Comptes Rendus
Partially hyperbolic geodesic flows are Anosov
[Les flots géodésiques partiellement hyperboliques sont Anosov]
Comptes Rendus. Mathématique, Volume 334 (2002) no. 7, pp. 585-590.

Considérons une action de ou 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=SU avec dimU=dimS. On montre alors que cette action est nécessairement hyperbolique.

We prove that if a or -action by symplectic linear maps on a symplectic vector bundle E has a weakly dominated invariant splitting E=SU 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 :
Publié le :
DOI : 10.1016/S1631-073X(02)02196-9

Gonzalo Contreras 1

1 Cimat, PO box 402, 36.000 Guanajuato GTO, México, Mexique
@article{CRMATH_2002__334_7_585_0,
     author = {Gonzalo Contreras},
     title = {Partially hyperbolic geodesic flows are {Anosov}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {585--590},
     publisher = {Elsevier},
     volume = {334},
     number = {7},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02196-9},
     language = {en},
}
TY  - JOUR
AU  - Gonzalo Contreras
TI  - Partially hyperbolic geodesic flows are Anosov
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 585
EP  - 590
VL  - 334
IS  - 7
PB  - Elsevier
DO  - 10.1016/S1631-073X(02)02196-9
LA  - en
ID  - CRMATH_2002__334_7_585_0
ER  - 
%0 Journal Article
%A Gonzalo Contreras
%T Partially hyperbolic geodesic flows are Anosov
%J Comptes Rendus. Mathématique
%D 2002
%P 585-590
%V 334
%N 7
%I Elsevier
%R 10.1016/S1631-073X(02)02196-9
%G en
%F CRMATH_2002__334_7_585_0
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

Cité par Sources :

Commentaires - Politique