Zeta functions and the periodic orbit structure of hyperbolic dynamics

This work studies a variety of problems concerned with the distribution of closed orbits of hyperbolic flows. Basic material from the theory of shifts of finite type and their suspensions is presented and the modelling role of these systems for hyperbolic flows is exploited. Spectral properties of the Ruelle operator are analysed and used to establish analytic properties of a dynamical zeta function which incorporates information about closed orbits. Classical techniques from number theory are then used to establish our main theorems. The general theory is applied, especially, to geodesic flows on surfaces of variable negative curvature.