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Entropy and closed geodesies

Published online by Cambridge University Press:  19 September 2008

A. Katok
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742, USA
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Abstract

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We study asymptotic growth of closed geodesies for various Riemannian metrics on a compact manifold which carries a metric of negative sectional curvature. Our approach makes use of both variational and dynamical description of geodesies and can be described as an asymptotic version of length-area method. We also obtain various inequalities between topological and measure-theoretic entropies of the geodesic flows for different metrics on the same manifold. Our method works especially well for any metric conformally equivalent to a metric of constant negative curvature. For a surface with negative Euler characteristics every Riemannian metric has this property due to a classical regularization theorem. This allows us to prove that every metric of non-constant curvature has strictly more close geodesies of length at most T for sufficiently large T then any metric of constant curvature of the same total area. In addition the common value of topological and measure-theoretic entropies for metrics of constant negative curvature with the fixed area separates the values of two entropies for other metrics with the same area.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1982

References

REFERENCES

[1]Anosov, D. V.. Geodesic flows on closed Riemannian manifolds of negative curvature. Proc. Steklov Inst. Math. 90 (1967).Google Scholar
[2]Berger, M.. Lectures on geodesies in Riemannian Geometry, Tata Institute, Bombay, 1965.Google Scholar
[3]Berger, M.. Some relations between volume, injectivity radius and convexity radius in Riemannian manifolds. Differential Geometry and Relativity. Reidel: Dordrecht-Boston, 1976, 3342.CrossRefGoogle Scholar
[4]Birkhoff, G. D.. Dynamical systems. A.M.S. Colloquium Publications 9. A.M.S.: New York, 1927.Google Scholar
[5]Bishop, R. & Crittenden, R.. Geometry of Manifolds. Academic Press: New York, 1964.Google Scholar
[6]Bowen, R.. Periodic orbits for hyperbolic flows. Amer. J. Math. 94 (1972), 130.CrossRefGoogle Scholar
[7]Denker, M., Grillenberger, C. & Sigmund, K.. Ergodic theory on compact spaces. Lecture Notes in Math. No 527, Springer: Berlin, 1976.Google Scholar
[8]Dinaburg, E. I.. On the relations among various entropy characteristics of dynamical systems. Math. USSR, Izv. 5 (1971), 337378.CrossRefGoogle Scholar
[9]Gromoll, D., Klingenberg, W. & Meyer, W.. Riemannische Geometrie in Groβen. Springer: Berlin, 1968.CrossRefGoogle Scholar
[10]Hedlund, G. A.. The dynamics of geodesic flows. Bull. Amer. Math. Soc. 45 (1939), 241260.CrossRefGoogle Scholar
[11]Helgason, S.. Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press: New York, 1978.Google Scholar
[12]Jenkins, J. A.. Univalent Functions and Conformal Mappings. Springer: Berlin, 1958.Google Scholar
[13]Katok, A.. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Publ. Math. IHES 51 (1980), 137173.CrossRefGoogle Scholar
[14]Katok, A.. Counting closed geodesies on surfaces, Mathematische Arbeitstagung, 1980, Universitat Bonn.Google Scholar
[15]Katok, A.. Closed geodesies and ergodic theory. London Math. Soc. Symp. on Ergodic Theory, Durham 1980. Abstracts, University of Warwick, 1980.Google Scholar
[16]Katok, A.. Lyapunov exponents, entropy, hyperbolic sets and ɛ-orbits. (In preparation.)Google Scholar
[17]Manning, A.. Topological entropy for geodesic flows. Ann. Math. 110 (1979), 567573.CrossRefGoogle Scholar
[18]Manning, A.. Curvature bound for the entropy of the geodesic flow on a surface. (Preprint, 1980.)Google Scholar
[19]Margulis, G. A.. Applications of ergodic theory to the investigation of manifolds of negative curvature. Fund. Anal. Appl. 3 (1969), 335336. (Translated from Russian.)CrossRefGoogle Scholar
[20]Margulis, G. A.. On some problems in the theory of U-systems Dissertation, Moscow State University, 1970. (In Russian.)Google Scholar
[21]Morse, M.. The calculus of variations in the large. A.M.S. Colloquium Publications 18 A.M.S.: New York, 1936.Google Scholar
[22]Morse, M.. A fundamental class of geodesies in any closed surface of genus greater than one. Trans. A.M.S. 26 (1924), 2561.CrossRefGoogle Scholar
[23]Pesin, Ja. B.. Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surveys 32 (1977), 4, 55114.CrossRefGoogle Scholar
[24]Poincaré, H.. Sur les lignes geodesigue des surfaces convexes. Trans. Amer. Math. Soc. 6 (1905), 237274.Google Scholar
[25]Ruelle, D.. An inequality for the entropy of differentiable map. Bol. Soc. Bras. Mat. 9 (1978), 8387.CrossRefGoogle Scholar
[26]Sarnak, P.. Entropy estimates for geodesic flows. Ergod. Th. & Dynam. Sys. 2 (1982).CrossRefGoogle Scholar
[27]Schiffer, M. & Spencer, D. C.. Functionals on Finite Riemannian Surfaces. Princeton Univ. Press: Princeton, 1954.Google Scholar
[28]Selberg, A.. Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Ind. Math. Soc. 20 (1956), 4787.Google Scholar
[29]Sinai, Ja. G.. The asymptotic behaviour of the number of closed geodesies on a compact manifold of negative curvature. Izv. Akad. NaukSSSR, Ser. Math. 30 (1966), 12751295.Google Scholar
English translation, A.M.S. Trans. 73 2 (1968), 229250.Google Scholar