DETERMINATION OF RESONANCE-SPECTRA FOR BOUND CHAOTIC SYSTEMS

We consider the computation of the eigenvalues of the evolution operator-the resonance spectrum-by means of the zeros of a zeta function. In particular we address the problems of applying this formalism to bound chaotic systems, caused by e.g. intermittency and non-completeness of the symbolic dynamics. For bound intermittent system we derive an approximation of the zeta function. With the aid of this zeta function it is argued that bound systems with long time tails have branch cuts in the zeta function and traces (of the evolution operator) approaching unity as a power law. We also show that the dominant time scale can be much longer than the period of the shortest periodic orbit, as is, for example, the case for the hyperbola billiard. Isolated zeros of the zeta function for the hyperbola billiard are evaluated by means of a cycle expansion. Crucial for the success of this approach is the identification of a sequence of periodic orbit responsible for a logarithmic branch cut in the zeta function, Semiclassical implications are briefly discussed.