DYNAMICAL ZETA-FUNCTIONS FOR ARTIN BILLIARD AND THE VENKOV-ZOGRAF FACTORIZATION FORMULA

Dynamical zeta functions are expected to relate the Schrodinger operat or's spectrum to the periodic orbits of the corresponding fully chaoti c Hamiltonian system. The relationship is exact in the case of surface s of constant negative curvature. The recently found factorization of the Selberg zeta function for the modular surface is known to correspo nd to a decomposition of the Schrodinger operator's eigenfunctions Int o two sets obeying different boundary condition on Artin's billiard. H ere we express zeta functions for Artin's billiard in terms of general ized transfer operators, providing thereby a new dynamical proof of th e above interpretation of the factorization formula. This dynamical pr oof is then extended to the Artin-Venkov-Zograf formula for finite cov erings of the modular surface.