Resonances of the Frobenius-Perron Operator for a Hamiltonian Map with a Mixed Phase Space

Resonances of the (Frobenius-Perron) evolution operator P for phase-space densities have recently attracted considerable attention, in the context of interrelations between classical and quantum dynamics. We determine these resonances as well as eigenvalues of P for Hamiltonian systems with a mixed phase space, by truncating P to finite size in a Hilbert space of phase-space functions and then diagonalizing. The corresponding eigenfunctions are localized on unstable manifolds of hyperbolic periodic orbits for resonances and on islands of regular motion for eigenvalues. Using information drawn from the eigenfunctions we reproduce the resonances found by diagonalization through a variant of the cycle expansion of periodic-orbit theory and as rates of correlation decay.