Christian Bonatti

Récurrence et généricité

Résumé.Nous montrons un lemme de connexion C1 pour les pseudo-orbites des difféomorphismes des variétés compactes. Nous explorons alors les conséquences pour les difféomorphismes C1-génériques. Par exemple, les difféomorphismes conservatifs C1-génériques (d’une variété connexe) sont transitifs. Abstract.We prove a C1-connecting lemma for pseudo-orbits of diffeomorphisms on compact manifolds. We explore some consequences for C1-generic diffeomorphisms. For instance, C1-generic conservative diffeomorphisms (on connected manifolds) are transitive.

Perturbations of the derivative along periodic orbits

We show that a periodic orbit of large period of a diffeomorphism or flow, either admits a dominated splitting of a prescribed strength, or can be turned into a sink or a source by a C1-small perturbation along the orbit. As a consequence we show that the linear Poincare flow of a C1-vector field admits a dominated splitting over any robustly transitive set.

Removing zero Lyapunov exponents

In an explicit family of partially hyperbolic diffeomorphisms of the torus T 3 , ShubandWilkinsonrecentlysucceededin perturbingthe Lyapunovexponentsofthe center direction. We present here a local version of their argument, allowing one to perturb the center Lyapunov exponents of any partially hyperbolic system, in any dimension and with arbitrary dimension of the center bundle.

Lyapunov exponents with multiplicity 1 for deterministic products of matrices

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 cocycles 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 Holder continuous potentials), all Oseledets subspaces are one-dimensional. 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.

ROBUST HETERODIMENSIONAL CYCLES AND

A diffeomorphism f has a heterodimensional cycle if there are (transitive) hyperbolic sets � andhaving different indices (dimension of the unstable bundle) such that the unstable manifold ofmeets the stable one ofand vice-versa. This cycle has co-index one if index (�) = index (�) ± 1. This cycle is robust if, for every g close to f, the continuations ofandfor g have a heterodimensional cycle. We prove that any co-index one heterodimensional cycle associated to 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 di- mension). 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.

C^1

Abstract We give the topological classification of three-dimensional closed orientable manifolds admitting Morse–Smale diffeomorphisms without heteroclinic curves: these manifolds are either the three-dimensional sphere or connected sums of S 2 × S 1 s and we give a formula relating the number of sinks, sources and saddle periodic points to the topology of the manifold.

-GENERIC DYNAMICS

We give a topological criterion for the minimality of the strong unstable (or stable) foliation of robustly transitive partially hyperbolic diffeomorphisms. As a consequence we prove that, for 3-manifolds, there is an open and dense subset of robustly transitive diffeomorphisms (far from homoclinic tangencies) such that either the strong stable or the strong unstable foliation is robustly minimal. We also give a topological condition (existence of a central periodic compact leaf) guaranteeing (for an open and dense subset) the simultaneous minimality of the two strong foliations.

Three-manifolds admitting Morse-Smale diffeomorphisms without heteroclinic curves

Abstract We give a complete invariant, called global scheme , of topological conjugacy classes of gradient-like diffeomorphisms, on compact 3-manifolds. Conversely, we can realize any abstract global scheme by such a diffeomorphism.

MINIMALITY OF STRONG STABLE AND UNSTABLE FOLIATIONS FOR PARTIALLY HYPERBOLIC DIFFEOMORPHISMS

Abstract We say that two hyperbolic periodic points of a diffeomorphism f are persistently connected if there is a neighbourhood of f having a dense subset of diffeomorphisms for which there is a transitive set containing these two points. We prove that two points are generically in the same transitive set if and only if they are persistently connected with their homoclinic class being equal. As a consequence, we get the local genericity of the Newhouses phenomenon (coexistence of infinitely many sinks or sources) for C 1 -diffeomorphisms of three manifolds.

Topological classification of gradient-like diffeomorphisms on 3-manifolds

This paper suggests a program for getting a global view of the dynamics of diffeomorphisms, from the point of view of the C 1 -topology. More precisely, given any compact manifold M , one splits Diff 1 ( M ) into disjoint C 1 -open regions whose union is C 1 -dense, and conjectures state that each of these open sets and their complements is characterized by the presence of: • either a robust local phenomenon; • or a global structure forbidding this local phenomenon. Other conjectures state that some of these regions are empty. This set of conjectures draws a global view of the dynamics, putting in evidence the coherence of the numerous recent results on C 1 -generic dynamics.