On dynamical zeta function

The dynamical zeta function is usually defined as an infinite (and divergent) product over all primitive periodic orbits. It is possible to show that as (h) over bar -> 0 it can be represented as det(1-T), where the operator T(q,q') defines the semiclassical Poincare map. Here, certain consequences of this representation for chaotic systems are discussed. In particular, it is shown that the zeta function can be expressed through a subset of specially selected orbits, the error of this approximation being small as (h) over bar -> 0. Assuming that the chosen Poincare surface of section is divided into small cells of phase-space area of 2 pi(h) over bar, these trajectories are uniquely characterized by the requirement that they never go twice through the same cell.