On some invariants of Birkhoff billiards under conjugacy

TL;DR

This paper investigates invariants of Birkhoff billiards under conjugacy, showing certain Taylor coefficients are invariant under smooth conjugations, while also characterizing conjugacy relations between elliptic billiard maps and ellipses.

Contribution

It establishes invariants of Birkhoff billiards under smooth conjugacy and characterizes conjugacy classes of elliptic billiard maps and ellipses.

Findings

Taylor coefficients of the Mather's β-function are invariants under smooth conjugacy.

Elliptic billiard maps are topologically conjugate near boundaries.

Ellipses are similar if their billiard maps are topologically conjugate.

Abstract

In the class of strictly convex smooth boundaries, each of which not having strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the "normalized" Mather's $\beta$-function are invariants under $C^\infty$-conjugacies. In contrast, we prove that any two elliptic billiard maps are $C^0$-conjugated near their respective boundaries, and $C^\infty$-conjugated in the open cylinder, near the boundary and away from a plain passing through the center of the underlying ellipse. We also prove that if the billiard maps corresponding to two ellipses are topologically conjugated then the two ellipses are similar.