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 $C^\infty$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.
 Publication:

arXiv eprints
 Pub Date:
 May 2021
 DOI:
 10.48550/arXiv.2105.14640
 arXiv:
 arXiv:2105.14640
 Bibcode:
 2021arXiv210514640K
 Keywords:

 Mathematics  Dynamical Systems;
 2020 numbers: 37C83;
 37E40;
 37J51
 EPrint:
 17 pages. Submitted