CLASSICAL AND SEMICLASSICAL ZETA-FUNCTIONS IN TERMS OF TRANSITION-PROBABILITIES

We propose a semiclassical quantization scheme for bound hyperbolic sy stems based on the properties of a single ergodic trajectory. The dyna mics of the system is approximated by transition probabilities between cells of a partition of the phase-space. We construct a transfer matr ix of the corresponding Markov graph which approaches the classical Fr obenius-Perron (transfer) operator in the limit of infinitesimal tesse lations of the phase-space. A semiclassical zeta function may be obtai ned as the determinant of an appropriately weighted transfer operator and leads to a product over the closed paths of the graph in close ana logy to the Gutzwiller-Voros zeta function which is a product over per iodic orbits.