Spectral invariants of convex billiard maps:a viewpoint of Mather's beta function

TL;DR

This paper develops a Birkhoff normal form for convex billiard maps, derives an explicit local formula for the beta function, and investigates the connection between spectral invariants and the beta function.

Contribution

It introduces a constructive method to obtain a Birkhoff normal form and links spectral invariants to Mather's beta function in convex billiards.

Findings

Derived a Birkhoff normal form for convex billiard maps

Obtained an explicit local formula for the beta function

Explored the relationship between spectral invariants and the beta function

Abstract

For strictly convex billiard maps of smooth boundaries, we get a Birkhoff normal form via a list of constructive generating functions. Based on this, we get an explicit formula for the beta function (locally), and explored the relation between the spectral invariants of the billiard maps and the beta function.