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.

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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

A diffeomorphism f has a heterodimensional cycle if there are (transitive) hyperbolic sets Λ and Σ having different indices (dimension of the unstable bundle) such that the unstable manifold of Λ meets the stable one of Σ and vice versa. This cycle has co-index 1 if index(Λ) = index(Σ) ± 1. This cycle is robust if, for every g close to f , the continuations of Λ and Σ for g have a heterodimensional cycle.We prove that any co-index 1 heterodimensional cycle associated with a pair of hyperbolic saddles generates C 1 -robust heterodimensioal cycles. Therefore, in dimension three, every heterodimensional cycle generates robust cycles.We also derive some consequences from this result for C 1 -generic dynamics (in any dimension). Two of such consequences are the following. For tame diffeomorphisms (generic diffeomorphisms with finitely many chain recurrence classes) there is the following dichotomy: either the system is hyperbolic or it has a robust heterodimensional cycle. Moreover, any chain recurrence class containing saddles having different indices has a robust cycle. 471Question 1.2. Let M be closed manifold. Does it exist a C 1 -open and dense subset O ⊂ Diff 1 (M ) such that every f ∈ O either verifies the Axiom A and the no-cycles condition or has a C 1 -robust heterodimensional cycle?Note that a positive answer to this question implies the C 1 -density of hyperbolic surface diffeomorphisms. See the discussion in § 1.3 about the Smale density conjecture. We will see that Theorem 1.14 gives a partial positive answer to this question for the so-called tame diffeomorphisms (diffeomorphisms finitely many homoclinic classes, see the preciseThe examples by Abraham-Smale of non-Axiom A diffeomorphisms involves a hyperbolic set Γ whose unstable manifold has dimension strictly greater than the dimension of its unstable bundle. Note that a normally hyperbolic extension of transitive Anosov diffeomorphisms on a torus T 2 gives an example of this configuration.The construction in [8] gives a slightly different mechanism for constructing non-Axiom A diffeomorphisms and robust heterodimensional cycles, based on the notion of blender. Roughly speaking, a blender is a hyperbolic set whose embedding in the ambient manifold verifies some specific geometric properties, whose effect is that, as in the Abraham-Smale example, the unstable manifold of a blender looks like a manifold of higher dimension. We review the construction and main properties of blenders in § 4.1.3. See also [15, Chapter 6.1] for a discussion of this notion.One of the goals of this paper is to show that blenders (and as a consequence robust heterodimensional cycles) appear in a natural way in the unfolding of heterodimensional cycles associated with two saddles. Definition 1.3 (heterodimensional cycle and co-index 1 cycle).A diffeomorphism f has a heterodimensional cycle (see Figure 1) associated with two hyperbolic periodic saddles P and Q of f if the saddles P and Q have different indices, the stable manifold of the orbit of P meets the unstable manifold of...

A diffeomorphism f has a C 1 -robust homoclinic tangency if there is a C 1 -neighbourhood U of f such that every diffeomorphism in g ∈ U has a hyperbolic set Λ g , depending continuously on g, such that the stable and unstable manifolds of Λ g have some non-transverse intersection. For every manifold of dimension greater than or equal to three, we exhibit a local mechanism (blender-horseshoes) generating diffeomorphisms with C 1 -robust homoclinic tangencies.Using blender-horseshoes, we prove that homoclinic classes of C 1 -generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings (of appropriate dimensions) display C 1 -robust homoclinic tangencies.keywords: chain recurrence set, dominated splitting, heterodimensional cycle, homoclinic class, homoclinic tangency, hyperbolic set.MSC 2000: 37C05, 37C20, 37C25, 37C29, 37C70.1 Introduction Framework and general settingA homoclinic tangency is a dynamical mechanism which is at the heart of a great variety of non-hyperbolic phenomena: persistent coexistence of infinitely many sinks [22], Hénon-like strange attractors [5,20], super-exponential growth of the number of periodic points [19], and non-existence of symbolic extensions [15], among others. Moreover, homoclinic bifurcations (homoclinic tangencies and heterodimensional cycles) are conjectured to be the main source of non-hyperbolic dynamics (Palis denseness conjecture, see [23]).In this paper, we present a local mechanism generating C 1 -robust homoclinic tangencies. Using this construction, we show that the occurrence of robust tangencies is a quite general phenomenon in the non-hyperbolic setting, specially when the dynamics does not admit a suitable dominated splitting. * This paper was partially supported by CNPq, Faperj, and PRONEX (Brazil) and the Agreement in Mathematics Brazil-France. We acknowledge the warm hospitality of I.M.P.A, Institute de Mathématiques de Bourgogne, and PUC-Rio during the stays while preparing this paper 1 Let us now give some basic definitions (in Section 2, we will state precisely the definitions involved in this paper). A transitive hyperbolic set Λ has a homoclinic tangency if there is a pair of points x, y ∈ Λ such that the stable leaf W s (x) of x and the unstable leaf W u (y) of y have some non-transverse intersection Given a hyperbolic set Λ of a diffeomorphism f , for g close to f , we denote by Λ g the hyperbolic set of g which is the continuation of Λ (i.e., Λ g is close to Λ and the dynamics of f on Λ and g on Λ g are conjugate). Definition 1.1 (Robust cycles).• Robust homoclinic tangencies: A transitive hyperbolic set Λ of a C r -diffeomorphism f has a C r -robust homoclinic tangency if there is a C r -neighborhood N of f such that for every g ∈ N the continuation Λ g of Λ for g has a homoclinic tangency.• Robust heterodimensional cycles: A diffeomorphism f has a C r -robust heterodimensional cycle if there are transitive hyperbolic sets Λ and Σ of f whose stable bundles have different dimensions and a C r -n...