An analogue of the prime number theorem for closed orbits of Axiom A flows

Abstract

For an Axiom A flow restricted to a basic set we extend the zeta function to an open set containing $\mathscr{R}(s) \ge h$ where $h$ is the topological entropy. This enables us to give an asymptotic formula for the number of closed orbits by adapting the Wiener-Ikehara proof of the prime number theorem.

For a geodesic flow on a $d$ dimensional compact manifold $M^d$ of constant negative curvature $(\kappa = -1)$, Margulis [9] announced a proof that the number of closed orbits $\tau$ with least period $\lambda(\tau)$ not exceeding $x$ is asymptotic to $e^{(d-1)x}/(d-1)x$. (See also Hejhal [7] who provides a proof based on the Selberg zeta function.). This is a special case of the thoerem which, according to Alexeev and Jacobson [2], appears in Margulis’s dissertation and which shows that
\[
\#\{\tau\colon \lambda(\tau) \le x\} \sim e^{hx}/hx
\]
for Anosov flows, where $h$ is the topological entropy. (Presumably the flow is assumed to be mixing.)

For Axiom A flows Bowen, [3] (see also Sinai [14]), proved the existence of positive constants $A$, $B$ such that
\[
A \cdot e^{hx}/hx \le \#\{\tau\colon \lambda(\tau) \le x\} \le B \cdot e^{hx}/hx,
\]
and conjectured Margulis’s more precise result for (topologically weak-mixing) flows of this type. Equivalently he conjectured that
$$
\pi(x) \equiv \#\{\tau\colon N_h(\tau) \le x\} \sim x/\log x \tag{0.1}
$$
where $N_h(\tau) = e^{\lambda(\tau)h}$.

Our aim is to prove (0,1) for (basic sets of) topologically weak-mixing Axiom A flows (Theorem 2). A modified asymptotic formula for Axiom A flows which are not topologically weak-mixing is also established.

Sarnak and Woo (cf. [13]) obtain very precise asymptotic estimates for $\pi(x)$ for geodesic flows on non-compact manifolds with finite volume, a case not covered by our results in their present form.

As one would expect in this area the proof of (0.1) depends very heavily on the work of Bowen [5]. We have stated the result in the form (0.1) to emphasise the analogy with the prime number theorem, and in fact our proof follows the main lines of the Wiener-Ikehara proof of that theorem once the relevant properties of an appropriate zeta function have been gathered. Especially relevant is the analytic extension of the zeta function to an open domain containing $\mathscr{R}(s)\ge h$, $s\ne h$ (Theorem 1). (When the flow is not topologically weak-mixing, points $h+ nit_0$, $n\in \mathbf{Z}$, have to be excluded, for some $t_0 > 0$.). The possibility of this extension almost answers a question raised by Ruelle in [7]. It will be clear that we utilise many of Ruelle’s thermodynamics ideas to establish properies ofo the zeta function.

We wish to thank C. Series, R. Spatzier and P. Walters for various enlightening remarks and references in connection with our work. Our special thanks go to D. Ruelle who helped us with the proof of his theorem (Proposition 11).

This work is a generalisation to Axiom A flows of results achieved for suspensions, by locally constant function, of shifts of finite type (cf. [11]).

Authors

William Parry

Mark Pollicott