INVARIANTS OF THE LENGTH SPECTRUM AND SPECTRAL INVARIANTS OF PLANAR CONVEX DOMAINS

This paper is concerned with a conjecture of Guillemin and Melrose tha t the length spectrum of a strictly convex bounded domain together wit h the spectra of the linear Poincare maps corresponding to the periodi c broken geodesics in OMEGA determine uniquely the billiard ball map u p to a symplectic conjugation. We consider continuous deformations of bounded domains OMEGA(s), s is-an-element-of [0, 1], with smooth bound aries and suppose that OMEGA0 is strictly convex and that the length s pectrum does not change along the deformation. We prove that OMEGA(s) is strictly convex for any s along the deformation and that for differ ent values of the parameter s 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 restrictio n of the billiard ball map on them are spectral invariants of the Lapl acian with Dirichlet (Neumann) boundary conditions for suitable deform ations of strictly convex domains.