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An analogue of the prime number theorem for closed orbits of Axiom A flows. (English) Zbl 0537.58038
The connection between the distribution of periods of the periodic orbits of a flow and the distribution of the primes in analytic number theory is the zeta function which, for a flow $$\phi$$, is defined as follows: Let $$\tau$$ denote a generic periodic orbit and $$\lambda$$ ($$\tau)$$ its least period. Set $$N(\tau)=e^{\lambda(\tau)}$$, then $\zeta(s)=\prod_{\tau}1/(1-N(\tau)^{-s}).$ The main result of the paper is the analogue of the prime number theorem for flows as conjectured by R. Bowen [Am. J. Math. 94, 413-423 (1972; Zbl 0249.53033)]. Let $$\phi$$ be an Axiom A flow restricted to a basic set with topological entropy h. If $$\phi$$ is topologically weak-mixing then $\Pi(x)=\#\{\tau:\quad e^{\lambda(\tau)}\leq x\}\sim x/\log x.$ A key indepently interesting step in the proof is, under the same hypotheses on $$\phi$$, that $$\zeta$$ has a nowhere vanishing analytic extension to an open neighborhood of R(s)$$\geq h$$ except for a simple pole at $$s=h$$. In addition, an asymptotic estimate for $$\Pi$$ (x) is given in the case when $$\phi$$ is not topologically weak-mixing.
Reviewer: C.Chicone

##### MSC:
 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics 11N05 Distribution of primes
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