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TL;DR

This paper investigates the recurrence times of a billiard system with periodic convex scatterers, establishing convergence rates and distributional limits for return times as the neighborhood size shrinks.

Contribution

It provides the first rigorous analysis of the asymptotic distribution and convergence rates of return times in a planar billiard with periodic convex obstacles.

Findings

Almost sure convergence rate of return times established.

Rescaled return times converge in distribution.

Results apply as the neighborhood radius approaches zero.

Abstract

We consider a billiard in the plane with periodic configuration of convex scatterers. This system is recurrent, in the sense that almost every orbit comes back arbitrarily close to the initial point. In this paper we study the time needed to get back in an r-ball about the initial point, in the phase space and also for the position, in the limit when r->0. We establish the existence of an almost sure convergence rate, and prove a convergence in distribution for the rescaled return times.