Spatio-temporal Poisson processes for visits to small sets

TL;DR

This paper introduces a new spatio-temporal Poisson process model for visits to small sets in measure-preserving dynamical systems, extending classical hitting time results to include position data.

Contribution

It defines a novel process capturing both visit times and positions, proving its convergence to a Poisson point process under decorrelation, with applications to hyperbolic maps and billiards.

Findings

Convergence of the process to a Poisson point process in time and space.

Application to hyperbolic maps and SRB measures.

Analysis of billiard systems like Sinai billiards and stadiums.

Abstract

For many measure preserving dynamical systems $(\Omega,T,m)$ the successive hitting times to a small set is well approximated by a Poisson process on the real line. In this work we define a new process obtained from recording not only the successive times $n$ of visits to a set $A$, but also the position $T^n(x)$ in $A$ of the orbit, in the limit where $m(A)\to0$. We obtain a convergence of this process, suitably normalized, to a Poisson point process in time and space under some decorrelation condition. We present several new applications to hyperbolic maps and SRB measures, including the case of a neighborhood of a periodic point, and some billiards such as Sinai billiards, Bunimovich stadium and diamond billiard.