Lyapunov Exponents for Open Billiards in the Exterior of Balls

TL;DR

This paper investigates the chaotic dynamics of billiard flows outside multiple small balls in three-dimensional space, proving that the system exhibits two distinct positive Lyapunov exponents under certain conditions.

Contribution

It establishes the inequality of the two positive Lyapunov exponents for billiard flows outside multiple balls, extending understanding of chaotic behavior in such open billiard systems.

Findings

Two positive Lyapunov exponents are different: λ₁ > λ₂ > 0

Results hold for any Gibbs measure on the non-wandering set

System exhibits strong chaotic properties under specified conditions

Abstract

In this paper we consider the billiard flow in the exterior of several (at least three) balls in $\R^3$ with centres lying on a plane. We assume that the balls satisfy the no eclipse condition (H) and their radii are small compared to the distances between their centres. We prove that with respect to any Gibbs measure on the non-wandering set of the billiard flow the two positive Lyapunov exponents are different: $\lambda_1 > \lambda_2 > 0$.