Records and occupation time statistics for area-preserving maps

A relevant problem in dynamics is to characterize how deterministic systems may exhibit features typically associated to stochastic processes. A widely studied example is the study of (normal or anomalous) transport properties for deterministic systems on a non-compact phase space. We consider here two examples of area-preserving maps: the Chirikov-Taylor standard map and the Casati-Prosen triangle map, and we investigate transport properties, records' statistics and occupation time statistics. While the standard map, when a chaotic sea is present, always reproduces results expected for simple random walks, the triangle map -- whose analysis still displays many elusive points -- behaves in a wildly different way, some of the features being compatible with a transient (non conservative) nature of the dynamics.


I. INTRODUCTION
One of the most remarkable advances in modern dynamics lies in the recognition that deterministic systems may exhibit statistical properties typical of purely stochastic processes: for instance such systems may display diffusion properties similar to random walks [1][2][3][4]. Area-preserving maps (see for instance [1]) represent a prominent example of Hamiltonian systems where subtle features of dynamics, as integrability vs chaotic properties, may be studied. In this context one of the most outstanding example is represented by the Chirikov-Taylor standard map (SM) (see [1,5] and references therein): we also mention the fundamental role of such a map in the development of quantum chaos, unveiling features like quantum dynamical localization [6]. Though the SM has been extensively explored by numerical simulations, very few rigorous results have been proven (see, for instance, the introduction in [7]): however it is generally believed that for large nonlinearity parameter this map typically exhibits good stochastic properties, and sensitive dependence upon initial conditions. Here a remark is due: such a map can be studied either on a 2-torus or on an (unbounded) cylinder: the latter representation is naturally adopted when transport properties are concerned, and analogies with random walks are taken into account [1,3,8,9]. While particular nonlinear parameters in the standard map can be tuned to generate strong anomalous diffusion [10], here we will only deal with the case in which diffusion is normal. Our findings will be confronted with those obtained for another area-preserving map, characterized by the lack of exponential instability: the so called Casati-Prosen Triangle Map (TM) [11], introduced by considering, in an appropriate limit, the Birkhoff dynamics of a triangular billiard: apart from its intrinsic interest, such a map is an ideal benchmark to test whether stochasticity properties, exhibited by strongly chaotic systems, are showcased also by systems lacking any exponential instability. It also turns out that many features about the TM are still debated, starting from basic properties like ergodicity and mixing (see for instance [12,13]).
More precisely we will compare different indicators for both map on the cylinder: though in principle further complications are added when one considers a noncompact phase space [14,15], this is the appropriate scenario to discuss transport properties and record statistics, and to check whether tools from infinite ergodic theory may enrich our understanding of such systems.
Our main findings are that the SM, in its typical chaotic regime, displays all stochastic properties of a purely stochastic system, while -as expected-results are far more complicated for the TM, even if we believe that some new insight is provided by our analysis, in particular as regards persistence behaviour, occupation time statistics and the relationship between transport properties and record statistics.
The paper is organized as follows.
In Sec. II, the Chirikov-Taylor standard map (1) and the triangle map (3) -our basic models-are presented and we also mention the main properties we analyze. Section III is dedicated to discuss transport properties, records' statistics and occupation time statistics. We end with a discussions in Sec. IV.

II. THE BASIC SETTING
We recall the definition of the SM K being the nonlinear parameter: when K is sufficiently big no KAM invariant circles bound the motion and one can study moments of the diffusing variable p ∈ R: The typical behaviour observed for the second moment in simulations is normal diffusion ν(2) = 1/2 [16,17], while, for certain parameter values, the existence of stable running orbits (accelerator modes) induces superdiffusion, ν(2) > 1/2) [18][19][20]. We point out that a finer description of anomalous transport is obtained by considering the full spectrum ν(q): if ν(q) = α · q, for some α = 1/2 one speaks about weak anomalous diffusion whereas the case of a nontrivial ν(q) is dubbed strong anomalous diffusion [10]. As far as the SM is concerned we will consider the case where transport in the stochastic sea is normal (even if the phase space exhibits a mixture of chaotic and elliptic components (see Figure 1). On the other side the TM is defined (on the cylinder) as: where ⇂ · · · ↿ denotes the nearest integer. It was introduced in [11] (see also [21]) as a limit case for the Birkhoff map of irrational triangular billiards: systems lacking exponential instability, whose ergodic properties are subtly related to irrationality properties of the angles [22][23][24][25]: we remark that polygonal billiards represent both a hard mathematical challenge [26][27][28][29], and a natural benchmark when trying to assess which microscopic dynamical features lead to macroscopic transport laws [30][31][32] ( see also [33,34]): in this respect it is worth mentioning that anomalous transport has been associated to scaling exponents of the spectral measure [35], and that generalized triangle maps have been investigated recently, both as connected to dynamical localization [36], and with respect to slow diffusion [37]. A typical phase portrait (on the torus) of the TM is shown in Figure 2. Before (golden mean). Here 100 randomly distributed initial conditions were used for x and p: each initial condition is iterated 5 × 10 4 times. Notice the typical filament structure in the phase space [23,24].
mentioning the numerical experiments we performed, a crucial observation is in order. When looking at transport properties (and records statistics), considering maps on the cylinder is quite natural, while from the ergodic point of view this perspective is somehow delicate, since no renormalizable invariant density exists [14,15], and the appropriate setting is infinite ergodic theory. When polygonal channels are considered, even establishing recurrent properties of the dynamics is a demanding task [38].
The first set of properties we investigated is more conventional, and a few results -as we will mention in the next section-have already been considered, especially as far as the SM is considered. We will look at transport properties, in particular through the first and the second moment of the diffusing variable We will also consider records' statistics, which recently has turned very popular (see [39,40] and references therein). Then we will study statistical properties like persistence probability and (generalized) arcsine law [41,42]: while motion in the stochastic sea for the SM will exhibit typical properties of a simple stochastic process like a random walk, our findings are quite different in the case of the TM.

III. RESULTS
We start by considering properties associated to the spreading of trajectories over the phase space, then we will consider occupation time statistics.

A. Diffusion
This is a warm-up exercise, since transport properties have been studied both for the SM [1,16,17] and for the TM [25]. We observe normal transport for the case of the SM (see panels (c) and (d) in Figure 3), while for the TM are results indicate a superdiffusion, with in agreement with [25]. We remark that by looking at the power-law exponents of the first two moments, we have that possibly anomalous diffusion is weak [10], namely if we consider the full spectrum of moments' asymptotics: we have a single scaling, in the sense that where normal diffusion is recovered when α = 1/2. This is reasonable since weak anomalous diffusion has been observed in polygonal billiards [43].

B. Average number of records
The statistics of records is very popular in the analysis of correlated and uncorrelated stochastic time sequences [39,40]: since this subject has not been explores thoroughly in the deterministic setting (with the remarkable exception of [44,45]), we briefly review the basic concepts. First of all let us recall the (straightforward) definition of a record: given a sequence of real data (we consider x 0 to be the first record). To the sequence of data points we associate the binary string The number of records up to time N is then are important tools to access the nature of the data sequence: as a matter of fact if the different x j are independent, identically distributed random variables, then, for large N we have [46,47]: where γ E = 0.5772 . . . is the Euler-Mascheroni constant, and We remark that both quantities are independent of the common distribution of the random variables: this universality is an important feature of record statistics in different contexts. Results are quite different for a correlated sequence, as when x j denotes the position of a random walker at time j: where the jumps are taken from a common distribution ℘(ξ). In this case the behaviour is [39,40]: so that the standard deviation is of the same order of magnitude as the average. Again this is a universal result, independent of the particular jump distribution ℘(ξ), as long as the distribution is continuous and symmetric. The crucial ingredient of the proof is that the process renews as soon as a new record is achieved and the appearance of the new record is related to the survival probability for the process, which is universal in view of Sparre-Andersen theorem [42,48,49] (see also [50]). Numerical results on records statistics are reported in Figures 3, 4, panels (a) and (b): for the SM our results are consistent with early investigations [44,45], and with the asymptotic behaviour of a random walk, while for the TM we observe anomalous scaling w.r.t. (14,15): the behaviour is related to transport properties, in the sense that data are consistent with the growths: A similar behaviour was observed in [44,45], for the SM in the presence of accelerator modes. We remark that, though in the following we will fix our attention of a particular parameter value for the TM, we checked that reported experiments do not depend on the particular parameter choice, as exemplified in Figure 5, where the growth of the averaged number of records is reported for different parameters of the TM. While a general, quantitative relationship (if any) be- tween transport exponents and statistical properties of records has not been fully developed, to the best of our knowledge, it is possible, in some cases, to connect φ(1) to the expected maximum of the walk [51,52], that, for random walk with unit jumps, coincides with the number of records. On the other side we mention that nonhomogeneous random walks offer examples where such relationship does not hold [53][54][55][56][57].

C. Occupation time statistics
When we consider the evolution on the cylinder, both for the SM and the TM, we are in the presence of infinitely ergodic systems [14,15], since, while Lebesgue measure is preserved, due to area conservation, the (constant) phase space is unbounded, so the invariant density cannot be normalized. This has a series of remarkable consequences, which originally have been considered in the context of stochastic processes, and then explored in the deterministic evolution framework.
One of the most striking property that has been investigated is the generalized arcsine law (see [41] for the standard formulation for stochastic processes): we briefly recall the main result that lies at the basis of our analysis, namely Lamperti's theorem [58]. The original formulation involves discrete stochastic processes, for which the infinite set of possible states can be separated into two sets A and B separated by a single site x 0 , such that a transition from one set to the other can only be achieved by passing through x 0 , which can be taken as the starting site, and is supposed to be recurrent (namely the probability of returning to it is 1). For instance we can think of one dimensional random walk on an integer lattice, with x 0 = 0 and A (B) consists of strictly positive (negative) lattice sites. We are interested in the limiting distribution of N (n)/n, the fraction of time spent in the positive semi-axis up to time n. The theorem states that such a distribution exists in the n → ∞ limit, and it is characterized by two parameters α and η. η is related to symmetry properties of the process, being the expectation value of the fraction of time spent in R + : for a symmetric process η = 1/2, and from now on we will only consider such a case. The other parameter α is instead connected to the behaviour of the generating function of first return probabilities to the starting site: it can be shown [59] that it can be related to the probability P n of being at the starting site after n steps in the following way: where H(n) is a slowly varying function, namely Under such conditions the density of ϕ = N (n)/n in the infinite time limit is given by Lamperti distribution: that reproduces the usual arcsine law when α = 1/2, in the universality class of Sparre-Andersen theorem. Deviations from standard arcsine law have been reported for a number of cases, in the framework of deterministic dynamics [60][61][62][63][64][65][66][67], mainly in the context of intermittent maps. Numerical experiments for the SM confirm the validity of the arcsine law, α = 1/2, see panel (a) in Figure 6: to our knowledge this is the first time such an indicator has been considered in the analysis of area preserving maps. The results, as expected, are quite different for the TM, and they suggest novel features exhibited by this map. In particular (see panel (b) in Figure 6) numerical results are well fitted by a Lamperti distribution (with α ≈ 0.42), thus different from an ordinary random walk), except for the endpoints, that present enhanced peaks. Intuitively such an additional contribution might be due to a fraction of orbits never returning to the origin: this would correspond, in stochastic language, to a transient random walk (we recall that, according to Pólya's theorem [68] a simple symmetric random walk is recurrent -so the return to the starting site is sure-in one and two dimensions, and transient in higher dimensions). Such a possibility is indeed not excluded for infinite polygonal channels [38]. Our last set of simulations concerns the survival probability [61]: When considering recurrent random walks, the asymptotic behaviour of the survival probability is again ruled by Lamperti exponent [58,59] (see also [69]): Once again SM simulations (see panel (a) in Figure 7) agree with expected behaviour for simple random walks (α = 1/2), while the situation is completely different for the TM, where the survival probability seems to tend to a finite limit for large n, see panel (b) in Figure 7. This is coherent with the transient nature of the TM, which we conjectured in the analysis of generalized arcsine law.

IV. DISCUSSION
We have performed a set of extensive numerical experiments on two paradigmatic area-preserving maps, the SM and the TM, focusing in the case where such maps are considered on a cylinder, namely a non compact phase space. Firstly we reproduced known results about normal diffusion for typical (chaotic) parameters of the SM, and superdiffusion for the TM. Then we explored records' statistics: numerical simulations again confirm that the SM behave like a simple random walk, while anomalous growth is exhibited by the TM. The most interesting results arise in the analysis of occupation times, like generalized arcsine law and survival probability. While once again normal stochastic properties are displayed by the SM, the TM presents more surprising results, which we conjecture are possibly connected to lack of conservativity [38] (or transient behaviour, in the language of random walks). This feature, which we think is worth of further investigations, might suggest new stochastic modeling of the TM (see [37]).