[Le théorème de densité de Pugh pour les flots géodésiques]
Étant donnée une variété riemannienne compacte sans bord, nous démontrons un théorème de densité -générique pour les flots géodésiques et, plus précisément, nous prouvons qu'il existe une partie -résiduelle de l'ensemble des flots géodésiques continus, telle que tout flot dans cette partie admet un ensemble dense d'orbites périodiques.
Given a closed Riemannian manifold, we prove the -general density theorem for continuous geodesic flows. More precisely, we prove that there exists a residual (in the -sense) subset of the continuous geodesic flows such that, in that residual subset, the geodesic flow exhibits dense closed orbits.
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@article{CRMATH_2015__353_6_545_0,
author = {Bessa, M\'ario and Torres, Maria Joana},
title = {The $ {C}^{0}$ general density theorem for geodesic flows},
journal = {Comptes Rendus. Math\'ematique},
pages = {545--549},
publisher = {Elsevier},
volume = {353},
number = {6},
year = {2015},
doi = {10.1016/j.crma.2015.03.012},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/j.crma.2015.03.012/}
}
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AU - Torres, Maria Joana
TI - The $ {C}^{0}$ general density theorem for geodesic flows
JO - Comptes Rendus. Mathématique
PY - 2015
SP - 545
EP - 549
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PB - Elsevier
UR - http://archive.numdam.org/articles/10.1016/j.crma.2015.03.012/
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Bessa, Mário; Torres, Maria Joana. The $ {C}^{0}$ general density theorem for geodesic flows. Comptes Rendus. Mathématique, Tome 353 (2015) no. 6, pp. 545-549. doi : 10.1016/j.crma.2015.03.012. http://archive.numdam.org/articles/10.1016/j.crma.2015.03.012/
[1] Bumpy metrics, Proc. Sympos. Pure Math. (Chern, S.S.; Smale, S., eds.), Volume vol. XIV (1970), pp. 1-3
[2] On generic properties of closed geodesics, Math. USSR, Izv., Volume 21 (1983), pp. 1-29
[3] Uniqueness of generating Hamiltonians for topological Hamiltonian flows, J. Symplectic Geom., Volume 11 (2013) no. 1, pp. 37-52
[4] Genericity of geodesic flows with positive topological entropy on , J. Differ. Geom., Volume 61 (2002) no. 1, pp. 1-49
[5] Chaotic homeomorphisms are generic, Topol. Appl., Volume 102 (2000), pp. 297-302
[6] Category theorems, Fundam. Math., Volume 42 (1955), pp. 276-288
[7] A note on generic properties of continuous maps, Ergodic Theory and Dynamical Systems, II, Prog. Math., vol. 21, Birkhäuser, Boston, MA, USA, 1982, pp. 97-101
[8] Differential Topology, Graduate Texts in Mathematics, vol. 33, Springer, 1976
[9] On proofs of the general density theorem, Proc. Amer. Math. Soc., Volume 124 (1996) no. 4, pp. 1305-1309
[10] Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, 1995
[11] Generic properties of geodesic flows, Math. Ann., Volume 197 (1972) no. 4, pp. 323-334
[12] Topology, vol. 1, Academic Press, 1966
[13] The group of Hamiltonian homeomorphisms and -symplectic topology, J. Symplectic Geom., Volume 5 (2007) no. 2, pp. 167-219
[14] Topological contact dynamics III. Uniqueness of the topological Hamiltonian and -rigidity of the geodesic flow, 2013 (preprint) | arXiv
[15] The group of Hamiltonian homeomorphisms and continuous Hamiltonian flows, Symplectic Topology and Measure Preserving Dynamical Systems, Contemp. Math., vol. 512, Amer. Math. Soc., Providence, RI, USA, 2010, pp. 149-177
[16] Geometric Theory of Dynamical Systems. An Introduction, Springer-Verlag, New York, Berlin, 1982
[17] Genericity theorems in topological dynamics, Dynamical Systems – Warwick 1974, Lecture Notes in Math., vol. 468, Springer, Berlin, 1975, pp. 241-250
[18] An improved closing lemma and a general density theorem, Amer. J. Math., Volume 89 (1967) no. 4, pp. 1010-1021
[19] The closing lemma, including Hamiltonians, Ergod. Theory Dyn. Syst., Volume 3 (1983), pp. 261-313
[20] On a generic property of geodesic flows, Math. Ann., Volume 298 (1994) no. 1, pp. 101-116
[21] Closing geodesics in C1 topology, J. Differ. Geom., Volume 91 (2012) no. 3, pp. 361-382
[22] On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows, Int. Math. Res. Not. (2006) Art. ID 34028, 9 p.; Erratum to: On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows, Int. Math. Res. Not. (2006), Art. ID 38784, 4 p
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