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January  2014, 8(1): 75-91. doi: 10.3934/jmd.2014.8.75

Topological entropy of minimal geodesics and volume growth on surfaces

Eva Glasmachers 1, , Gerhard Knieper 1, , Carlos Ogouyandjou 2, and  Jan Philipp Schröder 1,

1. 

Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany, Germany
2. 

Institut de Mathématiques et de Sciences Physiques (IMSP), Université d’Abomey-Calavi 01 BP 613 Porto-Novo, Benin

Received  August 2013 Revised  March 2014 Published  July 2014

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: volume growth., topological entropy, Geodesic flows on surfaces.
Mathematics Subject Classification: Primary: 37A35, 37D40; Secondary: 53D2.
Citation: Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75
References:
[1]

R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331. doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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/. Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[8]

W. Klingenberg, Geodätischer Fluss auf Mannigfaltigkeiten vom hyperbolischen Typ, Invent. Math., 14 (1971), 63-82. doi: 10.1007/BF01418743.  Google Scholar

[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.  Google Scholar

[10]

A. Manning, Topological entropy for geodesic flows, Annals of Math. (2), 110 (1979), 567-573. doi: 10.2307/1971239.  Google Scholar

[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.  Google Scholar

[12]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

show all references

References:
[1]

R. Bowen, Entropy-expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331. doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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/. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

W. Klingenberg, Geodätischer Fluss auf Mannigfaltigkeiten vom hyperbolischen Typ, Invent. Math., 14 (1971), 63-82. doi: 10.1007/BF01418743.  Google Scholar

[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.  Google Scholar

[10]

A. Manning, Topological entropy for geodesic flows, Annals of Math. (2), 110 (1979), 567-573. doi: 10.2307/1971239.  Google Scholar

[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.  Google Scholar

[12]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

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