Mather $\beta$-function for ellipses and rigidity

TL;DR

This paper derives an explicit formula for the Mather β-function for ellipses using a novel generating function approach and explores its application to rigidity problems in billiard dynamics.

Contribution

It provides a new, simplified derivation of the Mather β-function for ellipses and applies it to address rigidity issues in billiard systems.

Findings

Explicit formula for Mather β-function for ellipses

Simplified derivation using non-standard generating functions

Application to rigidity problems in billiard dynamics

Abstract

The goal of the first part of this note is to get an explicit formula for rotation number and Mather $\beta$-function for ellipse. This is done here with the help of non-standard generating function of billiard problem. In this way the derivation especially simple. In the second part we discuss application of Mather $\beta$-function to rigidity problem.