Entry and Return times distribution

This is a review article on the distributions of entry and return times in dynamical systems which discusses recent results for systems of positive entropy.Comment: To appear in "Dynamical Systems: An International Journal dedicated to the Statistical Properties of Dynamical Systems

Limiting entry times distribution for arbitrary null sets SETS

We describe an approach that allows us to deduce the limiting return times distribution for arbitrary sets to be compound Poisson distributed. We establish a relation between the limiting return times distribution and the probability of the cluster sizes, where clusters consist of the portion of points that have finite return times in the limit where random return times go to infinity. In the special case of periodic points we recover the known P\'olya-Aeppli distribution which is associated with geometrically distributed cluster sizes. We apply this method to several examples the most important of which is synchronisation of coupled map lattices. For the invariant absolutely continuous measure we establish that the returns to the diagonal is compound Poisson distributed where the coefficients are given by certain integrals along the diagonal.Comment: 33