Dynamical spectral rigidity among Z2 -symmetric strictly convex domains close to a circle

We show that any sufficiently (finitely) smooth Z(2) -symmetric strictly convex domain sufficiently close to a circle is dynamically spectrally rigid; i.e., all deformations among domains in the same class that preserve the length of all periodic orbits of the associated billiard flow must necessarily be isometric deformations. This gives a partial answer to a question of P. Sarnak.