Compound Poisson law for hitting times to periodic orbits in two-dimensional hyperbolic systems
We show that a compound Poisson distribution holds for scaled exceedances of observables uniquely maximized at a periodic point in a variety of two-dimensional hyperbolic dynamical systems with singularities , including the billiard maps of Sinai dispersing billiards in both the finite and infinite horizon case. The observable we consider is of form where is a metric defined in terms of the stable and unstable foliation. The compound Poisson process we obtain is a Pólya-Aeppli distibution of index . We calculate in terms of the derivative of the map . Furthermore if we define and by the maximal process satisfies an extreme value law of form . These results generalize to a broader class of functions maximized at , though the formulas regarding the parameters in the distribution need to be modified.
AMS classification numbers: 37D50, 37A25
1 Motivation and relevant works
The study of the statistical properties of 2-dimensional hyperbolic systems with singularities was motivated in large part by a desire to understand mathematical billiards with chaotic behavior. This model was introduced by Sinai in  and has been studied by many authors [2, 3, 51, 52, 17].
The statistical properties of chaotic dynamical systems are often described by probabilistic limit theorems. Let be a dynamical system, i.e., a transformation preserving a probability measure on . For any real-valued function on (often called an observable), let , for any . We consider the stochastic process generated by the time-series . We assume has a unique maximal point . For any fixed high threshold , rare events can be defined as the event of the exceedance , for some . Laws of rare events for chaotic dynamical systems have been investigated, with results on both hitting time statistics (HTS) and return time statistics (RTS) [24, 26, 34, 33, 37, 23, 28, 29]. Techniques from extreme value theory (EVT) have been used to understand the statistics of rare events [24, 28, 29, 37, 39, 40, 42, 27]. Extreme value theory concerns the distributional and almost sure limits of the derived time series of maxima . Since we are assuming has a unique maximum at there is a close relation between extreme values of and visits to shrinking neighborhoods of .
Let be a sequence of constants defined by the requirement that . The research cited above has shown that for a variety of chaotic dynamical systems, with this scaling,
where . The parameter is called the extremal index and roughly measures the clustering of exceedances of the maxima. In fact is the average cluster size of exceedances given that one exceedance occurs.
For certain one-dimensional uniformly expanding maps and Anosov toral automorphisms a strict dichotomy has been observed. The dichotomy is that if is not periodic and otherwise [42, 27, 29]. In this paper, we investigate the extremal index when is a periodic point in the setting of two-dimensional hyperbolic systems with singularities. Poisson return time statistics for generic points in a variety of billiard systems (both polynomially and exponentially mixing) were established in . Related results on Poisson return time statistics were obtained for Young Towers with polynomial tails in . However the results of both works [31, 47] were limited to a full measure set of generic points, which explicitly excluded periodic orbits. This paper extends the results of  to the case of periodic points in a large class of billiard systems, using somewhat different techniques. For periodic points we show a compound Poisson distribution for exceedances holds and that the extremal index is determined by the Jacobian of the map along the periodic orbit (a precise formula is given in the statement of our main theorem). In  a similar result was shown for smooth toral automorphisms, and in fact in that setting a strict dichotomy was shown ( if is not periodic and otherwise). We don’t expect a dichotomy to hold in the class of hyperbolic systems with singularities, for example chaotic billiards. We expect the presence of singularities to enable a variety of extremal distributions, for example at observations maximized on singular sets. The analysis of the statistical properties of hyperbolic systems with singularities is more difficult than for smooth uniformly hyperbolic systems and the arguments of  do not generalize in a straightforward way. The main difficulty is caused by the singularities and the resulting fragmentation of phase space during the dynamics, which slows down the global expansion of unstable manifolds. We are able to use a growth lemma  to overcome these difficulties under certain other assumptions on the dynamics. Our results apply in particular to the billiard mapping of Sinai dispersing billiards with finite and infinite horizon.
1.1 Main assumptions
Let be a 2-dimensional compact Riemannian manifold, possibly with boundary. Let be an open subset and let be a diffeomorphism of onto (here ). We assume that is a finite or countable union of smooth compact curves . Similarly, is a finite or countable union of smooth compact curves. If has boundary , it must be a subset of both and . We call and singularity sets for the maps and , respectively. We denote by , , the connected components of ; then are the connected components of . We assume that is time-reversible, and the restriction of the map to any component can be extended by continuity to its boundary , though the extensions to for need not agree. Similarly, for each the restriction of to any connected component can be extended by continuity to its boundary .
Next we assume that the map is hyperbolic, as defined by Katok and Strelcyn . This means that preserves a probability measure such that -a.e. point has two non-zero Lyapunov exponents: one positive and one negative. Also, the first and second derivatives of the maps and do not grow too rapidly near their singularity sets and , respectively, and the -neighborhood of the singularity set has measure ; this is to ensure the existence and absolute continuity of stable and unstable manifolds at -a.e. point. Let . is (mod 0) the union of all unstable manifolds, and we assume that the partition of into unstable manifolds is measurable, so that induces conditional distributions on -almost all unstable manifolds (see the definition and basic properties of conditional measures in [17, Appendix A]). Most importantly, we assume that the conditional distributions of on unstable manifolds are absolutely continuous with respect to the Lebesgue measure on . This means that is the so called Sinai-Ruelle-Bowen (SRB) measure.
We also assume that our SRB measure is ergodic and mixing. In physics terms, is an equilibrium state for the potential .
For , let
for each . Then the map is a diffeomorphism.
We next make more specific assumptions on the system to give
sufficient conditions for exponential decay rates of correlations, as well as for the coupling lemma. These assumptions have been made in other works in the literature [12, 15, 17, 22].
Hyperbolic cones for
1of . There exist two families of cones (unstable) and (stable) in the tangent spaces , for all , and there exists a constant , with the following properties:
and , wherever exists.
These families of cones are continuous on and the angle between and is uniformly bounded away from zero.
We say that a smooth curve is an unstable (stable) curve if at every point the tangent line belongs in the unstable (stable) cone (). Furthermore, a curve is an unstable (stable) manifold if is an unstable (stable) curve for all (resp. ).
Singularities. The boundary is transversal to both stable and unstable cones. Every other smooth curve (resp. ) is a stable (resp. unstable) curve. Every curve in terminates either inside another curve of or on the boundary . A similar assumption is made for . Moreover, there exists such that for any
and for any ,
Note that (2) implies that for -a.e. , there exists a stable and unstable manifold , such that and does not intersect , for any .
For every , define , the forward separation time of , to be the smallest integer such that and belong to distinct elements of . Fix , then defines a metric on . Similarly we define the backward separation time .
Regularity of stable/unstable curves. We assume that the following families of stable/unstable curves, denoted by are invariant under (resp., ) and include all stable/unstable manifolds:
Bounded curvature. There exist and , such that the curvature of any is uniformly bounded from above by , and the length of the curve .
Distortion bounds. There exist and such that for any unstable curve and any ,
where denotes the Jacobian of at with respect to the Lebesgue measure on the unstable curve .
Absolute continuity. Let be two unstable curves close to each other. Denote
The map defined by sliding along stable manifolds is called the holonomy map. We assume , and furthermore, there exist uniform constants and , such that the Jacobian of satisfies
Similarly, for any we can define the holonomy map
and then (4) and the uniform hyperbolicity (h1) imply
One-step expansion. We have
where the supremum is taken over regular unstable curves , denotes the length of , and , , denote the smooth components of .
These assumptions h4 are satisfied by the billiard map associated to Sinai dispersing billiards with finite and infinite horizon. Further systems which satisfy these assumptions are given in the last section on applications.
These assumptions h4 along with the Growth Lemma imply the existence of a Young Tower with exponential tails [22, Lemma 17].
1.2 Statement of the main results
We fix a hyperbolic periodic point with prime period , and let be given by
where is a metric on that will be specified later. We assume that iterates of do not lie on the singular sets . Our metric will be adapted to the chart given by the stable and unstable manifolds of , denoted as and . If the stable manifold of , denoted as intersects , say at a point , then we define , i.e. the distance of and measured along the unstable manifold . Note that if is very short and does not reach , then we may extend as an unstable curve, thus can be similarly defined. If the unstable manifold (or extension to an unstable curve) of , denoted as intersects at , we define . The foliation of stable and unstable curves of a sufficiently small neighborhood of will be Hölder continuous. Moreover, if both , lie in the same local chart determined by stable and unstable manifolds (or stable and unstable curves) so that , , we define
As in  we expect the form of the distribution for cluster size in the compound Poisson distribution we obtain to depend upon the metric used. With the dynamically adapted metric we use we obtain a geometric distribution with parameter equal to the extremal index , , where is the derivative of in the unstable direction at . With the usual Euclidean metric we would expect a different distribution along the lines of the (complicated) calculations in [36, Section 6].
The functional form of will determine the scaling constants in the extreme value distribution but results for one functional type are easily transformed into any other, see for example [39, 28]. Let . Since is invariant the process is stationary. Note that , where denotes the -ball centered at of radius . In order to obtain a nondegenerate limit for the distribution of , we choose a normalizing sequence such that
for any . We also define
Then we can check that
The extremal index, written as , takes values in , and can be interpreted as an indicator of extremal dependence, with indicating asymptotic independence of extreme events. On the other hand represents the case where we can expect strong clusterings of extreme events. In this case it is natural to expect excursions away from moderate values to extreme regions to persist for random times which have heavy-tailed distributions.
Note that the choice of is made so that the mean number of exceedances by is approximately constant. For the case when are iid, it was shown in  that (8) is equivalent to the fact that as . We are interested in knowing if there is a non-degenerate distribution such that the scaled process converges in distribution to as goes to infinity. Then we say that is an Extreme Value Distribution for . Note that by the Birkhoff Ergodic Theorem, we know that almost surely and hence we must scale as . We refer the event as an exceedance at time of level .
The assumption that is a hyperbolic -periodic point, the assumptions (h1)-(h4) and the definition of the -metric implies that there exists , such that
, where is the derivative of in the unstable direction at . The calculation is the same as in [36, Section 6].
Our first theorem is
Let be a periodic orbit of period , , and define
where is the metric adapted to the chart given by the stable and unstable foliation. Define . Then
Now we consider the related rare event point processes (REPP). Let be the number of exceedances of a level by the process . If is an i.i.d. sequence, it was proved in  that if satisfies as , then follows a Poisson distribution with intensity :
for any . On the other hand, for a stationary process with clustering of recurrence we expect that converges to a compound Poisson distribution. This behavior has been observed in a variety of one-dimensional expanding maps [29, 27, 42] as well as in two-dimensional hyperbolic linear toral automorphisms [34, 26, 36]. In fact in these papers a strict dichotomy has been observed in the extreme value statistics of functions maximized at periodic as opposed to non-periodic points. In this paper we show that a compound Poisson process governs the return times to periodic points for a large class of two-dimensional invertible hyperbolic systems. As we mentioned the precise form of the compound Poisson process depends upon the metric used. For simplicity we consider a dynamically adapted metric , which yields a geometric distribution for the cluster size of exceedances with parameter . Other distributions for the cluster size would be expected with different metrics, as in [36, Section 6].
To make precise our statements we need the following definitions. Denote by the ring generated by intervals of the type , for . Thus for every there are intervals , say with , such that . For and , we set and . Similarly, for , we define and .
Given and a sequence , the rare event point process (REPP) is defined by counting the number of exceedances (or events ) during the (re-scaled) time period . More precisely, for every , set
For the sake of completeness, we also define what we mean by a Poisson and a compound Poisson process.
Let be an iid sequence of random variables with common exponential distribution of mean . Let be another iid sequence of random variables, independent of , and with distribution function . Given these sequences, for define
where denotes the Dirac measure at . We say that is a compound Poisson process of intensity and multiplicity d.f. .
In our context we think of the random variable as determining the number of exceedances given that one has occurred in a certain time interval.
In our setting will always be integer valued, as it gives the size of a cluster. This means that is completely defined by the values , for every . Note that, if and , then is the standard Poisson process and, for every , the random variable has a Poisson distribution of mean .
Let be a periodic orbit of period , , and
where is the metric adapted to the chart given by the stable and unstable foliation. Define and . Let be a sequence satisfying for some and be given by . Consider the REPP as in Definition 2. Then the REPP converges in distribution to a compound Poisson process with intensity and geometric multiplicity d.f. given by , for all .
2 Extreme value theory scheme of proof.
Our proofs are based on ideas from extreme value theory. Let and define . For and a set , let
Next we introduce two conditions following .
Condition : We say that holds for the sequence if, for every
where is decreasing in and there exists a sequence such that and
We consider the sequence given by condition and let be another sequence of integers such that as ,
Condition : We say that holds for the sequence if there exists a sequence as above and such that
We note that, when , which corresponds to a non-periodic point, condition corresponds to condition from .
To establish Theorem 2 it suffices to establish Condition and a property related to Condition , which we call Condition  (see also the discussion in [36, Section 8.1]). We state for completeness, but its proof is a decay of correlations result which requires only very minor modifications to that of the proof of Condition . We will only prove Condition in this paper. The proof of Condition uses the assumption of a Young Tower to approximate the indicator function of a complicated set by a function constant on stable manifolds, which allows a decay rate to be bounded by the norm of the set rather than a Hölder norm. In fact the scheme of the proof of Condition is itself somewhat standard [31, 36] and the novelty of this paper is the proof of Condition in this setting, which uses the growth lemma of  applied to unstable curves.
Let be a periodic point of prime period . Recall and define the sequence of nested sets centered at given by
For , we define the following sets:
Note that . Furthermore, . Using our -metric the set centered at can be decomposed into a sequence of disjoint strips where are the most outward strips and the inner strips are sent outward by onto the strips , i.e.,
Definition 4 ().
We say that holds for the sequence if for any integers , and any with ,
where for each we have that is nonincreasing in and as , for some sequence .
3 Background on billiards: growth lemma and decay of correlations for hyperbolic billiards.
The use of standard pairs of unstable and stable curves  instead of standard pairs of unstable and stable manifolds simplifies our proof of Condition . We now sketch some background on the use of standard pairs.
For any , let be the set of all bounded real-valued functions such that for any and lying on one stable curve , such that
Similarly we define as the set of all real-valued functions such that for any and lying on one unstable curve such that
When we study autocorrelations of certain observables, we will need to require that the latter belong to the space , i.e., they are Hölder functions on every stable and unstable manifold. For correlations of two distinct functions , as above we always assume that and with some , unless otherwise specified. For every we define
By using coupling methods, we obtain the following bounds for the rate of decay of correlations for our class of hyperbolic systems with singularities, see .
For systems satisfy (h1)- (h4), there exist , , such that for any observables and on , with , for ,
for any .
Let and be the constants given in (3). Fix a large constant . Given an unstable curve and a probability measure on , the pair is said to be a standard pair if is absolutely continuous with respect to , such that the density function is regular in the sense that
Iterates of standard pairs require the following extension of standard pairs, see [17, 22]. Let be a family of standard pairs equipped with a factor measure on the index set . We call a standard family if the following conditions hold:
is a measurable partition into unstable curves;
There is a finite Borel measure supported on such that for any measurable set ,
For simplicity, we denote such a family by .
Given a standard family and , the forward image family is such that each is a connected component of for some , associated with (where is the pushforward of to ) and . It is not hard to see that is a standard family as well (cf. ).
Let be the collection of all standard families. We introduce a characteristic function on to measure the average length of unstable curves in a standard family. More precisely, we define a function by
where is as in assumption (h2). Let denote the class of such that .
It turns out that the value of decreases exponentially in until it becomes small enough. This fundamental fact, called the growth lemma, was first proved by Chernov for dispersing billiards in , and later proved in  under Assumptions (h1)-(h4).
There exist constants , , and such that for any ,
4 Checking condition
The proof of Condition uses decay of correlations and is fairly standard. The following analysis is taken from  but is included for completeness. The proof considers stable manifolds rather than stable curves, as well as ideas from Young’s Tower construction for billiard systems . Recall that assumptions along with the Growth Lemma imply the existence of a Young Tower with exponential tails [22, Lemma 17]. Throughout this section, to simplify notation, will denote an unspecified constant, whose value may vary from line to line.
We now briefly the structure of a Young Tower with exponential return time tails for a map of a Riemannian manifold equipped with Lebesgue measure .
A Young Tower has a base set with a hyperbolic product structure as in Young  with an return time function . There is a countable partition of so that is constant on each partition element . We denote by . The Young Tower is defined by
equipped with a tower map given by
We will refer to as the base of the tower and denote . Similarly we call , the th level of the tower. A key role is played by the return map by , which is uniformly expanding Gibbs-Markov.
We may form a quotiented tower (see  for details) by introducing an equivalence relation for points on the same stable manifold. This operation helps in our decay of correlations estimates, as it allows decay rates for the indicator function of complicated sets to be estimated in the norm.We now list the features of the Tower that we will use.
There exists an invariant measure for which has absolutely continuous conditional measures on local unstable manifolds in , with density bounded uniformly from above and below.
There exists an -invariant measure on which is given by for a measurable , and extended to the entire tower in the obvious way. There is a projection given by which semi-conjugates and , so that . The invariant measure , which is an SRB measure for , is given by . Denote by the local stable manifold through and by the local unstable manifold. Let denote the ball of radius centered at the point .
We lift a function to by defining (we keep the same symbol for on and on ).