The billiard ball problem and rotation numbers

TL;DR

This paper introduces rotation numbers and vectors for billiard maps using ergodic theory, aiming to analyze billiard dynamics and establish non-ergodicity in specific domains.

Contribution

It presents a novel application of ergodic theory concepts to billiard maps, providing tools for understanding their dynamical properties.

Findings

Defined rotation numbers and vectors for billiard maps

Proposed a method to analyze non-ergodicity in billiard systems

Established a theoretical framework for future studies

Abstract

We introduce the concepts of rotation numbers and rotation vectors for billiard maps. Our approach is based on the birkhoff ergodic theorem. We anticipate that it will be useful, in particular, for the purpose of establishing the non-ergodicity of billiard in certain domains.