Random circular billiards on surfaces of constant curvature: Pseudo integrability and mixing

TL;DR

This paper investigates the dynamical properties of random billiard maps on surfaces of constant curvature, demonstrating invariance of Liouville measure and establishing ergodicity in the case of random circular billiards.

Contribution

It introduces a new framework for random billiards on curved surfaces and proves ergodicity for the specific case of random circular billiards.

Findings

Liouville measure is invariant for the random billiard map.

Ergodicity is established for random circular billiards.

Framework applies to Euclidean, hyperbolic, and spherical surfaces.

Abstract

Given a random map (T_1, T_2, T_3, T_4, p_1, p_2, p_3, p_4), we define a random billiard map on a surface of constant curvature (Euclidean plane, hyperbolic plane, or the sphere). The Liouville measure is invariant for this billiard map. Finally, we show some dynamical properties such as ergodicity in the case of random circular billiards.