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Lev Buhovsky


Buhovsky L., Kaloshin V.
Nonisometric Domains with the Same Marvizi–Melrose Invariants
2018, vol. 23, no. 1, pp.  54-59
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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