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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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[1] DOI: 10.1007/BF02760669 · Zbl 0552.28020 · doi:10.1007/BF02760669
[2] DOI: 10.2307/2006982 · Zbl 0537.58038 · doi:10.2307/2006982
[3] DOI: 10.1112/blms/3.2.215 · Zbl 0219.58007 · doi:10.1112/blms/3.2.215
[4] DOI: 10.1215/S0012-7094-76-04338-6 · Zbl 0346.10010 · doi:10.1215/S0012-7094-76-04338-6
[5] Dunford, Linear Operators I (1967) · Zbl 0056.34601
[6] DOI: 10.2307/2373793 · Zbl 0282.58009 · doi:10.2307/2373793
[7] DOI: 10.2307/2374628 · Zbl 0249.53033 · doi:10.2307/2374628
[8] DOI: 10.2307/2373590 · Zbl 0254.58005 · doi:10.2307/2373590
[9] DOI: 10.1070/RM1967v022n05ABEH001228 · Zbl 0177.42002 · doi:10.1070/RM1967v022n05ABEH001228
[10] Wiener, The Fourier Integral and Certain of its Applications (1933) · Zbl 0006.05401
[11] DOI: 10.1070/RM1972v027n04ABEH001383 · doi:10.1070/RM1972v027n04ABEH001383
[12] Selberg, J. Indian Math. Soc. 20 pp 47– (1956)
[13] DOI: 10.1002/cpa.3160340602 · Zbl 0501.58027 · doi:10.1002/cpa.3160340602
[14] Ruelle, Thermodynamic Formalism (1978)
[15] DOI: 10.1007/BF01403069 · Zbl 0329.58014 · doi:10.1007/BF01403069
[16] Pollicott, Ergod. Th. & Dynam. Sys. 4 pp 135– (1984)
[17] Parry, Classification Problems in Ergodic Theory (1982) · Zbl 0487.28014 · doi:10.1017/CBO9780511629389
[18] Margulis, Func. Anal, i Ego Prilozhen 3 pp 89– (1969)
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