# American Institute of Mathematical Sciences

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

## Ergodicity and topological entropy of geodesic flows on surfaces

 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.
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

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##### 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
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