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Fully chaotic Hamiltonian systems possess an infinite number of classicalsolutions which are periodic, e.g. a trajectory ``p'' returns to its initialconditions after some fixed time tau_p. Our aim is to investigate the spectrumtau_1, tau_2, ... of periods of the periodic orbits. An explicit formula forthe density rho(tau) = sum_p delta (tau - tau_p) is derived in terms of theeigenvalues of the classical evolution operator. The density is naturallydecomposed into a smooth part plus an interferent sum over oscillatory terms.The frequencies of the oscillatory terms are given by the imaginary part of thecomplex eigenvalues (Ruelle--Pollicott resonances). For large periods,corrections to the well--known exponential growth of the smooth part of thedensity are obtained. An alternative formula for rho(tau) in terms of the zerosand poles of the Ruelle zeta function is also discussed. The results areillustrated with the geodesic motion in billiards of constant negativecurvature. Connections with the statistical properties of the correspondingquantum eigenvalues, random matrix theory and discrete maps are alsoconsidered. In particular, a random matrix conjecture is proposed for theeigenvalues of the classical evolution operator of chaotic billiards

Periodic orbit spectrum in terms of Ruelle-Pollicott resonances