  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.
