G.Popov
Invariants of the length spectrum and spectral
invariants of
planar convex domains.
(95K, LaTeX)
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.