On the statistical distribution of first--return times of balls and cylinders in chaotic systems

Details On the statistical distribution of first--return times of balls and cylinders in chaotic systems On the statistical distribution of first--return times of balls and cylinders in chaotic systems

2009-07-27

arxiv.org/abs/0907.4675v1

Int. J. of Bifurcations and Chaos, 20 (2010) 1845-1858

Nonlinear Sciences

Chaotic Dynamics

submitted to Int. J. Bifurcations and Chaos

Scientific paper

We study returns in dynamical systems: when a set of points, initially populating a prescribed region, swarms around phase space according to a deterministic rule of motion, we say that the return of the set occurs at the earliest moment when one of these points comes back to the original region. We describe the statistical distribution of these "first--return times" in various settings: when phase space is composed of sequences of symbols from a finite alphabet (with application for instance to biological problems) and when phase space is a one and a two-dimensional manifold. Specifically, we consider Bernoulli shifts, expanding maps of the interval and linear automorphisms of the two dimensional torus. We derive relations linking these statistics with Renyi entropies and Lyapunov exponents.

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