Distribution of Periodic Orbits of Symbolic and Axiom A Flows

Indeed, Margulis [lo] announced that for the geodesic flow on a d-dimensional compact manifold of curvature 1 the number of periodic orbits 7 with (minimal) period r(1) I x is asymptotic to ecd-‘jx/(d 1)x. This result bears a striking resemblance to the prime number theorem. Parry and Pollicott [12], following earlier work by Bowen [2, 41, generalized Margulis’ theorem to weakly mixing Axiom A flows, proving that # { 7: r(1) I x} eh*/hx, where h is the topological entropy of the flow. Sarnak [15] has related results for the horocycle flow. Bowen [3] and Parry [ll] proved analogues of the Dirichlet density theorem for mixing Axiom A flows, e.g., if 7(G) represents the integral of the continuous function G over one period of 7, then Z roj s ,7(G)/~(l) (eh”/hx)lG dji, where ji is the invariant probability measure of maximum entropy. This paper pursues an altogether different analogy, this between the distribution problems for periodic orbits of Axiom A and symbolic flows and those of classical probability theory. This analogy leads to theorems which apparently have no counterparts in number theory. Moreover, it leads to techniques quite different from those commonly used in studying periodic orbits: in particular, there is no use of zeta functions or any of the attendant Tauberian theorems. We do not believe that the main results of this paper can be obtained by analyzing zeta functions. These results do, however, make use of the groundwork done by Bowen in [4], which reduces