This article is cited in 2 papers

Alexey Borisov Memorial Volume

On Some Invariants of Birkhoff Billiards Under Conjugacy

Comlan E. Koudjinan, Vadim Kaloshin

Institute of Science and Technology Austria (ISTA), Am Campus 1, 3400 Klosterneuburg, Austria

Abstract: In the class of strictly convex smooth boundaries each of which has no strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the “normalized” Mather's $\beta$-function are invariant under $C^\infty$-conjugacies. In contrast, we prove that any two elliptic billiard maps are $C^0$-conjugate near their respective boundaries, and $C^\infty$-conjugate, near the boundary and away from a line passing through the center of the underlying ellipse. We also prove that, if the billiard maps corresponding to two ellipses are topologically conjugate, then the two ellipses are similar.

Keywords: Birkhoff billiard, integrability, conjugacy, Mather’s $\beta$-function, Marvizi – Melrose invariants.

MSC: 37C83, 37E40, 37J51

Received: 03.12.2021
Accepted: 08.09.2022

Language: English

DOI: 10.1134/S1560354722050021

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