Quantitative recurrence: spatio-temporal point process and applications

Françoise Péne (Université de Bretagne Occidentale)

We consider chaotic probability preserving dynamical systems. We study a spatio-temporal point pro- cess capturing the information both in time and in space on visits of orbits in a set of small measure. More precisely, we are interesting in the convergence of this process to a Poisson Point process when the measure of the set goes to 0. We will focus on a strategy to prove such a convergence and on consequences of such a result : convergence of Birkhoff sums to stable law, point process of visits in a neighbourhood of a hyperbolic periodic point, time spent in a small neighbourhood of some posi- tion of a point particle evolving in the Sinai billiard, etc. This talk is based on joint works with Benoît Saussol.