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Bowen's equidistribution theory and the Dirichlet density theorem

Published online by Cambridge University Press:  19 September 2008

William Parry
William Parry
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, England
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Abstract

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Let φ be an Axiom A flow restricted to a basic set, let g be a C∞ function and let , where λg(τ) is the g length of the closest orbit τ, λ(τ) is the period of τ and h is the topological entropy of φ. We obtain an asymptotic formula for πg 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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 4 , Issue 1 , March 1984 , pp. 117 - 134
Copyright
Copyright © Cambridge University Press 1984

References

REFERENCES

[1]Anosov, D. V. & Sinai, Ja. G.. Some smooth ergodic systems. Russian Math. Surveys No. 5, 22 (1967) 103–167.CrossRefGoogle Scholar
[2]Bowen, R.. Periodic orbits for hyperbolic flows. Amer. J. Math. 94 (1972) 1–30.CrossRefGoogle Scholar
[3]Bowen, R.. The equidistribution of closed geodesies. Amer. J. Math. 94 (1972) 413–423.Google Scholar
[4]Bowen, R.. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973) 429–450.CrossRefGoogle Scholar
[5]Dunford, N. & Schwartz, J. T.. Linear Operators I. John Wiley (Interscience): New York, 1967.Google Scholar
[6]Hejhal, D. A.. The Selberg trace formula and the Riemann zeta function. Duke Math. J. 43 (1976), 441–482.Google Scholar
[7]Manning, A.. Axiom A diffeomorphisms have rational zeta functions. Bull. London Math. Soc. 3 (1971) 215–220.CrossRefGoogle Scholar
[8]Margulis, G. A.. Applications of ergodic theory to the investigation of manifolds of negative curvature. Func. Anal, i Ego Prilozhen. 3 No. 4. (1969), 89–90.Google Scholar
[9]Parry, W.. An analogue of the prime number theorem for shifts of finite type and their suspensions. Israel J. Math. 45 (1983), 41–52.Google Scholar
[10]Parry, W. & Pollicott, M.. An analogue of the prime number theorem for closed orbits of Axiom A flows. Ann. of Math. 118 (1983), 573–591.CrossRefGoogle Scholar
[11]Parry, W. & Tuncel, S.. Classification Problems in Ergodic Theory. London Math. Soc. Lecture Notes 67, Cambridge University Press: Cambridge (1982).CrossRefGoogle Scholar
[12]Pollicott, M.. A complex Ruelle operator theorem and two counter examples. Ergod. Th. & Dynam. Sys. 4 (1984), 135–146.CrossRefGoogle Scholar
[13]Ruelle, D.. Zeta functions for expanding maps and Anosov flows. Invent. Math. 34 (1976), 231–242.CrossRefGoogle Scholar
[14]Ruelle, D.. Thermodynamic Formalism. Addison-Wesley: Reading (1978).Google Scholar
[15]Sarnak, P.. Asymptotic behaviour of periodic orbits of the horocycle flow and Eisenstein series. Comm. Pure and Applied Math. 34 (1981), 719–739.CrossRefGoogle Scholar
[16]Selberg, A.. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20 (1956) 47–87.Google Scholar
[17]Sinai, Ja. G.. Gibbs measures in ergodic theory. Russian Math. Surveys No. 4 (166) (1972), 21–64.Google Scholar
[18]Sunada, T.. Geodesic flows and geodesic random walks. To appear.Google Scholar
[19]Wiener, N.. The Fourier Integral and Certain of its Applications. Cambridge University Press: Cambridge, 1933.Google Scholar