Periodic orbit asymptotics for intermittent Hamiltonian systems

Abstract

We address the problems in applying cycle expansions to bound chaotic systems, caused by e.g. intermittency and incompleteness of the symbolic dynamics. We discuss zeta functions associated with weighted evolution operators and in particular a one-parameter family of weights relevant for the calculation of classical resonance spectra, semiclassical spectra and topological entropy. For bound intermittent system we discuss an approximation of the zeta function in terms of probabilities rather than cycle instabilities. This approximation provides a generalization of the fundamental part of a cycle expansion for a finite subshift symbolic dynamics. This approach is particularly suitable for determining asymptotic properties of periodic orbits which are essential for scrutinizing the semiclassical limit of Gutzwiller's semiclassical trace formula. The Sinai billiard is used as model system. In particular we develope a crude approximation of the semiclassical zeta function which turns out to possess non analytical features. We also discuss the contribution to the semiclassical level density from the neutral orbits. Finally we discuss implications of our findings for the spectral form factor and compute the asymptotic behaviour of the spectral rigidity. The result is found to be consistent with exact quantum mechanical calculations.