Poincaré recurrences as a tool to investigate the statistical properties of dynamical systems with integrable and mixing components

Abstract

Poincaré recurrences seem able to capture some of the fundamental properties of dynamical systems. In fact, the asymptotic distribution of Poincaré recurrences is exponential for a wide class of mixing systems, even if they are not uniformly hyperbolic. On the other hand, we found strong numerical evidences that for integrable systems such distribution follows an algebraic decay law, showing this behavior for a skew integrable map on a cylinder. For a mixed system, that is a system composed by two or more invariant regions, we proved that the statistics of Poincaré recurrences of points at the boundaries is a linear combination of the spectra characteristic of the various components. We think that these results could allow to understand the behavior of area-preserving maps in the mixed regions where integrable structures and chaotic components coexist. In this respect, the intense numerical studies performed by several authors suggest that in the thin stochastic layer surrounding a chain of islands the decay of Poincaré recurrences could follow a power law due to the sticking phenomenon, which is believed to be responsible for the anomalous diffusion modeled by Levy like processes. Furthermore, such a mixture of exponential and power law decays has been observed in a model of stationary flow with hexagonal symmetry, when the transport is anomalous. Some preliminary investigations show that, at least for the skew and for the mixing maps, the results obtained about the first return times spectra also hold for the successive Poincaré recurrences.