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/

[1] G. Contreras; R. Iturriaga Convex Hamiltonians without conjugate points, Ergodic Theory Dynamic Systems, Volume 19 (1999), pp. 901-952

[2] P. Eberlein When is a geodesic flow of Anosov type? I., J. Differential Geom., Volume 8 (1973), pp. 437-463

[3] J. Franks; C. Robinson A quasi-Anosov diffeomorphism that is not Anosov, Trans. Amer. Math. Soc., Volume 223 (1976), pp. 267-278

[4] M. Hirsch; C. Pugh; M. Shub Invariant Manifolds, Lecture Notes in Math., 583, Springer, 1977

[5] R. Mañé Quasi-Anosov diffeomorphisms and hyperbolic manifolds, Trans. Amer. Math. Soc., Volume 229 (1977), pp. 351-370

[6] C. Robinson A quasi-Anosov flow that is not Anosov, Indiana Univ. Math. J., Volume 25 (1976), pp. 763-767

[7] R.O. Ruggiero Persistently expansive geodesic flows, Comm. Math. Phys., Volume 140 (1991), pp. 203-215

[8] R.J. Sacker; G.R. Sell A note on Anosov diffeomorphisms, Bull. Amer. Math. Soc., Volume 80 (1974), pp. 278-280

[9] R.J. Sacker; G.R. Sell Existence of exponential dichotomies and invariant splitting I, II, III, J. Differential Equations, Volume 15 (1974), pp. 429-458 22 (1976) 478–496; 22 (1976) 497–522

[10] J. Selgrade Isolated invariant sets for flows on vector bundles, Trans. Amer. Math. Soc., Volume 203 (1975), pp. 359-390

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