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2013
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 Buhovsky L., Kaloshin V. Nonisometric Domains with the Same Marvizi–Melrose Invariants 2018, vol. 23, no. 1, pp.  54-59 Abstract 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 DOI:10.1134/S1560354718010057