Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-24T11:57:10.047Z Has data issue: false hasContentIssue false
  • English
  • Français

Entropy and closed geodesies

Published online by Cambridge University Press:  19 September 2008

A. Katok
A. Katok
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland 20742, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Information
Ergodic Theory and Dynamical Systems , Volume 2 , Issue 3-4 , December 1982 , pp. 339 - 365
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, 33–42.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), 1–30.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), 337–378.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), 241–260.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), 137–173.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), 567–573.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), 335–336. (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), 25–61.CrossRefGoogle Scholar
[23]Pesin, Ja. B.. Characteristic Lyapunov exponents and smooth ergodic theory. Russ. Math. Surveys 32 (1977), 4, 55–114.CrossRefGoogle Scholar
[24]Poincaré, H.. Sur les lignes geodesigue des surfaces convexes. Trans. Amer. Math. Soc. 6 (1905), 237–274.Google Scholar
[25]Ruelle, D.. An inequality for the entropy of differentiable map. Bol. Soc. Bras. Mat. 9 (1978), 83–87.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), 47–87.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), 1275–1295.Google Scholar
English translation, A.M.S. Trans. 73 2 (1968), 229–250.Google Scholar