  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.
