Resonance spectra and a periodic orbit sum rule for bound chaotic systems

Abstract

We consider the spectrum of the evolution operator for bound chaotic systems by evaluating its trace. This trace is known to approach unity as $t \rightarrow \infty$ for bound systems. It is written as the Fourier transform of the logaritmic derivative of a zeta function whose zeros are identified with the eigenvalues of the operator. Zeta functions are expected to be entire only for very specific systems, like Axiom-A systems. For bound chaotic systems complications arise due to e.g. intermittency and non completeness of the symbolic dynamics. For bound intermittent system an approximation of the zeta function is derived. It is argued that bound systems with {\em long time tails} have branch cuts in the zeta function and traces approaching unity as a powerlaw. Another feature of bound chaotic systems, relevant for the asymptotics of the trace, is that the dominant time scale can be much longer than the period of the shortest periodic orbit. This the case for e.g. 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 discussed at the end.