Vadim Kaloshin

Klosterneuburg, 3400 Austria
Institute of Science and Technology Austria


Koudjinan C. E., Kaloshin V.
On Some Invariants of Birkhoff Billiards Under Conjugacy
2022, vol. 27, no. 5, pp.  525-537
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
Citation: Koudjinan C. E., Kaloshin V.,  On Some Invariants of Birkhoff Billiards Under Conjugacy, Regular and Chaotic Dynamics, 2022, vol. 27, no. 5, pp. 525-537
Buhovsky L., Kaloshin V.
For any strictly convex planar domain $\Omega \subset \mathbb R^2$ with a $C^\infty$ boundary one can associate an infinite sequence of spectral invariants introduced by Marvizi–Merlose~\cite{MM}. These invariants can generically be determined using the spectrum of the Dirichlet problem of the Laplace operator. A natural question asks if this collection is sufficient to determine $\Omega$ up to isometry. In this paper we give a counterexample, namely, we present two nonisometric domains $\Omega$ and $\bar \Omega$ with the same collection of Marvizi–Melrose invariants. Moreover, each domain has countably many periodic orbits $\{S^n\}_{n \geqslant 1}$ (resp. $\{ \bar S^n\}_{n \geqslant 1}$) of period going to infinity such that $ S^n $ and $ \bar S^n $ have the same period and perimeter for each $ n $.
Keywords: convex planar billiards, length spectrum, Laplace spectrum, Marvizi–Melrose spectral invariants
Citation: Buhovsky L., Kaloshin V.,  Nonisometric Domains with the Same Marvizi–Melrose Invariants, Regular and Chaotic Dynamics, 2018, vol. 23, no. 1, pp. 54-59

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