# On Some Invariants of Birkhoff Billiards Under Conjugacy

*2022, Volume 27, Number 5, pp. 525-537*

Author(s):

**Koudjinan C. E., Kaloshin V.**

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.

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