Hyperbolic dynamics of Euler-Lagrange flows on prescribed energy levels

Paternain, Gabriel P.

Paternain, Gabriel P.

Séminaire de théorie spectrale et géométrie, Tome 15 (1996-1997), pp. 127-151.

Zbl

@article{TSG_1996-1997__15__127_0,
     author = {Paternain, Gabriel P.},
     title = {Hyperbolic dynamics of {Euler-Lagrange} flows on prescribed energy levels},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {127--151},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {15},
     year = {1996-1997},
     zbl = {0898.58041},
     language = {en},
     url = {http://www.numdam.org/item/TSG_1996-1997__15__127_0/}
}

TY  - JOUR
AU  - Paternain, Gabriel P.
TI  - Hyperbolic dynamics of Euler-Lagrange flows on prescribed energy levels
JO  - Séminaire de théorie spectrale et géométrie
PY  - 1996-1997
SP  - 127
EP  - 151
VL  - 15
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/item/TSG_1996-1997__15__127_0/
LA  - en
ID  - TSG_1996-1997__15__127_0
ER  -

%0 Journal Article
%A Paternain, Gabriel P.
%T Hyperbolic dynamics of Euler-Lagrange flows on prescribed energy levels
%J Séminaire de théorie spectrale et géométrie
%D 1996-1997
%P 127-151
%V 15
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/item/TSG_1996-1997__15__127_0/
%G en
%F TSG_1996-1997__15__127_0

Paternain, Gabriel P. Hyperbolic dynamics of Euler-Lagrange flows on prescribed energy levels. Séminaire de théorie spectrale et géométrie, Tome 15 (1996-1997), pp. 127-151. http://www.numdam.org/item/TSG_1996-1997__15__127_0/

Bibliographie
Cité par

[1] V.I. Arnold, A.B. Givental. - Symplectic Geometry, Dynamical Systems IV, Encyclopaedia of Mathematical Sciences, Springer Verlag: Berlin 1990 | MR | Zbl

[2] V. Bangert. - Minimal geodesics, Ergod. Th. and Dynam. Sys. 10 ( 1989) 263-286. | MR | Zbl

[3] Y. Benoist, P. Foulon, F. Labourie. - Flots d'Anosov a distributions stable et inestable différentiables, Journal of the AMS 5 ( 1992) 33-74. | MR | Zbl

[4] G. Besson, G. Courtois, S. Gallot. - Entropies et rigidités desespaces localement symétriques de courbure strictement négative, Geom. Funct. Anal. 5 ( 1995) 731-799. | MR | Zbl

[5] M. Blaly, L. Polterovich. - Hamiltonian diffeomorphisms and Lagrangian distributions, Geom. Funct. Anal. 2 ( 1992) 173-210 | MR | Zbl

[6] P. Boyland, C. Golé. - Lagrangian systems on hyperbolic manifolds, preprint. | MR | Zbl

[7] M. Brunella. - Expansive flows on Seifert manifolds and torus bundles, Bulletin of the Brazilian Math. Society 24 ( 1993) 89-104. | MR | Zbl

[8] C. Camacho, A. Lins Neto. - Geometric theory of foliations, Birkhauser Boston, Inc., Boston, Mass., 1985. | MR | Zbl

[9] G. Contreras, J. Delgado, R. Iturrlaga. - Lagrangian flows: the dynamics of globally minimizing orbits II, to appear in Bulletin of the Brazilian Math. Society, preprint available via internet at "http://www.ma.utexas.edu/mp-arc". | Zbl

[10] G. Contreras, R. Iturriaga. - Convex Hamiltonians without conjugate points, to appear in Ergod.Th and Dynam. Sys., preprint available via internet at "http:// www.ma.utexas.edu/mp-arc" | MR | Zbl

[11] G. Contreras, R. Iturriaga, G.P. Paternain, M. Paternain. - Lagrangian graphs, minimizing measures and Manés critical values, preprint 1997, available via internet at http://www.ma.utexas.edu/mp-arc\ | Zbl

[12] M.J. Dias Carneiro. - On minimizing measuresof the action of autonomous Lagrangians, Nonlineariry 8 ( 1995) 1077-1085. | MR | Zbl

[13] J.J. Duistermaat. - On the Morse index in variational calculus, Adv. in Math. 21 ( 1976) 173-195. | MR | Zbl

[14] A. Fathi. - Théorème KAM faible et Théorie de Mather sur les systems Lagrangiens, C.R. Acad. Sci. Paris, t. 324, Sériel ( 1997) 1043-1046. | MR | Zbl

[15] A. Fathi. - Solutions KAM faibles conjuguées et barrières de Peierls, preprint 1997. | MR | Zbl

[16] R. Feres, A. Katok. - Invariant tensor fields of dynamical systems with pinched liapunov exponents and rigidity of geodesic flows, Ergod. Th and Dynam. Sys. 9 ( 1989) 427-432. | MR | Zbl

[17] R. Feres, A. Katok. - Anosov flows with smooth foliations and rigidity of geodesie flows in three- dimensional manifolds of negative curvarture, Ergod. Th and Dynam. Sys. 10 ( 1990) 657-670. | MR | Zbl

[18] R. Feres. - Geodesic flows on manifolds with negative with smooth horospheric foliations, Ergod. Th and Dynam. Sys. 11 ( 1991) 653-686. | MR | Zbl

[19] L. Flaminio. - Local entropy rigidity for hyperbolic manifolds, Comm. in Analysis and Geom. 4 ( 1995) 555-596. | MR | Zbl

[20] P. Foulon. - Feuilletages des sphères et dynamiques Nord-Sud, C.R. Acad. Sci Paris, t. 318, Série I ( 1994) 1041-1042. | MR | Zbl

[21] E. Ghys. - Flots d'Anosov sur les3-variétés fibrées en cercles, Ergod. Th and Dynam. Sys. 4 ( 1984)67-80. | MR | Zbl

[22] E. Ghys. - Flots d'Anosov dont les feuilletages stables sont différentiables, Ann. Sci. École Norm. Sup. 20 ( 1987) 251-270. | Numdam | MR | Zbl

[23] U. Hamenstädt. - Invariant two-forms for geodesic flows, Math. Ann. 301 ( 1995) 677-698. | MR | Zbl

[24] B. Hasselblatt. - Regularityof the Anosov splitting and of horospherical foliations, Ergod. Th and Dynam. Sys. 14 ( 1994) 645-666. | MR | Zbl

[25] G.A. Hedlund. - Geodesics on a two-dimensional Riemannian manifold with periodic coefficients, Ann. of Math. 33 ( 1932) 719-739. | JFM | MR

[26] K. Hiraide. - Expansive homeomorphisms of compact surfaces are Pseudo-Anosov, Osaka J. Math. 27 ( 1990) 117-162. | MR | Zbl

[27] M. Hirsch, C. Pugh. - Smoothness of horocycle foliations, J. Diff. Geom. 10 ( 1975) 225-238. | MR | Zbl

[28] S. Hurder, A. Katok. - Differentiability, rigidity and Godbillon-Vey classes for Anosov flows, Publ. Math. IHES 72 ( 1990) 5-61 | Numdam | MR | Zbl

[29] T. Inaba, S. Matsumoto. - Nonsingular expansive flows on 3-manifolds and foliations with circle prong singularities, Japan J. of Math. 16 ( 1990) 329-340. | MR | Zbl

[30] M. Kanai. - Geodesic flows on negatively curved manifolds with smooth stable and unstable foliations, Ergod. Th and Dynam. Sys. 8 ( 1988) 215-240. | MR | Zbl

[31] W. Klingenberg. - Riemannian manifolds with geodesic flows of Anosov type, Ann. of Math., 99 ( 1974) 1-13. | MR | Zbl

[32] J. Lewowicz. - Expansive homeomorphisms on surfaces, Bulletin of the Brazilian Math. Society Vol 201 ( 1989)113-133. | MR | Zbl

[33] R. Mañé. - On a theorem of Klingenberg, Dynamical Systems and Bifurcation Theory, M. Camacho, M. Pacifico and F.Takens eds, Pitman Research Notes in Math. 160 ( 1987),319-345. | MR | Zbl

[34] R. Mané. - Global variational methods in conservative dynamics, 18 Colóquio Brasileiro de Maternatica.IMPA, 1991.

[35] R. Mané. - Lagrangian flows: the dynamics ofglobally minimizing orbits, International Congress on Dynamical Systems in Montevideo (a tribute to Ricardo Mafié), F.Ledrappier, J. Lewowicz, S. Newhouse eds, Pitman Research Notes in Math. 362 ( 1996) 120-131. | MR | Zbl

[36] R. Mané. - Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity 9 ( 1996) 273-310. | MR | Zbl

[37] J. Mather. - Action minimizing measures for positive definite Lagrangian systems, Math. Z. 207 ( 1991) 169-207. | MR | Zbl

[38] J. Mather. - Variational construction of connecting orbits, Ann. Inst Fourier 43 ( 1993) 1349-1386. | Numdam | MR | Zbl

[39] M. Morse. - A fondamental class of geodesics on any closed surface of genus greater than one, Trans. Amer. Math. Soc. 26 ( 1924) 25-60. | JFM | MR

[40] G.P. Paternain, M. Paternain. - On Anosou Energy Levels of Convex Hamiltonian Systems, Math. Z. 217 ( 1994) 367-376. | MR | Zbl

[41] G.P. Paternain, M. Paternain. - Expansivity for optical Hamiltonian systems with two degrees offreedom, C.R. Acad. Sci. Paris, t. 316, Série I ( 1993) 837-841. | MR | Zbl

[42] G. P. Paternain, M. Paternain. - Anosov geodesic flows and twisted symplectic structures, International Congress on Dynamical Systems in Montevideo (a tribute to Ricardo Marié), F. Ledrappier, J. Lewowicz, S. Newhouse eds, Pitman Research Notes in Math. 362 ( 1996) 132-145. | MR | Zbl

[43] G. P. Paternain, M. Paternain. - First derivative of topological entropy for Anosov geodesic flows in the presence of magnetic fields, Nonlinearity 10 ( 1997) 121-131. | MR | Zbl

[44] G. P. Paternain, M. Paternain. - Critical values of autonomous Lagrangian systems, Comment. Math. Helvetici 72 ( 1997), preprint available via internet at http://www.cmat.edu.uy/-gabriel". | MR | Zbl

[45] G. P. Paternain. - On Anosov energy levels of Hamiltonians on twisted cotangent bundles, Bulletin of the Brazilian Math. Society Vol 25 2 ( 1994) 207-211. | MR | Zbl

[46] G. P. Paternain. - On the regularity of the Anosov splitting for twisted geodesic flows, to appear in Math. Res. Lett., preprint available via internet at "http://www.cmat.edu.uy/-gabriel". | MR | Zbl

[47] M. Paternain. - Expansive geodesic flows on surfaces, Ergod. Th and Dynam. Sys. 13 ( 1993) 153-165. | MR | Zbl

[48] M. Paternain. - Expansive flows and the fondamental group, Bulletin of the Brazilian Math. SocietyVol 24 2 ( 1993)179-199. | MR | Zbl

[49] J. F. Plante. - Anosov flows, Amer. J. of Math. 94 ( 1972) 729-754 | MR | Zbl

[50] J. F. Plante. - Homology of closed orbits of Anosov flows, Proc. of the A.M.S. 37 ( 1973) 297-300. | Zbl

[51] R. Schwartzmann. - Asymptotic cycles, Ann. of Math. (2) 66 ( 1957) 270-284. | Zbl