Classical and semiclassical zeta functions in terms of transition probabilities

We propose a semiclassical quantization scheme for bound hyperbolic systems based on the properties of a single ergodic trajectory. The dynamics of the system is approximated by transition probabilities between cells of a partition of the phase-space. We construct a transfer matrix of the corresponding Markov graph which approaches the classical Frobenius-Perron (transfer) operator in the limit of infinitesimal tesselations of the phase-space. A semiclassical zeta function may be obtained as the determinant of an appropriately weighted transfer operator and leads to a product over the closed paths of the graph in close analogy to the Gutzwiller-Voros zeta function which is a product over periodic orbits.