Application of the convergence of the spatio-temporal processes for visits to small sets

TL;DR

This paper explores the convergence properties of spatio-temporal processes related to visits to small sets in dynamical systems, with applications to billiards and geodesic flows, including new distributional convergence results.

Contribution

It introduces new convergence results for spatio-temporal processes in billiard flows, extending previous work on recurrence and visits to small sets.

Findings

Distributional convergence of visits in Sinai billiard flows

Applications to recurrence and high record studies

Insights into flow behavior in small neighborhoods

Abstract

The goal of this article is to point out the importance of spatio-temporal processes in different questions of quantitative recurrence. We focus on applications to the study of the number of visits to a small set before the first visit to another set (question arising from a previous work by Kifer and Rapaport), the study of high records, the study of line processes, the study of the time spent by a flow in a small set. We illustrate these applications by results on billiards or geodesic flows. This paper contains in particular new result of convergence in distribution of the spatio temporal processes associated to visits by the Sinai billiard flow to a small neighbourhood of orbitrary points in the billiard domain.