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The Chebotarov theorem for Galois coverings of Axiom A flows

Published online by Cambridge University Press:  19 September 2008

William Parry  and
Mark Pollicott
William Parry
Affiliation:
Mathematics Institute, University of Warwick, Coventry, CV4 7AL, England
Mark Pollicott
Affiliation:
Department of Mathematics, University of Edinburgh, Edinburgh, EH9 3JZ, Scotland
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Abstract

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We consider G (Galois) coverings of Axiom A flows (restricted to basic sets) and prove an analogue of Chebotarev's theorem. The theorem provides an asymptotic formula for the number of closed orbits whose Frobenius class is a given conjugacy class in G. An application answers a question raised by J. Plante. The basic method is then extended to compact group extensions and applied to frame bundle flows defined on manifolds of variable negative curvature.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 1 , March 1986 , pp. 133 - 148
Copyright
Copyright © Cambridge University Press 1986

References

REFERENCES

[1]Bowen, R.. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc. 153 (1971), 401–414.CrossRefGoogle Scholar
[2]Bowen, R.. Periodic orbits for hyperbolic flows. Amer. J. Math. 94 (1972), 1–30.CrossRefGoogle Scholar
[3]Bowen, R.. Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95 (1973), 429–459.CrossRefGoogle Scholar
[4]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Maths. 470. Springer: Heidelberg, 1975.CrossRefGoogle Scholar
[5]Brauer, R.. On Artin's L-series with general group characters. Annals of Math. 48 (1947), 502–514.CrossRefGoogle Scholar
[6]Brin, M.. Ergodic theory of frame flows. In Ergodic theory and dynamical systems, II (ed. Katok, A.). Birkhauser: Basel, 1982.Google Scholar
[7]Gangolli, R.. Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank one. Illinois J. Math. 21 (1977), 1–41.CrossRefGoogle Scholar
[8]Hejhal, D. A.. The Selberg trace formula and Riemann zeta function. Duke Math. J. 43 (1976), 441–482.CrossRefGoogle Scholar
[9]Huber, H.. Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen, II. Math. Ann. 142 (1961), 385–398.CrossRefGoogle Scholar
[10]Manning, A.. Axiom A diffeomorphisms have rational zeta functions. Bull. London Math. Soc. 3 (1971), 215–220.CrossRefGoogle Scholar
[11]Margulis, G. A.. Applications of ergodic theory to the investigation of manifolds of negative curvature. Funkt. Anal. i Ego Pril. 3 (1969), 89–90.Google Scholar
[12]Parry, W.. An analogue of the prime number theorem for closed orbits of shifts of finite type and their suspensions. Israel J. Math. 45 (1983), 41–52.CrossRefGoogle Scholar
[13]Parry, W.. Bowen's equidistribution theory and the Dirichlet density theorem. Ergod. Th. & Dynam. Sys. 4 (1984), 117–134.CrossRefGoogle Scholar
[14]Parry, W. & Pollicott, M.. An analogue of the prime number theorem for closed orbits of Axiom A flows. Annals. of Math. 118 (1983), 573–591.CrossRefGoogle Scholar
[15]Plante, J. F.. Homology of closed orbits of Anosov flows. Proc. Amer. Math. Soc. 37 (1973), 297–300.CrossRefGoogle Scholar
[16]Pollicott, M.. A complex Ruelle-Perron-Frobenius theorem and two counter-examples. Ergod. Th. & Dynam. Sys. 4 (1984), 135–146.CrossRefGoogle Scholar
[17]Rudolph, D. J.. Classifying the isometric extensions of a Bernoulli shift. J. d' Analyse Mathematique 34 (1978), 36–60.CrossRefGoogle Scholar
[18]Ruelle, D.. Thermodynamic Formalism. Addison-Wesley: Reading, Mass., 1978.Google Scholar
[19]Sarnak, P.. Class numbers of indefinite binary quadratic forms. J. Number Theory 15 (1982), 229–247.CrossRefGoogle Scholar
[20]Selberg, A.. Harmonic analysis and discontinuous subgroups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc. 20 (1956), 47–87.Google Scholar
[21]Sunada, T.. Geodesic flows and geodesic random walks. Preprint.Google Scholar
[22]Wiener, N.. The Fourier integral and certain of its applications. C.U.P., Cambridge, 1967.Google Scholar