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Bowen’s equidistribution theory and the Dirichlet density theorem. (English) Zbl 0567.58014
Let $$\phi$$ be an axiom A flow restricted to a basic set, let g be a $$C^{\infty}$$ function and let $$\pi_ g(x)=\sum_{e^{\lambda (\tau)h}\leq x}(\lambda_ g(\tau)/\lambda (\tau))$$, where $$\lambda_ g(\tau)$$ is the g length of the closest orbit $$\tau$$, $$\lambda$$ ($$\tau)$$ is the period of $$\tau$$ and h is the topological entropy of $$\phi$$. We obtain an asymptotic formula for $$\pi_ g$$ which includes the ”prime number” theorem for closed orbits. This result generalizes Bowen’s theorem on the equidistribution of closed orbits. After establishing an analytic extension result for certain zeta functions the proofs proceed by orthodox number theoretical techniques.

##### MSC:
 37A99 Ergodic theory 37C10 Dynamics induced by flows and semiflows 28D20 Entropy and other invariants 11R45 Density theorems
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##### References:
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