Partial hyperbolicity and robust transitivity

Abstract

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,