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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

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