Poisson Limit Law for Dynamical Systems

Transcription

1 Poisson Limit Law for Dynamical Systems Marcello Seri, Filiz Tümel December 21, 2011 Abstract We give a summary of the results related to the study of asymptotic laws for rare events, in particular for the the Poisson Limit Law, in the framework of discrete time ergodic dynamical systems preserving a finite measure. 1 Introduction The Poisson Limit Law for arrays of independent Bernoulli random variables is a well known result in probability theory. Roughly speaking, it tells us that under certain conditions we can approximate the binomial distribution with a Poisson distribution. If we translate the same approach to the language of dynamical systems, it gives the distributional law of asymptotically rare events. Those are the events that in general have 0 probability to happen. However, by scaling the waiting time, i.e. the time that it takes to see the occurance of these events, in a best suited way it is possible in many cases to describe their asymptotic behaviour. It is quite remarkable that the construction of the theory necessary to study the problem has shown many deep and unexpected connections between the asymptotic statistics of the systems and their dynamical and geometrical properties. Recently, exhaustive analysis of these statistics was motivated not only by the teoretical interests around the problem, but also by its relevance in Department Mathematik, Universität Erlangen-Nürnberg, Cauerstr. 11, D Erlangen, Germany. Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato, 5, Bologna, Italy. Department of Mathematics, University of Houston, 4800 Calhoun Rd, Houston, Texas

2 applications in different areas such as entropy estimation, genome analysis, epidemology, nanotechnology, computer science, linguistics, among others. In view of current developments in these areas, the very classical problem is the transition from discrete to continuous dynamical description of particle systems. Emergence of long time scales from an ensemble of particles evolving on relatively fast time scales ends up on local relaxations of rates and mean first visiting times. Near continuum asymptotics can help to modify the models so that they can capture more global qualitative features. From the point of view of applications, capturing incorrectly the quantitative features of the discrete dynamics and the understanding of the rate of convergence of the limit law has become a fundamental task. The aim of this project is to give a summary of the results in this subject within the finite-measure preserving discrete dynamical systems framework. In the Preliminaries, we start with the Bernoulli random variables example and present its specific set-up. Then in the next two sections we introduce the general picture and give some motivation. Section 4 summarises the basic notions and results of thermodynamic formalism, so reader who is familiar with this subject can skip it. In Section 5 we give a panoramic of the known results in a choronological order, we point out the methods used for proofs and we discuss the general question by going over some examples. Lastly, in Section 6 we give some new directions for further studies. 2 Preliminaries We start with an easy example where the asymptotically rare sequence of events are Bernoulli trials and we show that visiting to the rare events converge to a Poisson distribution. Let X i denote the sequence of i.i.d. random variables with P{X i = 1} := p and P{X i = 0} := 1 p (2.1) for 0 < p < 1 and i N. Let S n := X 0 + X X n 1 for n 1. The process {S n } n=0 is called the Bernoulli random walk where the X i s are Bernoulli trials with state space A = {0, 1}, and its distribution is given by P(S n = k) = ( n k ) p k (1 p) n k. We can translate this random walk to a deterministic dynamical system by introducing the so called shift space. Let Ω := A N denote the sequences of possible trajectories of the random walk, Ω = {x = (x 0, x 1, x 2,...) : x i A}, 2

3 and consider the product topology on Ω induced by the discrete topology on A. The measure µ on Ω is defined uniquely by the measure on the cylinder sets given by where µ(c t [a 0, a 1,..., a m 1 ]) := p k (1 p) m k, k := m 1 i=0 a i, C t [a 0, a 1,..., a m 1 ] := {x Ω : x t = a 0, x t+1 = a 1,..., x t+m 1 = a m 1 }. Let σ : Ω be the shift map given by σ(x 0, x 1, x 2,...) = (x 1, x 2, x 3,...) and f : Ω {0, 1} be the projection map to the first coordinate given by f(x 0, x 1, x 2,...) = x 0 Now, we can represent the random variables on the probability space (Ω, µ) by X i (x 0, x 1, x 2,...) = f(σ i (x 0, x 1, x 2,...)), and the Bernoulli walk by where the distribution is n 1 S n (x 0, x 1, x 2,...) = f(σ i (x 0, x 1, x 2,...)) i=0 µ{(x 0, x 1, x 2,...) Ω : S n (x) = k} = ( ) n p k (1 p) n k. k Now if we assume that the probability of the event {X i = 1} depends on n, say P{X i = 1} = p n, in such a way that lim np n = λ, we obtain the classical n Poisson limit theorem: lim µ n{x Ω : S n (x) = k} = λk n k! e λ (2.2) where µ n is the measure on Ω induced by the Bernoulli distribution P{X i = 1} = p n and P{X i = 0} = 1 p n on A. However, the probability space (Ω, µ n ) is changing and this is not a suitable set up for the measure preserving dynamical systems. To fix the probability space, S n needs to be rescaled. Here is the right way to scale S n and phrase the classical Poisson limit theorem in Ergodic Theory [Pi91]: 3

4 Theorem 2.1 Let λ > 0. For almost each trajectory x Ω and n 1 λ/µ(cn(x)) lim µ n x Ω : χ Cn(x)(σ i (x)) = k = λk i=0 k! e λ where C n (x) = {y Ω : y 0 = x 0, y 1 = x 1,..., y n 1 = x n 1 } and χ Cn(x) is its characteristic function. The other very natural interest is the times of occurrence of some events. Let us continue with our Bernoulli random walk above and record the number of steps needed to come back to state 1s. For a random trajectory x = (x 0, x 1, x 2,...) we record the n > 0 values where x n = 1. We can consider the first time return to the event as a random variable, say N, and ask the probability of N being exactly k. This is obviously the event where x i = 0 for i = t + 1, t + 2,... t + k 1 and x t+k = 1 for some t N with x t = 1, so P(N = k) = (1 p) k 1 p, for k 1. This notion is called the return time, the first time at which a given process returns to a given subset of the state space, which is 1 in our example. Return times are examples of hitting times. Here we assume that at time N = t the process is at state 1 and we ask the probability of coming back to state 1 again in k steps. The next obvious question about the random variable N is the expected value: E(N) = k(1 p) k 1 p = p k=0 k(1 p) k 1 k=1 d = p (1 p) k d(1 p) ( k=0 ) d 1 = p d(1 p) 1 (1 p) p = (1 (1 p)) = 1 2 p. Let us rewrite everything by using our shift space notation. Let A denote the cylinder {x Ω : x 0 = 1}. For the trajectory x Ω with x i = 0 for i = 0, 1, 2,... k 1 and x k = 1, we can write σ k (x) A and that is the first time of this event happening. Again we can introduce the first entrance time 4

5 to A, say τ A. According to the invariant shift measure we introduced above we have, µ({x Ω : τ A (x) = k}) = µ({x : x i = 0 for i = 0, 1, 2,... k 1; x k = 1}) = (1 p) k 1 p And the expected value is again τ A dµ = k=1 kµ({x Ω : τ A(x) = k}) A = k=1 k(1 p)k 1 p = 1 p = 1 µ(a). Note that the shift map on the sequences of Bernoulli trials is ergodic with respect to the measure µ. 3 General setting We want now to define and rephrase what we have already seen in the examples of the previous section. Let (Ω, B, µ) be a probability space and T : Ω Ω a measure preserving map (that is µ(t 1 (A)) = µ(a) for every A B). Our dynamical system is the 4-tuple (Ω, B, µ, T ) (sometimes shortened with the triplet (Ω, T, µ)). For such systems the Poincaré Recurrence Theorem ensures that µ-almost every point in A B with positive measure will return to A after a certain number of iterations: µ {x A : n > 0 s.t. T n (x) A} = µ(a), where T n (x) := T (T n 1 (x)) and T 0 (x) := x. Thus it makes sense to talk about the time needed for a generic point x to come back to its neighborhood. Fix a set A B with positive measure, we call the first return time of x in A the function τ A : A N defined by τ A (x) := τ (1) A (x) := inf {n > 0 : T n (x) A}. Notice that τ A (x) < + for µ-a.e. x A. And since the Poincaré Recurrence guarantees that the point in A will almost surely come back to A we can define for j 2 the j-th return time of x in A, τ (j) A : A N, by { } τ (j) A (x) := inf n > τ (j 1) A (x) : T n (x) A. 5

6 Just as a side remark notice that to have the stronger result µ {x Ω : n > 0 s.t. T n (x) A} = 1, one needs to have an ergodic dynamical system, i.e. such that the only A Ω with µ(t 1 (A) A) = 0 are those with µ(a) = 0 or µ(a c ) = 0. At first sight the return times may seem simple deterministic functions, on the other hand we can imagine them as random variable and ask ourselves what is their expectation and what is their statistical behavior. Regarding the first question the following theorem can be proven. We will understand soon why it is particularly important. Theorem 3.1 (Kač s Lemma) Let A B be of positive measure. Call µ A the conditional probability measure defined by Then we have µ A (B) := E (τ A ) := A µ(a B), B B. µ(a) τ A dµ = µ({τ A < + }). In particular, when the system is ergodic we have E A (τ A ) := τ A dµ A = 1 µ(a). The first return map of x to the set A, A T A : A A, T A (x) := T τ A(x) (x), is defined almost everywhere on A and preserves by definition the restriction of µ on the set A and thus also the conditional probability measure µ A. In fact, the following theorem can be proven. Theorem 3.2 The dynamical system (A, T A, µ A ) is a well defined measure preserving dynamical system. It is ergodic if the original system is ergodic. Suppose now we have a sequence of positive measure subsets A m B, such that lim µ(a m) = 0. In general the time to wait for the occurrence m of A m is divergent with m, thus it is reasonable to call the events {A m } asymptotically rare, as in [Co97] and [HSV99]. Anyway, as can be understood by the the structure of Kač s Lemma, it s not always the case that ( ) lim E A m τ (j) m A m µ(a m ) = 1. 6

7 The main interest for this work is to understand the asymptotic distribution of the return times on this asymptotically rare family of events, i.e. the non-negative functions m (t) := µ x A m : ρ (j) τ (j) A m (x) ( ) > t. E Am τ (j) A m Looking back once more to the Kač s Lemma, considering the fact that the measure of A m is in general much easier to compute than the expected value of the return times, we will use that as a scale c m for this expected times. Thus the first problem becomes finding a sequence c m 0 such that the limit ({ ρ (j) (t) = lim m ρ(j) m (t) = lim µ x A m m exists for all t, j N +. Notice then that we have necessarily + 0 ( ) t dρ (j) (t) = lim c me Am τ (j) m A m, }) : c m τ (j) A m (x) > t, (3.3) and thus to have finite expectation of the limit distribution, we should choose the normalizing sequence so that the latter limit exists. Notice that this is in some sense equivalent to the assumption np n λ that one makes to obtain the Classical Poisson Limit Theorem for the Bernoulli process (see equation 2.2). Finally, in the case of ergodic systems we can define the visiting times, i.e. return times where the initial point x is taken in Ω instead of a fixed subset A, and the discussion follows more or less in the same way (for more details on this last observation see [Co97]). A natural question, now, may arise: how can we relate the present discussion with the summation that appears in Theorem 2.1? To clarify this point let us consider the ergodic situation, in this case we know that ( ) E Am τ (j) A m = 1 µ(a m ), and thus we have E Am ( τ (j) A m ) µ(a m ) = 1 and τ (j) A m (x) ( ) = τ (j) E Am τ (j) A m (x)µ(a m ). A m The following lemma (it may seem obvious for somebody) will give us the key for understanding the connection. 7

8 Lemma 3.3 Let x A m and let χ m be the characteristic function of A m. We have τ (k) A m (x)µ(a m ) > t [t/µ(a m)] i=1 χ m ( T i (x) ) k 1. In other words, the time requested to go back k times in the set A m is greater than t normalized with the measure of the set itself if and only if in such a time you have been in the set at least k 1 times. This is equivalent to say that if and only if [t/µ(a m)] i=1 χ m ( T i (x) ) = k τ (k+1) A m (x)µ(a m ) > t and τ (k) A m (x)µ(a m ) t. Therefore is essentially equivalent to analyze one setting or the other, and we are free to choose each time the easier situation. 3.1 Some first questions As we will briefly discuss later there are many examples of sequences of asymptotically rare events in certain dynamical systems, where the limit distribution ρ (1) (t) (or more generally ρ (j) (t)) is known to exist and it is computable. As we will see in most of the cases there are some common underlying structure and properties that motivates the presence of a particular limit distribution. In particular we would like to know under what conditions we will obtain in the limit the Poisson Law. Without entering yet too much in details, there are already some questions that deserves to be posed and some of them are still without a full answer. First of all, How does the limiting law depend on the choice of the sets A m and on the corresponding c m? Is it possible, playing with them, to obtain other kinds of limiting distribution? The answer to this questions is quite recent and can be found in a couple of papers by Lacroix et al. ([La02], [KL05]) or [DM04], where the authors prove that in case of an aperiodic ergodic dynamical system and for arbitrarily prescribed A n any possible limiting distributions can appear for almost any point. We should not consider this paper as a hint that the study of this subject is empty in meaning, instead it should be an indicator that we need 8

9 to direct our attention to the right class of sets. In fact the result of Lacroix needs to use a really peculiar set instead of the usual cylinder sets, the squares or the balls. We can regard the latter sets as in some sense dynamically regular sets. Moreover, the rare event statistic for balls and cylinders seems strictly related to other geometric and statistical properties of the system; for example in many cases it is strictly connected with the Lyapunov exponent and the local dimension (see [STV02] and references therein). Thus the following questions may be of greater interest Suppose that we fix a shape for the sequence A m, what is the underlying mechanism that will give rise to a limit law (say Poisson Limit Law) instead of chooing another shape? Are the rare event statistics on this shape connected to other dynamical and geometrical properties of the system? 4 Thermodynamic Formalism The contents of this section are required for later chapters to state the results. We describe here some of the basic notions and results of the thermodynamic formalism. We follow the class notes of a course given in Penn State - Gottingen Summer School at Gottingen University, August First, we will describe an important class of random transformations called subshifts of finite type, a key symbolic model for a lot of systems. Let A = {1, 2,..., N} be the symbol set for fixed N 2, let Σ := {(x i ) i Z : x i A} denote the set of two-sided sequences, and Σ + := {(x i ) i Z + : x i A} denote the one-sided sequences. Note that by using the natural projection Π : Σ Σ + any function on Σ +, say f + : Σ + C, can lift to a function on Σ, f : Σ C, by f(x) = f + (Π(x)). All other notions that we will describe here for one-sided sequences can be considered in a similar way for two-sided sequences, too, so from now on we will be working with onesided sequences only. Let σ : Σ + be the shift map to the left. We endow Σ with the product topology corresponding to the discrete topology on A. The sets C t [a 0, a 1,..., a n ] := {(x i ) i Z + Σ + : x t = a 0, x t+1 = a 1,..., x t+n = a n } are called the n-cylinders and they form a basis for the topology of Σ Expanding Systems Let (X, d) be a compact metric space, and T : X X be a continuous map for the following definitions. 9

10 Definition 4.1 The dynamical system (X, T ) is said to be expansive if there exists θ > 0 such that for any two points x, y X, for every n 1 d(t n (x), T n (y)) θ implies x = y. Thus, if we have x y, then for some n, d(t n (x), T n (y)) > θ, or we can say sup d(t n (x), T n (y)) > θ. The number θ > 0 in the definition is called n N the expansive constant for the map T. Example 4.2 Consider the shift space that is introduced in Section 2, Ω = {x = (x 0, x 1, x 2,...) : x i A} where A = {0, 1} with the metric d(x, y) = x i y i 2 i. The usual shift homeomorhism σ : Ω Ω is expansive since i=0 for x y there exists K N with x K y K, so d(σ K (x), σ K (y)) 1. Definition 4.3 The dynamical system (X, T ) is said to be expanding if there exists λ > 1 and δ > 0 such that for any two points x, y X d(t (x), T (y)) > λ d(x, y) whenever d(x, y) < δ for some metric d equivalent to the metric d on X. Proposition 4.4 Every expanding system is expansive. Proof. Let (X, T ) be an expanding dynamical system and λ > 1,δ > 0 as in Definition 4.3. Let x, y X with x y. If d(x, y) δ then we are done since sup d(t n (x), T n (y)) > δ 0 n N for some 0 < δ 0 < δ. If not, then images of x and y expands by λ. If d(t (x), T (y)) > δ, then again we re done. If not, d(t (x), T (y)) < δ implies d(t 2 (x), T 2 (y)) > λd(t (x), T (y)) > λ 2 d(x, y). Continuing this way, we can conclude that sup n N d(t n (x), T n (y)) > δ 0 since λ > 1, so for some N N we have λ N d(x, y) δ since d(x, y) > 0. Here δ 0 is an expansive constant. Theorem 4.5 A positively expansive map on (X, d) is an expanding map with respect to some metric. [CR80] Definition 4.6 The dynamical system (X, T ) is said to be R-expanding if it is expanding and if T is an open map. The first following lemma gives the characterization of R-expanding dynamical systems, then we are ready to give the lemma that provides the basis of thermodynamic formalism. 10

11 Lemma 4.7 The dynamical system (X, T ) is R-expanding if and only if there exists λ > 1 and δ > 0 such that for every x, y 0 X with d(t (x), y 0 ) < δ there exists unique y X with d(x, y) < δ and T (y) = y 0. Moreover, d(t (x), T (y)) > λd(x, y). Lemma 4.8 Let (X, T ) be an R-expanding dynamical system, then the preimages of T is uniformly bounded and locally constant, so the Perron-Frobenius operator of the continuous function ϕ C(X), L ϕ : C(X) C(X) f(x) is well defined, continuous and positive. 4.2 Pressure and Entropy y T 1 (x) f(y)e ϕ(y) Let (X, d) be a compact metric space, and T : X X a continuous map. For each n N, the function d n (x, y) = max d(t k (x), T k (y)) 0 k n 1 measures the maximum distance between the first n iterates of x and y. Each d n is a metric on X with d 1 = d, and d n d n 1. Moreover, the d n s all induce the same topology on X. Fix ɛ > 0. A subset A X is called (n, ɛ)-spanning if for every x X there exists y A such that d n (x, y) < ɛ and A is a minimal set with this property (Note that there exists finite (n, ɛ)-spanning sets by compactness). A subset A X is called (n, ɛ)-separated if for every x, y A we have d n (x, y) ɛ and A is a maximal set with this property. Note that the cardinality of an (n, ɛ)-spanning set is always less than or equal to the cardinality of an (n, ɛ)-separating set. Now, let {E n (ɛ)} be a family of (n, ɛ)-separating sets and {F n (ɛ)} be a family of (n, ɛ)-spanning sets for ɛ > 0, n 1. For a continuous function f C(X), let f n denote the n sum, f n (x) = f(x)+f(t (x))+...+f(t n 1 (x)). Then we can define the following limits: 1 P (T, f) = lim lim sup ɛ 0 n n log x E n(ɛ) e fn(x) 1 Q(T, f) = lim lim sup ɛ 0 n n log e fn(x) x F n(ɛ) 11

12 The limits exist sice by compactness the cardinality of {E n (ɛ)} and {F n (ɛ)} is always finite, and moreover they do not depend on the choices of spanning and separating sets. Definition 4.9 The limit P (T, f) is called the pressure of the continuous function f, and the map P (T,.) : C(X) R is called the pressure function. The pressure of the zero function f = 0 is called the topological entropy h top (T ) = P (T, 0) of the dynamical system (X, T ). Now, let s consider the dynamics (X, T ) equiped with an invariant probability measure µ, then we can also define the measure theoretic entropy. Let F be the Borel σ-field on X, and α be a measurable partition. The entropy of α and the measure µ is given by H µ (α) = A α µ(a) log(µ(a)). Let αt n denote the refinement α T 1 (α)... T (n 1) (α) and define the average of the entropy by 1 h µ (α, T ) = lim n n H µ(αt n ) Then the measure theoretic entropy of the map T is defined to be h µ (T ) = sup{h µ (α, T ) : H µ (α) < }. α The relation between the measure theoretic entropy and topological entropy is given by the Variational Principle: Theorem 4.10 For every continuous function f(x) C(X) { } P (X, f) = sup h µ (T ) + fdµ : µ M(X) where M(X) denotes the T -invariant measures. A T -invariant probability measure µ is called an equilibrium state if P (X, f) = h µ (T ) + fdµ. Note that for every continuous f, if (X, T ) is an expansive dynamical system there exists at least one equilibrum state [Bo75]. An equilibrium state is said to fulfil a supremum gap condition in the Perron-Frobenius operator if P (X, f) > sup f. 12

13 5 Known results In the beginning of the 1990s the study of the return times has been inaugurated by Pitskel [Pi91], Hirata [Hi93] and Collet [Col96] that were mainly focused on the case of uniformly hyperbolic dynamical systems. In the years that followed those works, at least five strategies have been developed that permits to understand what happens in many interesting cases. In this section we will briefly review the different approaches by entering a bit in the details of the ones that we think can bring some information on the underlying structure behind the Poisson processes and the convergence in law. From the very first works there have been actually many improvements; in particular now it s possible to study the behavior of a very large class of systems including certain parabolic and non-uniformly hyperbolic systems with a very deep and precise error control over the asymptotic convergence. Moreover, many links between the convergence and some other characteristic of the system (such as metric entropy, Lyapunov exponents) have been discovered. We will give some small details on this direction later. Chronologically speaking the first approach is due to Pitskel [Pi91] and dips its roots in an old probabilistic paper of Sevast yanov [Se72], where the following theorem is proved. Theorem 5.1 (Sevast yanov) Let {ηv n : v = 1,..., N(n)} for n 1 be an array of random 0, 1-valued variables and µ a probability measure. Put ζ n = N v=1 ηn v, and for v G r = { v Z r : 1 v 1 < v 2 < < v r N(n)} let b v n = µ(ηn v ), where ηn v = r s=1 ηn v s (in particular b n v = µ(ηv n )). Assume that there is a (monotonically to zero decreasing) sequence ɛ n so that the following five assumptions are satisfied. lim max n 1 v N bn v = 0, lim n N b n v = t > 0. v=1 Moreover, assume that there exist rare sets R r G r (r 1) such that lim b v n = 0, n v R r lim n v R r b n v 1 b n v r = 0, b n v lim 1 b n v r n b v n 13 = 1,

14 uniformly in v G r \ R r. Then lim µ(n r) = tr e t, n r! where N r = {y : ζ n (y) = r} is the r-level set of ζ n. Basically Pitskel applied this theorem to a mixing finite state stationary Markov chain (and used the cylinders structure to define the rare sets), and then extended the proof to hyperbolic automorphisms of the two-dimensional torus considering the fact that each of them is isomorphic to a Markov chain. This same scheme has been applied to Axiom-A diffeomorphisms and later by Haydn [Ha00] to the equilibrium states on Julia sets that present the supremum-gap condition in the Perron-Frobenius operator (see Lemma 4.8 and what follows). However the error control over the rate of convergence has been improved (and quantified) for rational maps and parabolic maps on the interval only later by Haydn and Vaienti [HV04]. A completely different approach, based on the properties of the Perron- Frobenius operator and on the theory of Ruelle zeta function, has been used by Hirata [Hi93] to study Axiom-A diffeomorphism f on smooth manifolds M. Assume that f restricted to its non-wandering set Ω is Axiom-A (i.e. Ω is hyperbolic and the fixed point of f are dense in Ω) and topologically mixing, and define a sequence of ɛ-ball B ɛ (x) centered in x with respect to the Riemannian structure on M. Hirata proved that for almost evety x Ω the sequence of return times in B ɛ (x) rescaled by the measure of the ball converges in law to a Poisson point process. The central point in the proof is that the eigenvalue 1 of the Perron- Frobenius operator must be, at least from a certain point in the limit, isolated from the rest of the spectrum. Thanks to this fact he can prove the convergence in law to Poisson point process for symbolic dynamics and the subsequent proof follows approximating the dynamics using a Markov partition. The paper states a general result that is valid for the return times of every order and the proof is certainly correct for the first return time. However, we must say that the argument seems incomplete for the higher times. Moreover, there is still no estimate for the control of the error over the rate of convergence by using the same approach. An interesting point about this paper is that the author also gives a class of typical counter-examples studying the limit law for periodic points z Ω. A similar analysis for counter-examples has been performed by Hirata and Vaienti [HV09]. The authors improve the result by showing how to construct processes on periodic points to obtain the convergence to any chosen asymptotic law. 14

15 Around the same time, Collet and Galves used the perturbation argument of the transfer operator and proved that the entrance time for expanding maps of the interval is asymptotically exponentially distributed, which leaded to the conclusion that under special hypothesis the sequence of visits to a small interval converges to a Poisson point process [CG95]. Another work on expanding circle maps during this period was the study of the approaching of two trajectories: in [CC94], Coelho and Collet proved that for f being a piecewise expanding circle map and x, y S 1 two arbitrary points, the point process recording the times n > 0 where f n (x) f n (y) < ɛ, scaled by ɛ 1 converges to a marked Poisson point process when ɛ 0. Some years after the pioneering works of Pitskel and Hirata, a new strategy has been proposed by Galves and Schmitt [GS97]. They developed a completely probabilistic method applicable to every ϕ-mixing system. In particular their work has been translated in the dynamical system language by Haydn [Ha99] for Julia sets, by Paccaut [Pa00] for a large class of non- Markovian maps of the interval, by Boubakri for a class of unimodal maps [Bo96] and by Abadi [Ab01] for α-mixing stationary processes. The interesting aspect of this last method is that it permitted to have a first control over the convergence rate and to discover a deep connection between the convergence in distribution of the return times and the Ornsten- Weiss formula [OW93], that connects the order of growth of the return times of patterns to the entropy of the system. In particular this led to other deep results, for example the repetition times for Gibbs measures have log-normal fluctuations (see [Pa00] and the references in [BSTV] for more information). The rate of convergence has been studed and improved by Hirata, Saussol and Vaienti [HSV99]. They consider a generic measure preserving dynamical system on a probability space and their main interest is to give general estimates for error between the distribution of the kth return times in a positive measure set and the Poisson law for measurable dynamical systems. The result is really powerful if one knows explicitly the rate of mixing of a dynamical system, in fact in that case the error can be explicitly computed. As an application they prove the almost sure Poisson limit law for a class of non-uniformly hyperbolic maps of the interval characterized by a parameter that measures the order of tangency at the neutral fxed point and controls the algebraic decay of correlations. In particular they prove sufficient and necessary conditions for the convergence to the Poisson Limit Law for the first return time and the first hitting time that does not depend on the ergodicity but only on the mixing rate. If we define c(k, A) := µ A ({x τ A (x) > k}) µ({x τ A (x) > k}) and c(a) := sup k c(k, A), then their condition can be reassumed in c(a n ) 0 for n. The proof works for any stationary measure µ and without require- 15

16 ments on the ergodicity of the system but, of course, with this generality one cannot expect to assure the convergence for c(a n ). Nice upper bounds in terms of the mixing rates are given in [HSV99] and sharpened in [Ab01]. A step forward in the study of non-uniformly hyperbolic maps on the interval has been made by Collet [Col01]. Using Young towers and extreme value theory the author proved the Poisson limit law for return times for non-uniformly hyperbolic exponentially mixing maps on the interval. These results have been improved and extended for a larger class of non-uniformly hyperbolic dynamical systems modeled by Young towers by the same author and Chazottes [CC10]. For what concerns uniformly partially hyperbolic systems, the Poisson limit law for rare events has been proven by Dolgopyat [Do04]. With all these results at hand, the work [BSTV] assume a great importance. The authors proved that the return time statistics of a dynamical system do not change if one passes to an induced map (i.e. the first return map). In this work the authors could prove the Poisson limit law for return times in cylinders for C 2 interval maps that preserves a conformal measure and such that the closure of the orbit of the critical points has zero measure, improving [Col01] and including maps like the Fibonacci-map (see [LM93]) and the Poisson limit law for return times in balls (approximated by cylinders) for polynomial maps on the Riemann sphere Ĉ restricted to their Julia set and such that the orbit of the critical points does not densely fill the Julia set, notice anyhow that they do not require the supremum gap condition that is a crucial hypothesis for the other papers about rational maps. 5.1 A counter -example Let R α : S 1 S 1 the standard rotation on the circle by angle α (0, 1), R α (x) := x + α (mod 1). The system (S 1, R α, dx) is a measure-preserving ergodic dynamical system for every α R \ Q, but differently from the previous examples it is not mixing and his entropy is 0 (see [Si08] or any other book on ergodic theory for more details). Denote by g : [0, 1) [0, 1) the Gauss map { } 1 G(x) := = 1 (mod 1). x x 16

17 If the continued fraction expansion of α is α = [a 1, a 2,...] = a a [ it is a standard fact of number theory that a n =, a i N +. 1 G n (α) Fix now a point x S 1. The sequence of best approximation of x by its R α orbit is the sequence of times k such that R k α (x) x < R j α (x) x for all j = 1, 2,..., k 1. It can be proven that this sequence is given by the convergence q n, i.e. the denominator of the truncated continued fraction expansion of α in its irreducible form: p n q n = [a 1, a 2,..., a n ]. Moreover, the points x and x := x + 1 (mod 1) divide the circle into two 2 connected components such that for every n > 1 the points f q n 1 (x) and f qn (x) are always in two different components. If we denote I n the interval with endpoints f qn (x) and x contained on the same component of the circle and we define J n := I n I n 1, we can state a result of Coelho and de Faria [CdF96] Theorem 5.2 The visiting times to J n, rescaled by µ(j n ) 1, converge in law under a subsequence {n i } of n, if and only if, either G n i (α) converges to 0 or Γ n i (α, α) converges to a point (β, τ) for some β (0, 1) and τ [0, 1), where Γ(α, t) : T 2 T 2 is the double Gauss map defined by Γ(α, t) := (G(α), [a 1, b 1, b 2,...]) with t = [b 1, b 2,...]. Moreover, for each choice of (β, τ), the corresponding limit point process are pairwise different. A consequence of this theorem is the following. Corollary 5.3 The visiting times to J n, rescaled by µ(j n ) 1, do not converge in law for Lebesgue a.e. rotation number α [0, 1). Basically this means that even considering ɛ-neighborhoods of a point, there is no universal law for rotations (or any other dynamical system that is semiconjugated with a rotation [CdF96]). More than being chocking, this results can be seen as an incentive to go deeper in the investigation of a universal 17 ].

18 theory of rare events. Is it the missing of mixing property that avoids the convergence in distribution? Is it the zero entropy? Are there some other kind of ergodic or statistic reasons behind this lack? These are, in lieu of this corollary, even more legitimate questions. An answer can be given at least to one of these questions. As remarked in [Wa, p ], on the set of all measure preserving transformations on a given space, most of the transformations are weakly mixing and have zero entropy, and of course they contain the strongly mixing transformations with zero entropy (that is a non-empty subset, see the Peterson subshifts [Pe70]). As we have already written at the beginning of this section, the Poisson Limit Law on balls holds for strong-mixing dynamical systems by using the result in [HSV99], thus the property of having zero entropy is not a sufficient condition for the lack of the Poisson Limit Law. Furthermore, an explicit example of a zero entropy weakly mixing dynamical system that present the Poisson Limit Law for cylinders has been recently presented by Grzegoreg and Kupza [GK11]. 5.2 Another special example It can be interesting, at this point, entering more in the detail of what was going wrong in the previous example. For this purpose let s introduce as in [HR04] the so called skew map on the cylinder C := T [0, 1] R : C C, R(x, y) := (x + y (mod 1), y). (C, R, λ) is a measure-preserving dynamical system with zero entropy where λ is the Lebesgue measure over C (see [HR04] for more details on this and what follows). On the other hand this system is non ergodic but shows a sort of local mixing property, i.e. if A ɛ (x, y) := [x, x + ɛ] [y, y + ɛ], 0 < ɛ < 1, it can be proven that λɛ (A ɛ (x, 0) R n (A ɛ (x, 0))) λ 2 ɛ(a ɛ (x, 0)) ( ) 1 = O, n where λ ɛ is the conditional measure defined by λ ɛ (B) := λ(b)/ɛ. This sort of local mixing property given by the decay of the autocorrelation seems to be enough to change the statistical behavior with respect to the standard rotation on the circle. In particular in [HR04] it is shown that even if the system is not ergodic ( ) E(τ Aɛ ) = λ R n (A ɛ (x, y)) n= λ(a ɛ (x, y)) = 1 ɛ,

19 that is incredibly similar to the Kač formula for ergodic systems. For this box sets of the form A ɛ (x, 0), the authors were able to prove that the first recurrence statistics as defined in equation 3.3 converges to the distribution ρ(t) = 1 if t = 0 1/2 if 0 < t < 1 1/2t 2 if t 1 i.e. we have in this case the convergence to a power law. For the more general case of y 0, there are for the moment only numerical results that seems to confirm that ρ(t) t 2. In addition, some upper and lower bounds for the limit statistics in terms of a power law has been produced for this example and also for similar Hamiltonian systems with mixed phase space (See the references in the previously cited paper). Concluding, even for this particular system (although not so different from a rotation) it is possible to find a limiting statistics for somewhat dynamically regular sets, even if this law is not exponential. This may support the idea that the limit laws for rare events on regular sets are produced by some kind of mixing effect and that the exponential behavior is somewhat related to a stronger form of mixing, such as (f, φ)-mixing introduced in one of the previous examples. 6 Questions, conjectures and generalizations Many results have been published and in the jungle of dynamical systems there is a certain uniformity in the appearance of the Poisson limit law for return times for balls and cylinders. Nevertheless there are still some big questions that deserve an answer and that are probably far from being answered: What are the the minimal conditions that identifies dynamical systems sharing Poisson statistics? Suppose that we have a measure preserving dynamical system and suppose that we know that it satisfies the Poisson limit law for rare events and eventually we can estimate the speed of convergence. Can this give some informations on the ergodic properties of the system? Can we estimate the mixing rate and the decay of correlation directly from the knowledge of the speed of convergence? Is there a general behavior for the rare event limit law for balls, squares and cylinders around non-generic points? If yes, does the coupled one with the generic limit law give a finer characterization of the system? 19,

20 To give a full answer to these questions, a lot of work still has to be done and more advances are required. In what follows, we will discuss briefly possible directions for the research and we briefly describe other fields in which the analisys of limit laws for rare events can be generalized. 6.1 Random dynamical systems Before defining the return time for random dynamical systems and ask our question about the Poisson Limit Law, let us first recall some basic results of random dynamical systems. Let Ω be a metric space with B its Borel σ- algebra and (Ω, B, P) be a probability space. Let ι : Ω Ω be a P-preserving map that represents the time evolution. Ω indexes the maps {T ω } ω Ω which are defined on X, T ω : X X and X is a compact Riemannian manifold. Now, we are ready to define the random dynamical system T on X over (Ω, B, P, ι). For x X, and ω Ω we require that the mapping (ω, x) is measurable so that the evolution of the random system makes sense: Tω n+m (x) = Tι n m (ω) Tω m (x) For a given initial state x X and for a realization ω Ω, the corresponding random orbit {x ω n} is given by x ω n := T ι n 1(ω)... T ι(ω) T ω (x). The skew product realization of the random system can be given by S : Ω X Ω X (ω, x) (ι(ω), T ω (x)) where S is a deterministic dynamical systems so we can consider an invariant measure of the skew product map S to understand the random dynamics (which is given by the second component of the skew product). If we choose the maps independently according to a distribution P, then we can consider P = P N and the time evolution ι by a shift map σ on Ω = Λ N where the measure space Λ is the index space for maps. Then for λ Λ, say λ = (λ 0, λ 1,...), we have similarly x λ n := T λn 1... T λ1 T λ0 (x), and S : Λ X Λ X (λ, x) (σ(λ), T λ0 (x)). Let ν be an invariant measure of S, we are interested in the measures such that ν = P µ and we call µ the stationary measure of the random dynamical system (See [MR09] for reasons). The following proposition characterizes the stationary measures of random systems: 20

21 Proposition 6.1 Let {T λ } be an indexed family of sets on X and P be a probability distribution on Λ. If (λ n ) n 0 is an i.i.d. stochastic process, then an invariant measure µ is stationary if and only if µ(t 1 λ (B))dP(λ) = µ(b) for any Borel measurable subset B in X. Λ Since we have a deterministic map S on Λ X, we can define the first return time to the set A B Λ X in a usual way, so say for (λ, x) A B, τ A B (λ, x) := inf{n > 0 : S n (λ, x) A B}. If we choose A = Λ, then S n (λ, x) Λ B if and only if x λ n B. Now, we are ready to give some definitions for random return times: Definition 6.2 For fixed λ Λ, the first quenched random return time to the set B X is defined by where x B. τ λ B(x) := inf{n > 0 : x λ n B} The Poisson Limit Law seems mostly a result of an aperiodicity of the systems if we look at the known results, so here is what it means a random dynamical system to be aperiodic: Definition 6.3 A random dynamical system with maps {T λ } λ Λ on X, with distribution P on Ω = Λ N and a stationary measure ν is called random aperiodic if P ν({(λ, x) : n N such that T λn 1... T λ1 T λ0 (x) = x}) = 0 Conjecture 6.4 An aperiodic random dynamical system {T λ } λ Λ on X with stationary measure ν satisfies quenched Poisson Limit Law, meaning it is possible to find a scaling function R n defined on the phase space X and a sequence of dynamically meaningful shrinking measurable sets {C n } n>0 such that Rn(x) ν x X : χ Cn T λn 1... T λ2 T λ1 (x) = k α αk e k! i=1 for almost every realization λ Ω and for every k N when n, where lim n R n(x)ν(c n ) α > 0. 21

22 6.2 Other settings Even if some of the techniques that we have previously cited have been (or were ready to be) extended to maps on spaces of higher dimension (d > 1), for this class of systems not as many results as for one dimensional maps are known. A good starting point can be the discussion in [Sa09] or the paper by Denker and Kan [DeK07]. The study of limit laws for non-hyperbolic dynamical systems is a nontrivial open problem that is central for many physical applications. In particular the statistical characterization of chaotic orbits for Hamiltonian dynamical systems and symplectic maps plays a central role. These kind of systems usually show the coexistence of regions of chaotic and regular motion in the phase space and this structure affects strongly the behavior of the chaotic orbits that stays trapped near the regular (periodic and quasiperiodic) regions for long times, changing drastically the dynamic properties of the system (for example the decay of correlations) and thus the statistics for rare events, that in general are expected to dacay as power-law and not exponentially. For maps like the Farey maps or the Pomeau-Manneville maps, where the structure is created by a marginal point (localized), the analisys is rather doable and it can be proved that the return times in sets not containing the marginal point scales as power-low (see for example [BSTV]). On the other hand, for higher dimensional maps and hamiltonian systems there is almost no rigorous result (the main difficulty arise from the fact that there is no more a marginal point to characterize the phase space and it is extremely difficult to describe the structure of the stability islands in the chaotic sea). For the interested reader a good starting point for further investigations in this subject can be [M92]. Finally, another possible direction for research is given by the generalization of the problem to the systems preserving an infinite measure. Differently from the previous discussions, these systems would require a deep work of redefining of the concepts presented on this survey. For more information one can relate to some of the following works and the references therein [Aa81], [GKP] and [Zw08]. Aknowledgments Marcello Seri and Filiz Tümel express their grateful thanks to Prof. M. Denker, Prof. A. Knauf and Dr. G. Cristadoro for their useful discussions and support. 22

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