American Institute of Mathematical Sciences

Advanced
2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147

Ergodicity and topological entropy of geodesic flows on surfaces

Jan Philipp Schröder 1,

1. 

Faculty of Mathematics, Ruhr University Bochum, Universitätsstraße 150, 44780 Bochum, Germany

Received  February 2015 Revised  May 2015 Published  August 2015

We consider reversible Finsler metrics on the 2-sphere and the 2-torus, whose geodesic flow has vanishing topological entropy. Following a construction of A. Katok, we discuss examples of Finsler metrics on both surfaces with large ergodic components for the geodesic flow in the unit tangent bundle. On the other hand, using results of J. Franks and M. Handel, we prove that ergodicity and dense orbits cannot occur in the full unit tangent bundle of the 2-sphere, if the Finsler metric has conjugate points along every closed geodesic. In the case of the 2-torus, we show that ergodicity is restricted to strict subsets of tubes between flow-invariant tori in the unit tangent bundle. The analogous result applies to monotone twist maps.
Keywords: dense orbit, Finsler metrics, monotone twist map, 2-sphere, topological entropy, invariant sets., 2-torus, ergodic component.
Mathematics Subject Classification: Primary: 37J35; Secondary: 37E99, 37A2.
Citation: 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
References:
[1]

S. Alpern and V. S. Prasad, Typical Dynamics of Volume Preserving Homeomorphisms,, Cambridge Tracts in Mathematics, (2000).   Google Scholar

[2]

S. Angenent, Parabolic equations for curves on surfaces: Part II. Intersections, blow-up and generalized solutions,, Ann. of Math. (2), 133 (1991), 171.  doi: 10.2307/2944327.  Google Scholar

[3]

S. Angenent, A remark on the topological entropy and invariant circles of an area preserving twistmap,, in Twist Mappings and Their Applications, (1992), 1.   Google Scholar

[4]

S. Angenent, Self-intersecting geodesics and entropy of the geodesic flow,, Acta Math. Sin. (Engl. Ser.), 24 (2008), 1949.  doi: 10.1007/s10114-008-6439-2.  Google Scholar

[5]

V. Bangert, On the existence of closed geodesics on two-spheres,, Internat. J. Math., 4 (1993), 1.  doi: 10.1142/S0129167X93000029.  Google Scholar

[6]

D. Bao, S.-S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry,, Graduate Texts in Mathematics, (2000).  doi: 10.1007/978-1-4612-1268-3.  Google Scholar

[7]

P. Bernard and C. Labrousse, An entropic characterization of the flat metrics on the two torus,, to appear in Geometriae Dedicata, (2015).  doi: 10.1007/s10711-015-0098-0.  Google Scholar

[8]

G. D. Birkhoff, Dynamical Systems,, American Mathematical Society Colloquium Publications, (1927).   Google Scholar

[9]

A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology, Classification,, Chapman & Hall/CRC, (2004).  doi: 10.1201/9780203643426.  Google Scholar

[10]

M. Bonino, Around Brouwer's theory of fixed point free planar homeomorphisms,, Notes de cours de l'École d'été, (2006).   Google Scholar

[11]

M. Brown, A new proof of Brouwer's lemma on translation arcs,, Houston J. Math., 10 (1984), 35.   Google Scholar

[12]

E. I. Dinaburg, On the relations among various entropy characteristics of dynamical systems,, Math. USSR Izv., 5 (1971), 337.  doi: 10.1070/IM1971v005n02ABEH001050.  Google Scholar

[13]

V. J. Donnay, Geodesic flow on the two-sphere. II. Ergodicity,, in Dynamical Systems, (1342), 112.  doi: 10.1007/BFb0082827.  Google Scholar

[14]

H. Duan and Y. Long, A remark on the existence of closed geodesics on symmetric Finsler 2-spheres,, 2012. Available from: , (): 201202.   Google Scholar

[15]

J. Franks, Geodesics on $\mathbbS^2$ and periodic points of annulus homeomorphisms,, Invent. Math., 108 (1992), 403.  doi: 10.1007/BF02100612.  Google Scholar

[16]

J. Franks and M. Handel, Entropy zero area preserving diffeomorphisms of $\mathbbS^2$,, Geom. Topol., 16 (2012), 2187.  doi: 10.2140/gt.2012.16.2187.  Google Scholar

[17]

E. Glasmachers and G. Knieper, Characterization of geodesic flows on $\mathbbT^2$ with and without positive topological entropy,, Geom. Funct. Anal., 20 (2010), 1259.  doi: 10.1007/s00039-010-0087-2.  Google Scholar

[18]

E. Glasmachers and G. Knieper, Minimal geodesic foliation on $\mathbbT^2$ in case of vanishing topological entropy,, J. Topol. Anal., 3 (2011), 511.  doi: 10.1142/S1793525311000623.  Google Scholar

[19]

M. A. Grayson, Shortening embedded curves,, Ann. of Math. (2), 129 (1989), 71.  doi: 10.2307/1971486.  Google Scholar

[20]

A. Harris and G. P. Paternain, Dynamically convex Finsler metrics and $J$-holomorphic embedding of asymptotic cylinders,, Ann. Global Anal. Geom., 34 (2008), 115.  doi: 10.1007/s10455-008-9111-2.  Google Scholar

[21]

G. A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients,, Ann. of Math. (2), 33 (1932), 719.  doi: 10.2307/1968215.  Google Scholar

[22]

M. W. Hirsch, Differential Topology,, Graduate Texts in Mathematics, (1976).   Google Scholar

[23]

A. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems,, Math. USSR Izv., 7 (1973), 535.   Google Scholar

[24]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137.   Google Scholar

[25]

A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems,, Encyclopedia of Mathematics and its Applications, (1995).  doi: 10.1017/CBO9780511809187.  Google Scholar

[26]

G. P. Paternain, Entropy and completely integrable Hamiltonian systems,, Proc. Amer. Math. Soc., 113 (1991), 871.  doi: 10.1090/S0002-9939-1991-1059632-7.  Google Scholar

[27]

J. P. Schröder, Invariant tori and topological entropy in Tonelli Lagrangian systems on the 2-torus,, to appear in Ergodic Theory and Dynamical Systems, (2015).  doi: 10.1017/etds.2014.137.  Google Scholar

[28]

J. P. Schröder, Global minimizers for Tonelli Lagrangians on the 2-torus,, J. Topol. Anal., 7 (2015), 261.  doi: 10.1142/S1793525315500090.  Google Scholar

[29]

Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285.  doi: 10.1007/BF02766215.  Google Scholar

show all references

References:
[1]

S. Alpern and V. S. Prasad, Typical Dynamics of Volume Preserving Homeomorphisms,, Cambridge Tracts in Mathematics, (2000).   Google Scholar

[2]

S. Angenent, Parabolic equations for curves on surfaces: Part II. Intersections, blow-up and generalized solutions,, Ann. of Math. (2), 133 (1991), 171.  doi: 10.2307/2944327.  Google Scholar

[3]

S. Angenent, A remark on the topological entropy and invariant circles of an area preserving twistmap,, in Twist Mappings and Their Applications, (1992), 1.   Google Scholar

[4]

S. Angenent, Self-intersecting geodesics and entropy of the geodesic flow,, Acta Math. Sin. (Engl. Ser.), 24 (2008), 1949.  doi: 10.1007/s10114-008-6439-2.  Google Scholar

[5]

V. Bangert, On the existence of closed geodesics on two-spheres,, Internat. J. Math., 4 (1993), 1.  doi: 10.1142/S0129167X93000029.  Google Scholar

[6]

D. Bao, S.-S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry,, Graduate Texts in Mathematics, (2000).  doi: 10.1007/978-1-4612-1268-3.  Google Scholar

[7]

P. Bernard and C. Labrousse, An entropic characterization of the flat metrics on the two torus,, to appear in Geometriae Dedicata, (2015).  doi: 10.1007/s10711-015-0098-0.  Google Scholar

[8]

G. D. Birkhoff, Dynamical Systems,, American Mathematical Society Colloquium Publications, (1927).   Google Scholar

[9]

A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology, Classification,, Chapman & Hall/CRC, (2004).  doi: 10.1201/9780203643426.  Google Scholar

[10]

M. Bonino, Around Brouwer's theory of fixed point free planar homeomorphisms,, Notes de cours de l'École d'été, (2006).   Google Scholar

[11]

M. Brown, A new proof of Brouwer's lemma on translation arcs,, Houston J. Math., 10 (1984), 35.   Google Scholar

[12]

E. I. Dinaburg, On the relations among various entropy characteristics of dynamical systems,, Math. USSR Izv., 5 (1971), 337.  doi: 10.1070/IM1971v005n02ABEH001050.  Google Scholar

[13]

V. J. Donnay, Geodesic flow on the two-sphere. II. Ergodicity,, in Dynamical Systems, (1342), 112.  doi: 10.1007/BFb0082827.  Google Scholar

[14]

H. Duan and Y. Long, A remark on the existence of closed geodesics on symmetric Finsler 2-spheres,, 2012. Available from: , (): 201202.   Google Scholar

[15]

J. Franks, Geodesics on $\mathbbS^2$ and periodic points of annulus homeomorphisms,, Invent. Math., 108 (1992), 403.  doi: 10.1007/BF02100612.  Google Scholar

[16]

J. Franks and M. Handel, Entropy zero area preserving diffeomorphisms of $\mathbbS^2$,, Geom. Topol., 16 (2012), 2187.  doi: 10.2140/gt.2012.16.2187.  Google Scholar

[17]

E. Glasmachers and G. Knieper, Characterization of geodesic flows on $\mathbbT^2$ with and without positive topological entropy,, Geom. Funct. Anal., 20 (2010), 1259.  doi: 10.1007/s00039-010-0087-2.  Google Scholar

[18]

E. Glasmachers and G. Knieper, Minimal geodesic foliation on $\mathbbT^2$ in case of vanishing topological entropy,, J. Topol. Anal., 3 (2011), 511.  doi: 10.1142/S1793525311000623.  Google Scholar

[19]

M. A. Grayson, Shortening embedded curves,, Ann. of Math. (2), 129 (1989), 71.  doi: 10.2307/1971486.  Google Scholar

[20]

A. Harris and G. P. Paternain, Dynamically convex Finsler metrics and $J$-holomorphic embedding of asymptotic cylinders,, Ann. Global Anal. Geom., 34 (2008), 115.  doi: 10.1007/s10455-008-9111-2.  Google Scholar

[21]

G. A. Hedlund, Geodesics on a two-dimensional Riemannian manifold with periodic coefficients,, Ann. of Math. (2), 33 (1932), 719.  doi: 10.2307/1968215.  Google Scholar

[22]

M. W. Hirsch, Differential Topology,, Graduate Texts in Mathematics, (1976).   Google Scholar

[23]

A. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems,, Math. USSR Izv., 7 (1973), 535.   Google Scholar

[24]

A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137.   Google Scholar

[25]

A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems,, Encyclopedia of Mathematics and its Applications, (1995).  doi: 10.1017/CBO9780511809187.  Google Scholar

[26]

G. P. Paternain, Entropy and completely integrable Hamiltonian systems,, Proc. Amer. Math. Soc., 113 (1991), 871.  doi: 10.1090/S0002-9939-1991-1059632-7.  Google Scholar

[27]

J. P. Schröder, Invariant tori and topological entropy in Tonelli Lagrangian systems on the 2-torus,, to appear in Ergodic Theory and Dynamical Systems, (2015).  doi: 10.1017/etds.2014.137.  Google Scholar

[28]

J. P. Schröder, Global minimizers for Tonelli Lagrangians on the 2-torus,, J. Topol. Anal., 7 (2015), 261.  doi: 10.1142/S1793525315500090.  Google Scholar

[29]

Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285.  doi: 10.1007/BF02766215.  Google Scholar

[1]

Ping-Liang Huang, Youde Wang. Periodic solutions of inhomogeneous Schrödinger flows into 2-sphere. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1775-1795. doi: 10.3934/dcdss.2016074
[2]

Tifei Qian, Zhihong Xia. Heteroclinic orbits and chaotic invariant sets for monotone twist maps. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 69-95. doi: 10.3934/dcds.2003.9.69
[3]

Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81
[4]

Sergei Agapov, Alexandr Valyuzhenich. Polynomial integrals of magnetic geodesic flows on the 2-torus on several energy levels. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6565-6583. doi: 10.3934/dcds.2019285
[5]

Charles Pugh, Michael Shub. Periodic points on the $2$-sphere. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1171-1182. doi: 10.3934/dcds.2014.34.1171
[6]

Pedro Duarte, Silvius Klein. Topological obstructions to dominated splitting for ergodic translations on the higher dimensional torus. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5379-5387. doi: 10.3934/dcds.2018237
[7]

Marie-Claude Arnaud. A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map. Journal of Modern Dynamics, 2011, 5 (3) : 583-591. doi: 10.3934/jmd.2011.5.583
[8]

Qiudong Wang. The diffusion time of the connecting orbit around rotation number zero for the monotone twist maps. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 255-274. doi: 10.3934/dcds.2000.6.255
[9]

Francisco Balibrea, J.L. García Guirao, J.I. Muñoz Casado. A triangular map on $I^{2}$ whose $\omega$-limit sets are all compact intervals of $\{0\}\times I$. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 983-994. doi: 10.3934/dcds.2002.8.983
[10]

Lei Wang, Quan Yuan, Jia Li. Persistence of the hyperbolic lower dimensional non-twist invariant torus in a class of Hamiltonian systems. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1233-1250. doi: 10.3934/cpaa.2016.15.1233
[11]

Yujun Ju, Dongkui Ma, Yupan Wang. Topological entropy of free semigroup actions for noncompact sets. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 995-1017. doi: 10.3934/dcds.2019041
[12]

Patrick Foulon, Vladimir S. Matveev. Zermelo deformation of finsler metrics by killing vector fields. Electronic Research Announcements, 2018, 25: 1-7. doi: 10.3934/era.2018.25.001
[13]

Luigi Ambrosio, Federico Glaudo, Dario Trevisan. On the optimal map in the $ 2 $-dimensional random matching problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7291-7308. doi: 10.3934/dcds.2019304
[14]

Ser Peow Tan, Yan Loi Wong and Ying Zhang. The SL(2, C) character variety of a one-holed torus. Electronic Research Announcements, 2005, 11: 103-110.

[15]

Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104
[16]

José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781
[17]

Dean Crnković, Bernardo Gabriel Rodrigues, Sanja Rukavina, Loredana Simčić. Self-orthogonal codes from orbit matrices of 2-designs. Advances in Mathematics of Communications, 2013, 7 (2) : 161-174. doi: 10.3934/amc.2013.7.161
[18]

Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517
[19]

José Ginés Espín Buendía, Víctor Jiménez Lopéz. A topological characterization of the $\omega$-limit sets of analytic vector fields on open subsets of the sphere. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1143-1173. doi: 10.3934/dcdsb.2019010
[20]

Ali Messaoudi, Rafael Asmat Uceda. Stochastic adding machine and $2$-dimensional Julia sets. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5247-5269. doi: 10.3934/dcds.2014.34.5247

2018 Impact Factor: 0.295

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences