Differentiability of the largest Lyapunov exponent for planar open billiards

TL;DR

This paper proves that the largest Lyapunov exponent in planar open billiards varies smoothly with small changes in the billiard shape, providing insights into the stability and chaos of such dynamical systems.

Contribution

It establishes the differentiability of the largest Lyapunov exponent concerning billiard deformations, a novel result in the study of billiard dynamics.

Findings

Largest Lyapunov exponent is differentiable with respect to billiard shape changes

Provides a mathematical foundation for understanding stability in open billiards

Enhances analysis of chaotic behavior in planar billiard systems

Abstract

In this paper, we estimate the largest Lyapunov exponent for open billiards in the plane. We show that the largest Lyapunov exponent is differentiable with respect to a billiard deformation.