HAL : hal-00322876, version 1

arXiv : 0809.3309

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Nonuniform hyperbolicity for C^1-generic diffeomorphisms

Flavio Abdenur 1, Christian Bonatti 2, Sylvain Crovisier 3

(18/09/2008)

We study the ergodic theory of non-conservative C^1-generic diffeomorphisms. First, we show that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we show that generic (for the weak topology) ergodic measures of C^1-generic diffeomorphisms are nonuniformly hyperbolic: they exhibit no zero Lyapunov exponents. Third, we extend a theorem by Sigmund on hyperbolic basic sets: every isolated transitive set L of any C^1-generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set L. In addition, confirming a claim made by R. Ma~né in 1982, we show that hyperbolic measures whose Oseledets splittings are dominated satisfy Pesin's Stable Manifold Theorem, even if the diffeomorphism is only C^1.

1 :	Departamento de Matematica
	PUC-RIO
2 :	Institut de Mathématiques de Bourgogne (IMB)
	CNRS : UMR5584 – Université de Bourgogne
3 :	Laboratoire Analyse, Géométrie et Application (LAGA)
	CNRS : UMR7539 – Université Paris XIII - Paris Nord – Université Paris VIII - Vincennes Saint-Denis

Domaine

:

Mathématiques/Systèmes dynamiques

Mots Clés : Dominated splitting – nonuniform hyperbolicity – generic dynamics – Pesin theory

hal-00322876, version 1

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Contributeur : Sylvain Crovisier <>

Soumis le : Jeudi 18 Septembre 2008, 22:13:15

Dernière modification le : Lundi 22 Septembre 2008, 13:11:50