Ergodicity and topological entropy of geodesic flows on surfaces

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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

Abstract
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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).
[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.
[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.
[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.
[5]	V. Bangert, On the existence of closed geodesics on two-spheres,, Internat. J. Math., 4 (1993), 1. doi: 10.1142/S0129167X93000029.
[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.
[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.
[8]	G. D. Birkhoff, Dynamical Systems,, American Mathematical Society Colloquium Publications, (1927).
[9]	A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology, Classification,, Chapman & Hall/CRC, (2004). doi: 10.1201/9780203643426.
[10]	M. Bonino, Around Brouwer's theory of fixed point free planar homeomorphisms,, Notes de cours de l'École d'été, (2006).
[11]	M. Brown, A new proof of Brouwer's lemma on translation arcs,, Houston J. Math., 10 (1984), 35.
[12]	E. I. Dinaburg, On the relations among various entropy characteristics of dynamical systems,, Math. USSR Izv., 5 (1971), 337. doi: 10.1070/IM1971v005n02ABEH001050.
[13]	V. J. Donnay, Geodesic flow on the two-sphere. II. Ergodicity,, in Dynamical Systems, (1342), 112. doi: 10.1007/BFb0082827.
[14]	H. Duan and Y. Long, A remark on the existence of closed geodesics on symmetric Finsler 2-spheres,, 2012. Available from: , (): 201202.
[15]	J. Franks, Geodesics on $\mathbbS^2$ and periodic points of annulus homeomorphisms,, Invent. Math., 108 (1992), 403. doi: 10.1007/BF02100612.
[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.
[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.
[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.
[19]	M. A. Grayson, Shortening embedded curves,, Ann. of Math. (2), 129 (1989), 71. doi: 10.2307/1971486.
[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.
[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.
[22]	M. W. Hirsch, Differential Topology,, Graduate Texts in Mathematics, (1976).
[23]	A. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems,, Math. USSR Izv., 7 (1973), 535.
[24]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137.
[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.
[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.
[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.
[28]	J. P. Schröder, Global minimizers for Tonelli Lagrangians on the 2-torus,, J. Topol. Anal., 7 (2015), 261. doi: 10.1142/S1793525315500090.
[29]	Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285. doi: 10.1007/BF02766215.

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References:

[1]	S. Alpern and V. S. Prasad, Typical Dynamics of Volume Preserving Homeomorphisms,, Cambridge Tracts in Mathematics, (2000).
[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.
[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.
[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.
[5]	V. Bangert, On the existence of closed geodesics on two-spheres,, Internat. J. Math., 4 (1993), 1. doi: 10.1142/S0129167X93000029.
[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.
[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.
[8]	G. D. Birkhoff, Dynamical Systems,, American Mathematical Society Colloquium Publications, (1927).
[9]	A. V. Bolsinov and A. T. Fomenko, Integrable Hamiltonian Systems. Geometry, Topology, Classification,, Chapman & Hall/CRC, (2004). doi: 10.1201/9780203643426.
[10]	M. Bonino, Around Brouwer's theory of fixed point free planar homeomorphisms,, Notes de cours de l'École d'été, (2006).
[11]	M. Brown, A new proof of Brouwer's lemma on translation arcs,, Houston J. Math., 10 (1984), 35.
[12]	E. I. Dinaburg, On the relations among various entropy characteristics of dynamical systems,, Math. USSR Izv., 5 (1971), 337. doi: 10.1070/IM1971v005n02ABEH001050.
[13]	V. J. Donnay, Geodesic flow on the two-sphere. II. Ergodicity,, in Dynamical Systems, (1342), 112. doi: 10.1007/BFb0082827.
[14]	H. Duan and Y. Long, A remark on the existence of closed geodesics on symmetric Finsler 2-spheres,, 2012. Available from: , (): 201202.
[15]	J. Franks, Geodesics on $\mathbbS^2$ and periodic points of annulus homeomorphisms,, Invent. Math., 108 (1992), 403. doi: 10.1007/BF02100612.
[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.
[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.
[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.
[19]	M. A. Grayson, Shortening embedded curves,, Ann. of Math. (2), 129 (1989), 71. doi: 10.2307/1971486.
[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.
[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.
[22]	M. W. Hirsch, Differential Topology,, Graduate Texts in Mathematics, (1976).
[23]	A. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems,, Math. USSR Izv., 7 (1973), 535.
[24]	A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms,, Inst. Hautes Études Sci. Publ. Math., 51 (1980), 137.
[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.
[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.
[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.
[28]	J. P. Schröder, Global minimizers for Tonelli Lagrangians on the 2-torus,, J. Topol. Anal., 7 (2015), 261. doi: 10.1142/S1793525315500090.
[29]	Y. Yomdin, Volume growth and entropy,, Israel J. Math., 57 (1987), 285. doi: 10.1007/BF02766215.

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