93-5 G.Popov

Invariants of the length spectrum and spectral invariants of planar convex domains. (95K, LaTeX) Jan 9, 93

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Abstract. This paper is concerned with a conjecture of V.Guillemin and R. Melrose that the length spectrum of a strictly convex bounded domain together with the spectra of the linear Poincar\'{e} maps corresponding to the periodic broken geodesics in $\Omega$ determine uniquely the billiard ball map up to a symplectic conjugation. We consider continuous deformations of bounded strictly convex domains $\Omega_s,\ s\in [0,1]$, with smooth boundaries. If the length spectrum does not change along the deformation, we prove that the invariant KAM circles of the corresponding billiard ball map $B_s$ with rotation numbers in a suitable Cantor set of a positive Lebesgue measure as well as the restriction of $B_s,\ s\in [0,1]$, on their union are symplectically equivalent to each other. 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.

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