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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
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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.

Information

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

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