Topological entropyof minimal geodesics  and volume growth on surfaces

Eva Glasmachers; Gerhard Knieper; Carlos Ogouyandjou; Jan Philipp Schröder

doi:10.3934/jmd.2014.8.75

Article Contents

2014, Volume 8, Issue 1: 75-91. Doi: 10.3934/jmd.2014.8.75

This issue Previous Article Loci in strata of meromorphic quadratic differentials with fully degenerate Lyapunov spectrum Next Article Minimal yet measurable foliations

Topological entropy of minimal geodesics and volume growth on surfaces

1.
Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum
2.
Institut de Mathématiques et de Sciences Physiques (IMSP), Université d’Abomey-Calavi 01 BP 613 Porto-Novo

Received: August 2013

Revised: March 2014

Published: July 2014

Abstract / Introduction Related Papers Cited by

Abstract

Let $(M,g)$ be a compact Riemannian manifold of hyperbolic type, i.e $M$ is a manifold admitting another metric of strictly negative curvature. In this paper we study the geodesic flow restricted to the set of geodesics which are minimal on the universal covering. In particular for surfaces we show that the topological entropy of the minimal geodesics coincides with the volume entropy of $(M,g)$ generalizing work of Freire and Mañé.

Keywords:

Mathematics Subject Classification: Primary: 37A35, 37D40; Secondary: 53D25.

Citation:

Related Papers

Cited by

References

[1]	R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.doi: 10.1090/S0002-9947-1972-0285689-X.
[2]	V. Bangert, Mather sets for twist maps and geodesics on tori, in Dynamics Reported, Vol. 1, Dynam. Report. Ser. Dynam. Systems Appl., 1, Wiley, Chichester, 1988, 1-56.
[3]	G. Contreras, R. Iturriaga, G. P. Paternain and M. Paternain, Lagrangian graphs, minimizing measures and Mañé's critical values, Geometric and Functional Analysis, 8 (1998), 788-809.doi: 10.1007/s000390050074.
[4]	A. Freire and R. Mañé, On the entropy of the geodesic flow in manifolds without conjugate points, Invent. Math., 69 (1982), 375-392.doi: 10.1007/BF01389360.
[5]	E. Glasmachers, Characterization of Riemannian Metrics on $T^2$ with and without Positive Topological Entropy, Ph.D thesis, Ruhr-Universität Bochum, 2007. Available from: http://www-brs.ub.ruhr-uni-bochum.de/netahtml/HSS/Diss/GlasmachersEva/.
[6]	G. A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. of Math. (2), 33 (1932), 719-739.doi: 10.2307/1968215.
[7]	A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.doi: 10.1017/CBO9780511809187.
[8]	W. Klingenberg, Geodätischer Fluss auf Mannigfaltigkeiten vom hyperbolischen Typ, Invent. Math., 14 (1971), 63-82.doi: 10.1007/BF01418743.
[9]	G. Knieper, Hyperbolic dynamics and riemannian geometry, in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 453-545.doi: 10.1016/S1874-575X(02)80008-X.
[10]	A. Manning, Topological entropy for geodesic flows, Annals of Math. (2), 110 (1979), 567-573.doi: 10.2307/1971239.
[11]	M. Morse, A fundamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc., 26 (1924), 25-60.doi: 10.1090/S0002-9947-1924-1501263-9.
[12]	P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.