Approximate zeta functions for the Sinai billiard and related systems

Abstract

We discuss zeta functions, and traces of the associated weighted evolution operators for intermittent Hamiltonian systems in general and for the Sinai billiard in particular. The intermittency of this billiard is utilized so that the zeta functions may be approximately expressed in terms of the probability distribution of laminar lengths. In particular we study a one-parameter family of weights. Depending on the parameter the trace can be dominated by branch cuts in the zeta function or by isolated zeros. In the former case the time dependence of the trace is dominated by a powerlaw and in the latter case by an exponential. A phase transition occurs when the leading zero collides with a branch cut. The family considered is relevant for the calculation of resonance spectra, semiclassical spectra and topological entropy.