Zeta functions and the periodic orbit structure of hyperbolic dynamics / William Parry and Mark Pollicott

As the authors state in the introduction: “Axiom A diffeomorphisms and flows, introduced by Smale, are generalizations of Anosov systems which in turn are based on the prototypical hyperbolic toral automorphisms and geodesic flows on surfaces of constant negative curvature”. To study these systems one usually models them by introducing Markov partitions, shifts and suspensions. In this work the emphasis is on problems associated with periodic orbits. After introducing basic concepts, a.o. the Ruelle operator and entropy, the authors discuss the relation between zeta functions and periodic points. One of the important points is the relation between the spectra of Ruelle operators (real and complex) and the poles of zeta functions. Among the various themes of the book is the proof of temporal, spatial and symmetrical distribution theorems. Five appendices have been added with basic material. (Zentralblatt)

Bibliogr. p. 259-266

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 is applied, especially, to geodesic flows on surfaces of variable negative curvature. (SMF)