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Regular and Chaotic Dynamics

, Volume 23, Issue 1, pp 54–59 | Cite as

Nonisometric Domains with the Same Marvizi – Melrose Invariants

  • Lev BuhovskyEmail author
  • Vadim Kaloshin
Article

Abstract

For any strictly convex planar domain Ω ⊂ R2 with a C boundary one can associate an infinite sequence of spectral invariants introduced by Marvizi–Merlose [5]. 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 Ω up to isometry. In this paper we give a counterexample, namely, we present two nonisometric domains Ω and \(\bar \Omega \) with the same collection of Marvizi–Melrose invariants. Moreover, each domain has countably many periodic orbits {Sn}n≥1 (resp. \({\left\{ {{{\bar S}^n}} \right\}_{n \geqslant 1}}\)) of period going to infinity such that Sn 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 

MSC2010 numbers

37D50 35P30 37E40 37J50 

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Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityRamat Aviv, Tel AvivIsrael
  2. 2.Department of MathematicsUniversity of MarylandCollege ParkUSA

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