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The “prime number theorem” for the periodic orbits of a Bernoulli flow

as x -s oo. Margulis' proof, unfortunately, has never been published in English. In 1976, Hejhal [6] gave a proof of Margulis' theorem based on the Selberg trace formula. Since Margulis' announcement statistical regularities in the distributions of periodic orbits have been discovered for various flows. For a weakly mixing Axiom A flow restricted to a basic set, Parry and Pollicott [10] (following earlier work by Bowen [1], [2]) proved that the number of periodic orbits with minimal period not exceeding x is asymptotic to ehx/hx as x -s o, where h is the topological entropy of the flow. Sarnak [11] proved a similar result for the horocycle flow. These results closely resemble the prime number theorem. This is no accident: the proofs in [6], [10], and [11] all use suitable zeta functions together with some of the standard machinery of analytic number theory. In a recent paper Parry [9] has taken this approach one step further and produced an analogue of the Dirichlet density theorem for periodic orbits of Axiom A flows. In this article I shall present a similar result for a very simple flow, the so-called Bernoulli flow. I shall use only elementary techniques of asymptotic analysis: no zeta functions, no Tauberian theorems, no heavy machinery. This approach has the advantage that it leads to some interesting results concerning the distributions of individual periodic orbits, results which have no analogues in analytic number theory. In another paper [7] I have shown that this elementary method can be adapted to the general Axiom A flow, giving similar results. No knowledge of geometry or dynamical systems is necessary to understand this paper.