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

Topological entropy of minimal geodesics and volume growth on surfaces

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.
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.

[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.

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.

[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.

[1]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147

[2]

Bryce Weaver. Growth rate of periodic orbits for geodesic flows over surfaces with radially symmetric focusing caps. Journal of Modern Dynamics, 2014, 8 (2) : 139-176. doi: 10.3934/jmd.2014.8.139

[3]

Keith Burns, Katrin Gelfert. Lyapunov spectrum for geodesic flows of rank 1 surfaces. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1841-1872. doi: 10.3934/dcds.2014.34.1841

[4]

David Ralston, Serge Troubetzkoy. Ergodic infinite group extensions of geodesic flows on translation surfaces. Journal of Modern Dynamics, 2012, 6 (4) : 477-497. doi: 10.3934/jmd.2012.6.477

[5]

Katrin Gelfert. Non-hyperbolic behavior of geodesic flows of rank 1 surfaces. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 521-551. doi: 10.3934/dcds.2019022

[6]

César J. Niche. Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 577-580. doi: 10.3934/dcds.2004.11.577

[7]

Radu Saghin. Volume growth and entropy for $C^1$ partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3789-3801. doi: 10.3934/dcds.2014.34.3789

[8]

Vladislav Kruglov, Dmitry Malyshev, Olga Pochinka. Topological classification of $Ω$-stable flows on surfaces by means of effectively distinguishable multigraphs. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4305-4327. doi: 10.3934/dcds.2018188

[9]

Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273

[10]

Vladislav Kibkalo, Tomoo Yokoyama. Topological characterizations of Morse-Smale flows on surfaces and generic non-Morse-Smale flows. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 4787-4822. doi: 10.3934/dcds.2022072

[11]

François Ledrappier, Omri Sarig. Fluctuations of ergodic sums for horocycle flows on $\Z^d$--covers of finite volume surfaces. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 247-325. doi: 10.3934/dcds.2008.22.247

[12]

Ilesanmi Adeboye, Harrison Bray, David Constantine. Entropy rigidity and Hilbert volume. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 1731-1744. doi: 10.3934/dcds.2019075

[13]

François Ledrappier, Seonhee Lim. Volume entropy of hyperbolic buildings. Journal of Modern Dynamics, 2010, 4 (1) : 139-165. doi: 10.3934/jmd.2010.4.139

[14]

Marcelo R. R. Alves. Positive topological entropy for Reeb flows on 3-dimensional Anosov contact manifolds. Journal of Modern Dynamics, 2016, 10: 497-509. doi: 10.3934/jmd.2016.10.497

[15]

Dieter Mayer, Fredrik Strömberg. Symbolic dynamics for the geodesic flow on Hecke surfaces. Journal of Modern Dynamics, 2008, 2 (4) : 581-627. doi: 10.3934/jmd.2008.2.581

[16]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete and Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555

[17]

Jaume Llibre. Brief survey on the topological entropy. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363

[18]

Wooyeon Kim, Seonhee Lim. Notes on the values of volume entropy of graphs. Discrete and Continuous Dynamical Systems, 2020, 40 (9) : 5117-5129. doi: 10.3934/dcds.2020221

[19]

Alfonso Artigue. Expansive flows of surfaces. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 505-525. doi: 10.3934/dcds.2013.33.505

[20]

Tomoo Yokoyama. Refinements of topological invariants of flows. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2295-2331. doi: 10.3934/dcds.2021191

2021 Impact Factor: 0.641

Metrics

  • PDF downloads (264)
  • HTML views (0)
  • Cited by (4)

[Back to Top]