Upgrading the Local Ergodic Theorem for planar semi-dispersing billiards

TL;DR

This paper proves the Local Ergodic Theorem for planar semi-dispersing billiards without relying on the Chernov-Sinai Ansatz, simplifying the verification process for ergodicity in these systems.

Contribution

It provides a new proof of the Local Ergodic Theorem for 2D billiards that removes the need for the difficult Chernov-Sinai Ansatz assumption.

Findings

Proves the Local Ergodic Theorem without the Chernov-Sinai Ansatz

Simplifies verification of ergodicity in planar billiards

Applicable to physically relevant models like hard ball gases

Abstract

The Local Ergodic Theorem (also known as the `Fundamental Theorem') gives sufficient conditions under which a phase point has an open neighborhood that belongs (mod 0) to one ergodic component. This theorem is a key ingredient of many proofs of ergodicity for billiards and, more generally, for smooth hyperbolic maps with singularities. However the proof of that theorem relies upon a delicate assumption (Chernov-Sinai Ansatz), which is difficult to check for some physically relevant models, including gases of hard balls. Here we give a proof of the Local Ergodic Theorem for two dimensional billiards without using the Ansatz.