Crossing the Entropy Barrier of Dynamical Zeta Functions

Dynamical zeta functions are an important tool to quantize chaotic dynamical systems. The basic quantization rules require the computation of the zeta functions on the real energy axis, where their Euler product representations running over the classical periodic orbits usually do not converge due to the existence of the so–called entropy barrier determined by the topological entropy of the classical system. We show that the convergence properties of the dynamical zeta functions rewritten as Dirichlet series are governed not only by the well–known topological and metric entropy, but depend crucially on subtle statistical properties of the Maslov indices and of the multiplicities of the periodic orbits that are measured by a new parameter for which we introduce the notion of a third entropy. If and only if the third entropy is nonvanishing, one can cross the entropy barrier; if it exceeds a certain value, one can even compute the zeta function in the physical region by means of a convergent Dirichlet series. A simple statistical model is presented which allows to compute the third entropy. Four examples of chaotic systems are studied in detail to test the model numerically.