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 made 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 R\'enyi entropies and Lyapunov exponents.
On the statistical distribution of first--return times of balls and cylinders in chaotic systems
MANTICA, GIORGIO DOMENICO PIO;
2010-01-01
Abstract
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 made 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 R\'enyi entropies and Lyapunov exponents.
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