Lorenzo J. Díaz

Partial hyperbolicity and robust transitivity

Throughout this paper M denotes a three-dimensional boundaryless compact manifold and Diff(M) the space of gl-diffeomorphisms defined on M endowed with the usual Cl-topology. A ~-invariant set A is transitive if A=w(x) for some xEA. Here w(x) is the forward limit set of x (the accumulation points of the positive orbit of x). The maximal invariant set of ~ in an open set U, denoted by A~(U), is the set of points whose whole orbit is contained in U, i.e. A ~ ( U ) = ~ i e z ~i(U). The set A~(U) is robustly transitive if Ar is transitive for every diffeomorphism r CLclose to ~. A diffeomorphism ~EDiff(M) is transitive if M=w(x) for some xEM, i.e. if A ~ ( M ) = M is transitive. Analogously, ~ is robustly transitive if every r gLclose to also is transitive, i.e. if A ~ ( M ) = M is robustly transitive. In this paper we focus our attention on forms of hyperbolicity (uniform, partial and strong partial) of a maximal invariant set A~(U) derived from its robust transitivity. Observe that U can be equal to M, and then ~ is robustly transitive. On one hand, in dimension one there do not exist robustly transitive diffeomorphisms: the diffeomorphisms with finitely many hyperbolic periodic points (Morse~ Smale) are open and dense in Diff(S1). On the other hand, for two-dimensional diffeomorphisms, every robustly transitive set A~(U) is a basic set (i.e. A~(U) is hyperbolic, transitive, and the periodic points of ~ are dense in A~(U)). In particular, every robustly transitive surface diffeomorphism is Anosov and the unique surface which supports such diffeomorphisms is the torus T 2. These assertions follow from [M3] and [M4]. In dimension bigger than or equal to three, besides Anosov (hyperbolic) diffeomorphisms there are robustly transitive diffeomorphisms of nonhyperbolic type. As far as we know, three types of such diffeomorphisms have been constructed: skew products,

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

We describe an open set A of arcs of diffeomorphisms bifurcating through the creation of heterodimensional cycles for which every diffeomorphism after the bifurcation is nonhyperbolic or unstable. We also prove that generically in A the borning nonwandering set is transitive and local maximal for a full (Lebesgue) set of parameter values.

-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.

Robust nonhyperbolic dynamics and heterodimensional cycles

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.

MINIMALITY OF STRONG STABLE AND UNSTABLE FOLIATIONS FOR PARTIALLY HYPERBOLIC DIFFEOMORPHISMS

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 setg, depending contin- uously on g, such that the stable and unstable manifolds ofg have some non-transverse intersection. For every manifold of dimension greater than or equal to three, we exhibit a lo- cal 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.

Connexions hétéroclines et généricité d'une infinité de puits et de sources

The authors study arcs of diffeomorphisms that bifurcate through the creation of a nonconnected heterodimensional cycle. They exhibit two open sets of such arcs; in the first one, after the bifurcation, the set of parameter values for which the corresponding diffeomorphism is stable (stable set) is empty, in the second one Omega -stability has some degree of prevalence, i.e. the relative measure of the Omega -stable set is bounded away from zero.

Abundance of C 1 -robust homoclinic tangencies

In this paper, we propose a model for the destruction of three-dimensional horseshoes via heterodimensional cycles. This model yields some new dynamical features. Among other things, it provides examples of homoclinic classes properly contained in other classes and it is a model of a new sort of heteroclinic bifurcations we call generating .

Nonconnected heterodimensional cycles: bifurcation and stability

We prove that there is a residual subset in Diff?1(M) such that, for every , any homoclinic class of f with invariant one-dimensional central bundle containing saddles of different indices (i.e. with different dimensions of the stable invariant manifold) coincides with the support of some invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of f.

Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes

We construct an open set A of arcs of diffeomorphisms bifurcating through the creation of heterodimensional cycles (i.e. there are points in the cycle having different indices) being robustly nonhyperbolic after the unfolding of the cycle: every diffeomorphism is not hyperbolic. We also prove that the arcs in A exhibit cycles persistently. Finally, for generic arcs in A and for a full (Lebesgue) set of parameters values we prove that the resulting nonwandering set is transitive and local maximal.