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

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.