Periodic Orbits and Zeta Functions

0. Introduction 1 1. Twisted orbits and zeta functions 3 2. Axiom A diieomorphisms 4 3. Symbolic dynamics and rationality 5 4. Zeta functions and for interval maps 9 5. The Ruelle zeta function 14 6. Zeta functions for hyperbolic ows 21 7. Zeta functions for analytic Anosov ows 25 8. L-functions and special values 26 9. counting closed orbits for ows 27 10. L-functions and homology 32 11. Pole free regions for the zeta function 35 12. References 37 0. Introduction The study of periodic orbits for dynamical systems dates back to the very origins of the subject. In this survey we shall try to give an overview of some of the main results, without any claims to being exhaustive. For other perspectives, we refer the reader to Ba], DP], PP1] and Ru5]. We begin by mentioning two results which illustrate an important theme in the subject. Given a diieomorphism f : M ! M of a compact manifold it is a classical problem to count the number of periodic orbits fx; fx; : : : ; f n?1 xg, with f n x = x. For each n 1, we denote by N f (n) the number of periodic points of period n. An important approach to studying the number N f (n) of isolated periodic points was proposed by Artin and Mazur AM]. They deened a function of a single complex variable z as follows.