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On the existence of periodic orbits for magnetic systems on the two-sphere
Ergodicity and topological entropy of geodesic flows on surfaces
1. | Faculty of Mathematics, Ruhr University Bochum, Universitätsstraße 150, 44780 Bochum, Germany |
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). Google Scholar |
[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. Google Scholar |
[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. |
show all references
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). Google Scholar |
[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. Google Scholar |
[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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