Computing Mather's $\beta$-function for Birkhoff billiards

Abstract

This article is concerned with the study of Mather's $\beta$-function associated to Birkhoff billiards. This function corresponds to the minimal average action of orbits with a prescribed rotation number and, from a different perspective, it can be related to the maximal perimeter of periodic orbits with a given rotation number, the so-called Marked length spectrum . After having recalled its main properties and its relevance to the study of the billiard dynamics, we stress its connections to some intriguing open questions: Birkhoff conjecture and the isospectral rigidity of convex billiards. Both these problems, in fact, can be conveniently translated into questions on this function. This motivates our investigation aiming at understanding its main features and properties. In particular, we provide an explicit representation of the coefficients of its (formal) Taylor expansion at zero, only in terms of the curvature of the boundary. In the case of integrable billiards, this result provides a representation formula for the $\beta$-function near $0$. Moreover, we apply and check these results in the case of circular and elliptic billiards.