Periodic orbit spectrum in terms of Ruelle--Pollicott resonances
Résumé
Fully chaotic Hamiltonian systems possess an infinite number of classical solutions which are periodic, e.g. a trajectory ``p'' returns to its initial conditions after some fixed time tau_p. Our aim is to investigate the spectrum tau_1, tau_2, ... of periods of the periodic orbits. An explicit formula for the density rho(tau) = sum_p delta (tau - tau_p) is derived in terms of the eigenvalues of the classical evolution operator. The density is naturally decomposed into a smooth part plus an interferent sum over oscillatory terms. The frequencies of the oscillatory terms are given by the imaginary part of the complex eigenvalues (Ruelle--Pollicott resonances). For large periods, corrections to the well--known exponential growth of the smooth part of the density are obtained. An alternative formula for rho(tau) in terms of the zeros and poles of the Ruelle zeta function is also discussed. The results are illustrated with the geodesic motion in billiards of constant negative curvature. Connections with the statistical properties of the corresponding quantum eigenvalues, random matrix theory and discrete maps are also considered. In particular, a random matrix conjecture is proposed for the eigenvalues of the classical evolution operator of chaotic billiards.