Hyperbolicity of asymmetric lemon billiards

TL;DR

This paper proves that asymmetric lemon billiards become hyperbolic when the outer radius is sufficiently large, confirming a conjecture about their chaotic behavior in certain geometric configurations.

Contribution

It establishes the hyperbolicity of asymmetric lemon billiards for large R, advancing understanding of their dynamical properties and confirming a key conjecture.

Findings

Asymmetric lemon billiards are hyperbolic for large R

Confirmed the conjecture about hyperbolicity in specific geometric conditions

Enhanced understanding of billiard dynamics in asymmetric convex domains

Abstract

Asymmetric lemon billiards was introduced in [CMZZ], where the billiard table $Q(r,b,R)$ is the intersection of two round disks with radii $r\le R$, respectively, and $b$ measures the distance between the two centers. It is conjectured [BZZ] that the asymmetric lemon billiards is hyperbolic when the arc $\Gamma_r$ is a major arc and $R$ is large. In this paper we prove this conjecture for sufficiently large $R$.