On some invariants of Birkhoff billiards under conjugacy

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 $C<sup>\infty$-conjugacies. In contrast, we prove that any two elliptic billiard maps are $C<sup>0$-conjugated near their respective boundaries, and $C<sup>\infty$-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.