Dynamical zeta functions for Artin's billiard and the Venkov--Zograf factorization formula

Abstract

Dynamical zeta functions are expected to relate the Schr\"odinger operator's spectrum to the periodic orbits of the corresponding fully chaotic Hamiltonian system. The relationsship is exact in the case of surfaces of constant negative curvature. The recently found factorisation of the Selberg zeta function for the modular surface is known to correspond to a decomposition of the Schr\"odinger operator's eigenfunctions into two sets obeying different boundary condition on Artin's billiard. Here we express zeta functions for Artin's billiard in terms of generalized transfer operators, providing thereby a new dynamical proof of the above interpretation of the factorization formula. This dynamical proof is then extended to the Artin--Venkov--Zograf formula for finite coverings of the modular surface.