Multiple returns for some regular and mixing maps.

Abstract

We study the distributions of the number of visits for some noteworthy dynamical systems, considering whether limit laws exist by taking domains that shrink around points of the phase space. It is well known that for highly mixing systems such limit distributions exhibit a Poissonian behavior. We analyze instead a skew integrable map defined on a cylinder that models a shear flow. Since almost all fibers are given by irrational rotations, we at first investigate the distributions of the number of visits for irrational rotations on the circle. In this last case the numerical results strongly suggest the existence of limit laws when the shrinking domain is chosen in a descending chain of renormalization intervals. On the other hand, the numerical analysis performed for the skew map shows that limit distributions exist even if we take domains shrinking in an arbitrary way around a point, and these distributions appear to follow a power law decay of which we propose a theoretical explanation. It is interesting to note that we observe a similar behavior for domains wholly contained in the integrable region of the standard map. We also consider the case of two or more systems coupled together, proving that the distributions of the number of visits for domains intersecting the boundary between different regions are a linear superposition of the distributions characteristic of each region. Using this result we show that the real limit distributions can be hidden by some finite-size effects. In particular, when a chaotic and a regular region are glued together, the limit distributions follow a Poisson-like law, but as long as the measure of the shrinking domain is not zero, the polynomial behavior of the regular component dominates for large times. Such an analysis seems helpful to understand the dynamics in the regions where ergodic and regular motions are intertwined, as it may occur for the standard map. Finally, we study the distributions of the number of visits around generic and periodic points of the dissipative Henon map. Although this map is not uniformly hyperbolic, the distributions computed for generic points show a Poissonian behavior, as usually occurs for systems with highly mixing dynamics, whereas for periodic points the distributions follow a different law that is obtained from the statistics of first return times by assuming that subsequent returns are independent. These results are consistent with a possible rapid decay of the correlations for the Henon map.