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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 |
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. |
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