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Communications in Mathematical Physics

, Volume 161, Issue 2, pp 335–364 | Cite as

Invariants of the length spectrum and spectral invariants of planar convex domains

  • Georgi Popov
Article

Abstract

This paper is concerned with a conjecture of Guillemin and Melrose that the length spectrum of a strictly convex bounded domain together with the spectra of the linear Poincaré maps corresponding to the periodic broken geodesics in Ω determine uniquely the billiard ball map up to a symplectic conjugation. We consider continuous deformations of bounded domainsΩ s ,s∈[0, 1], with smooth boundaries and suppose thatΩ0 is strictly convex and that the length spectrum does not change along the deformation. We prove thatΩ0 is strictly convex for anys along the deformation and that for different values of the parameters the corresponding billiard ball maps are symplectically equivalent to each other on the union of the invariant KAM circles. We prove as well that the KAM circles and the restriction of the billiard ball map on them are spectral invariants of the Laplacian with Dirichlet (Neumann) boundary conditions for suitable deformations of strictly convex domains.

Keywords

Boundary Condition Neural Network Statistical Physic Complex System Nonlinear Dynamics 
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Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • Georgi Popov
    • 1
    • 2
  1. 1.Institute of MathematicsBulgarian Academy of SciencesBG-SofiaBulgaria
  2. 2.Fachbereich MathematikTechnische Hochschule DarmstadtDarmstadtGermany

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