]. For dynamical systems preserving a probability measure, the distributional properties of are known in some cases [39]. For certain dynamical systems, almost sure growth rates of have also been investigated [32, 34]. In this article, we give precise quantification on the almost sure behaviour of for a general class of infinite observables. The process is indeed strongly related to the hitting time . In the case , for some monotone decreasing function , then the event

 {Mn(x)≤u} (4)

corresponds to the event with . Hence all three processes are interlinked. This allows us to transfer (almost sure) statistical information from any one of these processes, to the other two. The relation between and is explained in a very general setting and in more detail in Section 2.3. This construction allows us to establish new results on the almost sure growth of for a more general class observables relative to those considered in e.g. [11, 21, 34]. In particular these results have relevance to the case where is a physical observable, see [35, 39].

Similarly, for the hitting time function , we also establish new results on the speed of convergence to the limit for the logarithm law, building upon e.g. [24], see Sections 2.3 and 2.5. For convergence in distribution, a link between the limit laws for maxima and hitting times was established in e.g. [18, 28]. Here, we complement these results via an almost sure version.

#### Overview of the main results on Birkhoff sums of infinite observables in the present paper

In this article, we study the behaviour of by using two main tools: using information on hitting time behaviour, as explained in the previous sections, following the approach of [21, 23, 28] and methods based on almost-sure recurrence (as also adopted in [11, 32, 33, 34]) via the theory of dynamical Borel–Cantelli lemmas. The main results about maxima and Birkhoff sums we prove are stated in Theorem 2.5 and Proposition 2.8 and require as a main assumption, the fast decay of correlation of the system or the presence of a Markov structure. Our approach complements those used in [11, 21], extending these kind of results to a general class of observables, not related to the distance from a given point and to systems having an invariant measure which is not absolutely continuous (including the case of measures having a non integer local dimension).

### 1.2. Systems preserving an infinite measure

Based on the findings on the behaviour of Birkhoff sums and maxima of an infinite observable, we are able to address a number of relevant topics relating to systems preserving an infinite measure. We formulate a general framework, and show application to the the celebrated family of “intermittent” maps studied by Manneville–Pomeau in [40], and by Liverani–Saussol–Vaienti map in [38]. We focus on the case where is conservative, ergodic, and is -finite, with . The main idea here is to analyse a map induced over a finite part of the infinite system. The dynamical behaviour of the finite induced system is then easier to study and the findings can be applied to the original system, which can be seen as a suspension of the induced one (the construction is outlined at the beginning of Section 2.2). The suspension in our case will have an associated infinite observable which plays the role as the “return time function.” The results motivated in the previous sections give important information in this construction, such as understanding the Birkhoff sums of this observable.3 We find that the behaviour of the infinite observable gives a kind of “time rescaling” factor which is important in the behaviour of several quantitative ergodic aspects of the dynamics of such infinite systems. In particular we find this “anomalous scaling” in the following aspects:

#### The behaviour of the hitting time to small targets, and logarithm laws.

Here we are interested in the time needed for a typical trajectory of the system to hit a small target which could be seen as an extreme event. Let be a sequence of targets of measure going to zero and consider the hitting time to the -th target

 τ(f,x,An):=inf{n≥0:fn(x)∈An}.

It is proved (see [21, 22, 23, 25, 26, 27] and references therein) that in a wide variety of systems preserving a probability measure, a logarithm law holds

 limn→∞log(τ(f,x,An))−log(μ(An))=1, (5)

provided the target sets are regular enough, and the system is sufficiently chaotic (a precise statement of this kind is shown in Proposition 3.1). For infinite systems having a fast mixing first return map on a finite subspace, we show that the ratio in (5) converges to a number that depends on the return time associated to the return map. Hence we obtain an anomalous behaviour in a wide class of infinite systems (see Proposition 2.10 for a precise statement).

#### Almost sure scaling laws for the statistics of extremes in systems preserving an infinite measure.

For infinite systems which are conservative, ergodic and is -finite, we consider the behaviour of maxima of a given observable , see (3). As discussed in the Section 1.1.2 (see (4)), this is naturally related to the hitting times. In Proposition 2.13 we show a precise, quantitative link between maxima and hitting time behaviour. As a consequence we obtain an estimate for the behaviour of in infinite systems (see Corollary 2.15). As obtained for the hitting time problems and logarithm laws, we show that the scaling behaviour of depends on the return time statistics associated to the infinite measure system (in a way we make precise in Section 2), as well as on the local regularity of the observable function . This is unlike the behaviour of in the probability measure preserving case. This will be done in Section 2.5. We apply our theory to a family of intermittent maps in Section 2.6.

#### Dynamical Borel–Cantelli laws for infinite measure preserving systems.

Consider a measure preserving dynamical system , and let be a sequence of observables. Furthermore let , with . Now suppose that , but . A dynamical Borel–Cantelli problem is the problem to show existence (or otherwise) of a sequence with , -almost surely. In the case , we are just in a strong law of large numbers type of situation. Hence, we aim to generalise this concept in a non-stationary setting, i.e. where the observable changes with time. We address this problem for the system , where is a -finite (infinite) invariant measure. In the probability preserving case, this problem has been widely studied, and forms the basis of dynamical Borel–Cantelli Lemma results, see [30, 32, 33, 36]. For such systems it is shown that is the typical scaling law, and this is consistent with the corresponding theory for i.i.d. random variables, see [15, 20]. For infinite systems, we show that this scaling sequence is not the appropriate one to use, and we derive the corresponding scaling law. Such a result is new, and we apply our methods to obtain shrinking target (Borel–Cantelli) results for the intermittent map family described in [38] for the -finite (infinite) invariant measure case. As a further application we consider infinite systems modelled by Young towers, see Section 9.

#### Dynamical run-length problems for infinite measure preserving systems.

Suppose further that the measure preserving dynamical system admits a countable or finite partition on (with an index set), and each is coded with the sequence , by if and only if . The dynamical run length function of digit is defined by

 ξ(j)n(x):=max{0≤k≤n:∃0≤i≤n−k,εi+1(x)=⋯=εi+k(x)=j}. (6)

In the setting of successive experiments of coin tossing, corresponds to the longest length of consecutive terms of “heads/tails” up to -times experiments [16, 43]. Thus, the studies of dynamical run length functions is concerned with quantifying the asymptotic growth behaviour of for -typical . Such studies admit various applications in DNA sequencing [3], finance and non-parametric statistics [5, 6, 7, 44], reliability theory [44], Diophantine approximation theory to -expansions of real numbers [10, 17, 45], and Erdős–Rényi strong law of large numbers [13, 14, 16, 31].

We analyse the dynamical run length function in the case where is conservative, ergodic, and is -finite. In particular, we explicitly estimate the growth rate for for a family of intermittent maps in the -finite measure case in Section 7. In contrast to probability measure preserving systems (e.g. uniformly hyperbolic Gibbs–Markov systems; logistic-like maps satisfying the Collet–Eckmann condition; and families of intermittent maps preserving a.c.i.p. [12, 13, 14, 31, 45]), we show that apart from the local dimension, there is an additional scaling contribution, arising from the asymptotics of the return time function associated to the induced transformation, which needs to be taken into account in the growth rate for of infinite systems. As the reader will realize, our proof is based on a natural link between the dynamical run length function, hitting time, and growth of maximum for the return time functions. We are not aware of any such links, previously established in the literature of this subject.

### 1.3. Outline of the paper.

We structure the paper as follows. In Section 2 we state the main theoretical results. This includes results on the growth of Birkhoff sums for rapidly mixing systems, on the link between hitting time laws and growth of extremes, and dynamical Borel–Cantelli Lemma results for systems preserving a -finite infinite measure. In Sections 3 and 5 we prove these results, and then consider several independent topics which relate to our theory. This includes a result on the almost sure growth rates of extremes and hitting times for infinite systems, see Section 2.5. We then apply our theory to an intermittent map case study in Section 2.6, which includes a study of dynamical run-length problems in Section 2.6.3. We then describe situations in which the Birkhoff sums can wildly oscillate in Section 8. Finally we consider Borel–Cantelli results for general Markov extensions, such as Young towers (Section 9).

## 2. Statement of main results

### 2.1. Birkhoff sums, maxima, and hitting time statistics

Consider a dynamical system preserving a probability measure , together with an observable function with . Let us recall the notation used for Birkhoff sums and respectively maxima of an observable .

 Sn(x):=n−1∑k=0ϕ(fk(x)),Mn(x):=max0≤k≤n−1ϕ(fk(x)).

In specific contexts, we sometimes emphasize the dependence on , and write for (and similarly for maxima).

As noted before (see ) it is impossible to get precise estimates for the asymptotic behaviour of as increases. However, under suitable assumptions on ergodicity and on the chaotic properties of the system, coarser estimates on asymptotic growth rates are possible.

We show that we can achieve estimates for the scaling behaviour of both and for systems which are super-polynomially mixing, and for quite a large class of observables having some regularity. The regularity we need is a kind “Lipschitz” regularity of the suplevels of the observable . This is explained in the next definition. Essentially we ask that the suplevels of are regular enough that they could be sublevels of a Lipschitz function.

###### Definition 2.1.

Let be an unbounded function. Consider the suplevels of , defined by

 An={x∈X:ϕ(x)≥n}.

We say that has regular suplevels if the following holds:

1. We have , there is a constant satisfying eventually as increases, and there is , such that

 αϕ=limn→∞logμ(An)−log(n).
2. There is and a Lipschitz function such that

 An={x∈X:~ϕ(x)≤(μ(An))β}.

This assumption is verified by a large class of observables, including observables related to the distance from a point.

###### Example 2.2.

Suppose be a Riemannian manifold with boundary, and the Riemannian distance. For consider an observable of the form with in a neighbourhood of , and suppose exists and . Such conditions are verified for almost each in a wide class of uniformly and non-uniformly hyperbolic systems, see [42]. Then we have that is a ball of radius , and hence a regular set. In this case

 αϕ:=limn→∞logμ({x∈X:ϕ(x)≥n})−log(n)=dμ(~x)−k,

and for each , eventually Consider such that , and defined as

 ~ϕ(x)=μ(Ai)β+μ(Ai−1)β−μ(Ai)βri−1−ri(d(~x,x)−ri),

when . Since is bounded for our choice of , the function is Lipschitz.

Other examples include cases where the suplevels correspond to tubes or other sets, see [22, 25] for results about hitting times on targets which are suplevels of a Lipschitz function, applied to the geodesic flow, in which the targets relate to “cylinders” in the tangent bundle instead of balls. Thus the regular sublevels assumption of Definition 2.1 holds for a wide class of dynamical systems and observable geometries. We now consider the notion of decay of correlations.

###### Definition 2.3.

Let be functions in a Banach space . A measure preserving system is said to have decay of correlations in with rate function , if for each such , we have

 ∣∣∣∫φ∘fnψdμ−∫φdμ∫ψdμ∣∣∣≤∥φ∥∥ψ∥Θ(n).

Here stands for the norm on .

Usually decay of correlations is proved for a particular space , and a specified rate .

###### Definition 2.4 (Condition (SPDC)).

We say that satisfies condition (SPDC) (Super-Polynomial Decay of Correlations) if for all , we have and is the space of Lipschitz continuous functions.

This condition is quite general, and many systems having some form of piecewise hyperbolic behaviour satisfy it. See [4] for a survey containing a list of classes of examples having exponential or stretched exponential decay of correlations. When this kind of decay holds for Lipschitz observables these examples satisfy (SPDC). We remark that if a system has a certain decay of correlations with respect to Hölder observables, then it will have the same or faster speed when smoother observables (such as Lipschitz ones) are considered.

Now, suppose the non-integrable observable , has regular suplevels as in Definition 2.1. The following theorem concerns the growth of maxima and Birkhoff sums of .

###### Theorem 2.5.

Let be a probability measure preserving system on a metric space , satisfying condition (SPDC). Let , and be as in Definition 2.1, with . If , then for each and -a.e. , there exists such that for all ,

 n1α+ε≤Mn(x)

If , then for -a.e.

 limn→∞log(Sn(x))logn=limn→∞log(Mn(x))logn=∞.

This theorem therefore applies to a wide class of observable geometries. In the case where is related to the distance from a point, or for particular dynamical systems, see [21, 28] or [34, Theorem 2.5], or [11, Section 2.3]. Paarticular systems that are captured by the theory include Hénon maps [8], and certain Poincaré return maps for Lorenz attractors [27], to name a few. The proof of Theorem 2.5 can be found in Section 3.

###### Notation 1.

A statement of the form as means that there is a constant such that

 c−1≤v(n)u(n)≤c

holds for all large enough .

###### Remark 2.6.

In the above setting, if we make the stronger assumption we can get the more precise upper estimate that holds eventually for every and -a.e. .

#### Gibbs–Markov systems.

Theorem 2.5 shows how, with some strong assumptions on the system, we can get information on the scaling behaviour of . We will see (Proposition 2.8) that if we assume some even stronger assumptions on the system, as the presence of a Gibbs–Markov structure, we can get even more precise estimates.

Consider again a transformation and an observable with . We say that is a Gibbs–Markov system [2] if we have the following set up.

1. is an interval and there is a countable Markov partition such that contains a union of elements of , and there exists such that . Let .

2. There exists , such that for all we have for .

3. Uniform bounded distortion estimates hold on . That is, there exist , , such that for all , and ,

 ∣∣ ∣∣log(|f′(x)||f′(y)|)∣∣ ∣∣≤Cτn.
4. The measure is the unique invariant probability measure which is absolutely continuous with respect to Lebesgue measure.

For a Gibbs–Markov system satisfying 2.1.12.1.1, we will use the following assumptions on the observable .

1. For any the restriction of to is constant.

2. The observable satisfies the following asymptotics: there exists such that

 μ{x∈X:ϕ(x)=n}∼n−β−1.
###### Remark 2.7.

Note that assumption 2.1.1 implies that the observable is non-integrable, .

If is a Gibbs–Markov maps satisfying assumptions 2.1.12.1.1, we are able to obtain a result on the asymptotic speed of the typical growth of Birkhoff sums of , which we will now state. Our result below is similar to a result by Carney and Nicol [11, Theorem 4.1], and the proofs are also similar. Carney and Nicol assumed that the system satisfies a strong Borel–Cantelli lemma, but we do not assume this explicitly.

###### Proposition 2.8.

Let satisfy assumptions 2.1.12.1.1 , with as in assumption 2.1.1. Then each , and -almost all there is an such that for all :

 n1β(logn)−1β−ε≤Mn(x)

#### Oscillating Birkhoff sums

Proposition 1.1 gives us a general upper bound on the increase of . It does not depend on quantitative properties of the dynamical system but appears to be near to an optimal estimate in many strongly chaotic systems, see [11] for a discussion. However, there are chaotic systems for which the bound we obtain from (2) is far from the actual behaviour of , and there are examples in which their Birkhoff sum is strongly oscillating. These examples take the form of a skew product map of the form

 f(x,t)=(T(x),t+θφ(x)), (7)

where is a uniformly expanding interval map, is the circle, an irrational number, and a specified “skewing” function. We state the following Theorem, whose proof, and precise form of is described in Section 8.

###### Theorem 2.9.

There are measure preserving systems of the skew product form (7) which preserve a probability measure and have polynomial decay of correlations. Moreover for an infinite observable of the kind , for some , we have

 liminfn→∞logSn(x)logn

for -a.e. . Furthermore there are measure preserving systems with polynomial decay of correlations where even the limsup, and power law behaviour of the Birhkhoff sums does not follow the ratio suggested by (2), even along subsequences. In such systems for -a.e.

 limsupn→∞logSn(x)logn

### 2.2. Application to extreme events and hitting times in systems having a fast mixing return map.

The estimates on Birkhoff sums of infinite observables are useful to investigate quantitative aspects of the dynamics of systems preserving an infinite measure. Consider an infinite system , where is assumed to be infinite but -finite. A classical approach to study such an infinite system is by inducing the dynamics on a subset of positive (finite) measure. Let be the return time function to the domain , that is for :

 R(x)=min{n>0:fn(x)∈Y}.

Then defines a dynamical system which preserves the measure . (We shall now denote by ). The system is called the induced system. It is a finite measure preserving system, and its dynamics gives information on the original infinite system. The original system can be seen as a suspension on the induced system, that is if we define by

 ^X={(x,n)∈Y×N:0≤n

and

 ^f(x,n)={(x,n+1)if n+1

then is a suspension of and is isomorphic to if is defined in the natural way, e.g. see [50].

Here a major role is played by the return time function . In this case will be a non-integrable observable on , and to this situation we can apply the findings of the previous section. We remark that the observable is not necessarily related to the distance from a certain point. In particular if we are interested in hitting time or extreme problems then the asymptotic behaviour of the Birkhoff sums of the return time in the induced system is particularly important. We show in the following Sections 2.2.1 and 2.3 that the behaviour of implies anomalous scaling behaviour for the hitting time to small targets, and for growth of extreme events.

#### Logarithm law and the anomalous hitting time behaviour in infinite systems.

Here, we derive a limit (logarithm) law for the hitting time function. First, recall some definitions relating to the hitting time to general targets and logarithm laws in this context. Consider a dynamical system on a metric space Let be a decreasing sequence of targets; let us consider the hitting time of the orbit starting from to the target

 τ(f,x,Bn)=min{n≥0 s.t. fn(x)∈Bn}

(in case the map considered is obvious from the context instead of we may write for simplicity).

The classical logarithm law results relates the hitting time scaling behaviour to the measure of the targets. In many cases when preserves a probability measure and the system is chaotic enough, or it has generic arithmetic properties then the following holds for -a.e.

 limn→∞logτ(f,x,Bn)logn=limn→∞log(μ(Bn))−logn. (8)

In words: the hitting time scales as the inverse of the measure of the targets (compare with (5)).

We see that in systems preserving an infinite measure this law does not hold anymore, but under some chaoticity assumptions we can replace the equality (8), with a rescaled version of it. In fact, the rescaling factor depends on the return time behaviour of the system on some subset containing the target sets

Suppose preserves an infinite measure , is such that , and consider . The following holds.

###### Proposition 2.10.

Let be a dynamical system preserving an infinite measure . Let be the induced system over a domain of finite positive measure, preserving a probability measure , and with return time function . Suppose has regular suplevels with associated exponent (see Definition 2.1). Suppose that satisfies Condition (SPDC). Let be a decreasing sequence of targets also satisfying items (i) and (ii) of Definition 2.1. Consider , such that Then for -a.e. ,

 limn→∞logτ(f,x,Bn)logn=ααR.

The proof of Proposition 2.10 is in Section 4.1. In the next section achieve a corresponding statement for the maxima process .

### 2.3. On the link between almost sure growth of maxima and hitting time laws.

Suppose , and consider a sequence of functions . For consider the following maximum function sequence and corresponding hitting time function sequence defined by

 ~Mn(x):=max0≤k

Examples include the case where we have a probability space , with the -algebra of subsets of , a probability measure, and a sequence of random variables. Another case includes that of a measure preserving system , where we set , with specified observable function . In this latter case, coincides with the usual definition of given in (3).

In this section, we derive a precise link between the growth rate of (as ), and the growth rate of (as ). We’ll assume further that either as , or as .

First we make the basic observation that the event is the same as . We state our first elementary result.

###### Proposition 2.11.

Suppose , and consider the sequence of functions .

1. Suppose that are monotone increasing functions, such that , as . Suppose for given , there exists , such that for all we have . Then there exists , such that for all we have

 ℓ−12(u−1)≤τu(x)≤ℓ−11(u+1).
2. Suppose that are monotone increasing functions, such that , as . Suppose that for given , there exists , such that for all , we have Then there exists , such that

 ^ℓ−12(n)≤~Mn(x)≤^ℓ−11(n).
###### Remark 2.12.

In the statement of Proposition 2.11, we do not assume that is a measure space. In the case where is a stationary process, defined on a suitable measure space, then Proposition 2.11 asserts that almost sure bounds for imply almost sure bounds for , and vice versa. We will use this fact in our dynamical systems applications.

Proposition 2.11 is proved in Section 4.2. We now consider specific applications of this result. For a measure preserving dynamical system define

 τϕu(x)=inf{n:ϕ(fn(x))≥u},

and put , where we recall . Here is an observable function. Examples include or , for a given , but we have seen that our theory allows us to consider much more general cases.

#### The logarithm law for hitting times and maxima

We now consider the logarithm law, especially for the hitting time function. We show via Proposition 2.11 that a logarithm law for hitting time implies a logarithm law for maxima (and conversely). Again this is a pointwise result. See [28, Proposition 11] for a similar statement.

###### Proposition 2.13.

Consider a dynamical system . Suppose that , and . Then we have the following implications.

 limsupn→∞log[Mn(x)]logn =a1⟺liminfu→∞log[τϕu(x)]logu=1a1, liminfn→∞log[Mn(x)]logn =a2⟺limsupu→∞log[τϕu(x)]logu=1a2.

Moreover, if , then

 limu→∞log[τϕu(x)]logu=(limn→∞log[Mn(x)]logn)−1

provided the corresponding limits exist at .

###### Remark 2.14.

In Proposition 2.13, the logarithm function diminishes any behaviour associated to sub-polynomial corrections associated to the growth of (as ), or to that of (as ). In certain cases, this subpolynomial growth can be further quantified as we discuss below.

Proposition 2.13 is proved in Section 4.2. For infinite systems, we state the following corollary concerning the almost sure behaviour of the maxima process.

###### Corollary 2.15.

Let be a dynamical system preserving an infinite measure . Let and be as in Proposition 2.10. Consider , a function which is not bounded and that is bounded. Suppose that also has regular suplevels. Then

 limn→∞log[Mn(x)]logn=αRαϕ

holds for -a.e.

The proof of Corollary 2.15 can be found in Section 4.2. In Corollary 2.15, we have assumed (SPDC) for . In the case where satisfies stronger hypotheses, such as being Gibbs–Markov, then we can obtain stronger bounds on almost sure behaviour of the maxima function (as ), and also the hitting time function via Proposition 2.11. We remark further that Corollary 2.15 gives almost sure bounds on the maxima process in the case of observables having general geometries (beyond functions of distance to a distinguished point). Thus if we know (almost sure) bounds on the hitting time function, then we get corresponding bounds for the maxima process via Proposition 2.11 (or 2.13). This result allows us to address a question posed in e.g. [34, Section 6] concerning the existence of an almost sure behaviour of maxima for general observables (that are not solely a function of distance to a distinguished point).

#### On finding precise asymptotics on the maxima and hitting time functions

For certain stationary processes (or dynamical systems), the rate functions as appearing in Proposition 2.11 can be optimised. For i.i.d. processes , optimal expressions for these functions are given in e.g. [15, 20]. For dynamical systems having exponential decay of correlations, higher order corrections to the almost sure maxima function growth (beyond that given by a standard logarithm law in Proposition 2.13) are discussed in e.g. [32, 34]. To translate such results to almost sure behaviour of hitting times, then inversion of the functions is required. This we now discuss via an explicit example. Generalisations just depend on an analysis of the functional forms of and .

Consider the tent map , , and the observable function . It is shown that there exist explicit constants such that for Lebesgue-a.e.

 logn−c1loglogn≤Mn(x)≤logn+c2loglogn,

eventually in for Lebesgue-a.e. , see [34]. We deduce the following asymptotic for the hitting time function. A proof is given in Section 4.2.

###### Lemma 2.16.

Consider the tent map , and observable . Then for all , and -a.e. , there exists such that for all

 logτϕu(x)=u+O(logu).

Here the constant depends on , and on .

###### Remark 2.17.

We immediately deduce a logarithm law for entrance to balls with an error rate. In particular if we let , then . Then by Lemma 2.16, we obtain for -a.e.  that

 logτϕu(r)(x)=−logr+O(loglogr).

In the example above, we used a higher order asymptotic on the growth rate for the maxima to deduce a similar asymptotic for the hitting time function. We note that a converse result applies if we have knowledge of such asymptotics for the almost sure growth of the hitting time function, but no apriori bounds for the growth of the maxima function. For either the hitting time function, or maxima function almost sure growth rates are usually deduced via Borel–Cantelli arguments, see [20, 32, 34] for maxima, and [24] for hitting times. We elaborate in Section 2.5. Thus, Proposition 2.11 allows us to translate limit laws between maxima and hitting times without too much extra work, except for estimating inverses of the corresponding rate functions. We remark that in the case of distributional limits for maxima and hitting times (as opposed to almost sure bounds), a relation between their limit laws is described in [18, 19].

### 2.4. Dynamical Borel–Cantelli Lemmas for infinite measure preserving systems

For a (probability) measure preserving dynamical system , a dynamical Borel–Cantelli Lemma result asserts that for a sequence of sets with , we have

 μ(∞⋂i=1∞⋃n=i{x:fn(x)∈Bn})=1,

i.e. . A quantitative version leads to having the strong Borel–Cantelli (SBC) property defined as follows. Given a sequence of sets with , let .

###### Definition 2.18.

We say that satisfies the strong Borel–Cantelli property (SBC) if for -a.e.

 limn→∞Sn(x)En=1,

where , and denotes the indicator function on the set .

For dynamical systems preserving a probability measure , (SBC) results are now known to hold for various systems, see [32, 33, 34, 36]. Here, we derive corresponding Borel–Cantelli results for infinite systems , with a -finite measure, and .

We consider a conservative, ergodic system , and suppose there exists for which the induced system is Gibbs–Markov (see Sections 2.1, 2.2 for conventions), but now the return time function is not integrable with respect to . In the case of integrable return times, [36, Theorem 3.1] established strong Borel–Cantelli results for the system assuming strong Borel–Cantelli results for the induced system . Formally, consider a function sequence with , where . We say that the strong Borel–Cantelli property holds for this sequence, with respect to , if (necessarily) as and

 ∑nj=1pQ(j,x)(fjY(x))∑