Comptes Rendus
no. 7
Partially hyperbolic geodesic flows are Anosov
[Les flots géodésiques partiellement hyperboliques sont Anosov]
Gonzalo Contreras1
1 Cimat, PO box 402, 36.000 Guanajuato GTO, México, Mexique
Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 585-590.

Considérons une action de ℝ ou ℤ sur un fibré vectoriel muni d'une struture symplectique par des applications linéaires préservant cette structure symplectique, et supposons que cette action possède une décomposition invariante faiblement dominée E=S⊕U avec dimU=dimS. On montre alors que cette action est nécessairement hyperbolique.

We prove that if a ℤ or ℝ-action by symplectic linear maps on a symplectic vector bundle E has a weakly dominated invariant splitting E=S⊕U with dimU=dimS, then the action is hyperbolic. In particular, contact and geodesic flows with a dominated splitting with dimS=dimU are Anosov.

Reçu le :
Accepté le :
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DOI : 10.1016/S1631-073X(02)02196-9

Gonzalo Contreras 1

1 Cimat, PO box 402, 36.000 Guanajuato GTO, México, Mexique 
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Gonzalo Contreras. Partially hyperbolic geodesic flows are Anosov. Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 585-590. doi : 10.1016/S1631-073X(02)02196-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02196-9/

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  • Rafael O. Ruggiero; Ludovic Rifford; Ayadi Lazrag Franks' lemma for C2-Mañé perturbations of Riemannian metrics and applications to persistence, Journal of Modern Dynamics, Volume 10 (2016), pp. 379-411 | DOI:10.3934/jmd.2016.10.379 | Zbl:1345.93044
  • Fernando Carneiro; Enrique Pujals Partially hyperbolic geodesic flows, Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, Volume 31 (2014) no. 5, pp. 985-1014 | DOI:10.1016/j.anihpc.2013.07.009 | Zbl:1298.53089
  • José Antônio Gonçalv Miranda Positive topological entropy for magnetic flows on surfaces, Nonlinearity, Volume 20 (2007) no. 8, p. 2007 | DOI:10.1088/0951-7715/20/8/011
  • Hiromichi Nakayama Fiberwise divergent orbits of projective flows with exactly two minimal sets, Pacific Journal of Mathematics, Volume 229 (2007) no. 2, p. 469 | DOI:10.2140/pjm.2007.229.469

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