No abstract

No abstract

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 Λ of any C 1 -generic diffeomorphism f exhibits many ergodic hyperbolic measures whose supports coincide with the whole set Λ.In addition, confirming a claim made by R. Mañé 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 .

We show that, for every compact n-dimensional manifold, n ≥ 1, there is a residual subset of Diff 1 (M ) of diffeomorphisms for which the homoclinic class of any periodic saddle of f verifies one of the following two possibilities: Either it is contained in the closure of an infinite set of sinks or sources (Newhouse phenomenon), or it presents some weak form of hyperbolicity called dominated splitting (this is a generalization of a bidimensional result of Mañé [Ma3]). In particular, we show that any C 1 -robustly transitive diffeomorphism admits a dominated splitting. RésuméGénéralisant un résultat de Mañé sur les surfaces [Ma3], nous montrons que, en dimension quelconque, il existe un sous-ensemble résiduel de Diff 1 (M ) de difféomorphismes pour lesquels la classe homocline de toute selle périodique hyperbolique possède deux comportements possibles : ou bien elle est incluse dans l'adhérence d'une infinité de puits ou de sources (phénomène de Newhouse), ou bien elle présente une forme affaiblie d'hyperbolicité appelée une décomposition dominée. En particulier nous montrons que tout difféomorphisme C 1 -robustement transitif possède une décomposition dominée.

No abstract

No abstract

We exhibit an explicit criterion for the simplicity of the Lyapunov spectrum of linear cocycles, either locally constant or dominated, over hyperbolic (Axiom A) transformations. This criterion is expressed by a geometric condition on the cocycle's behaviour over periodic points and associated homoclinic orbits. It allows us to prove that for an open dense subset of dominated linear cocycles over a hyperbolic transformation and for any invariant probability with continuous local product structure (including all equilibrium states of Hölder continuous potentials), all Oseledets subspaces are onedimensional. Moreover, the complement of this subset has infinite codimension and, thus, is avoided by any generic family of cocycles described by finitely many parameters.This improves previous results of Bonatti, Gomez-Mont and Viana where it was shown that some Lyapunov exponent is non-zero, in a similar setting and also for an open dense subset. 1296C. Bonatti and M. Viana exponents often permit a very precise description of the dynamics at the ergodic level (Sinai-Ruelle-Bowen measures) [1, 12]; and the study of certain transversely projective foliations, where non-zero exponents imply the uniqueness of the harmonic measure on the leaves [10, 11]. Several methods have been devised for proving the existence of non-zero exponents: let us mention Furstenberg and Kesten [16, 17], Herman [21], Kotani [25], in various contexts of linear cocycles and Jakobson [23] and Benedicks and Carleson [5], for smooth transformations. The list is, of course, very far from complete. Furstenberg's [17]results about the products of iid random matrices suggest that nonzero exponents might be typical for linear cocycles, in great generality. However, recent work by Bochi [6] shows that this cannot be true without additional assumptions: he proves that generic (Baire second category subset) continuous SL(2, R)-cocycles have zero Lyapunov exponents or else they are uniformly hyperbolic. In fact, he obtains the same conclusion within cocycles given by the derivatives of area-preserving diffeomorphisms, which is much more delicate. Moreover, these results have been extended to arbitrary dimension by Bochi and Viana [7,8].Here we require a stronger regularity, starting from Hölder continuity, as well as domination: the map induced by the cocycle on the projective bundle is partially hyperbolic (this implies the hyperbolicity of the base dynamics). The latter condition is motivated by the applications mentioned earlier to partially hyperbolic systems and to transversely projective foliations. In this setting, it was proved in [11] that an explicit condition about the cocycle over some periodic point and some homoclinic orbit associated to it suffices to ensure the existence of at least one non-zero Lyapunov exponent.Our main result in the present paper states that a slightly stronger form of this condition, also satisfied by the vast majority of these cocycles, implies that all Lyapunov exponents are distinct. By vast majority we mean an open dens...

International audienceWe give explicit C (1)-open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy. Actually, our criterion provides the existence of a partially hyperbolic compact set with one-dimensional center and positive topological entropy on which the center Lyapunov exponent vanishes uniformly. The conditions of the criterion are met on a C (1)-dense and open subset of the set of diffeomorphisms having a robust cycle. As a corollary, there exists a C (1)-open and dense subset of the set of non-Anosov robustly transitive diffeomorphisms consisting of systems with nonhyperbolic ergodic measures with positive entropy. The criterion is based on a notion of a blender defined dynamically in terms of strict invariance of a family of discs