Dynamical Approach to the Zeta Function

Abstract

The definition ( 2.23) of Selberg’s zeta function as a product over the length spectrum was in some sense very convenient, because of the link between conjugacy classes of Γ and closed geodesics furnished by Proposition 2.25. On the other hand, existence of a meromorphic continuation was far from obvious and controlling the growth of the zeta function was quite difficult (see the proof of Theorem 10.1). There is an alternative framework for zeta functions which is much more general than Selberg’s original definition, and basic properties such as meromorphic continuation are much easier to prove from this point of view. The essential idea is to associate the zeta function to the dynamics of the geodesic flow on the surface (rather than to the geometry of the surface). This viewpoint leads to a definition which can be applied to any dynamical system under certain conditions on the flow. The main drawback to this approach, in the context of hyperbolic surfaces, is the fact that these methods do not apply when the surface has cusps because of the non-compactness of the convex core. In this section we restrict our attention to geometrically finite hyperbolic surfaces without cusps (convex cocompact Fuchsian groups).