GENERIC DIFFEOMORPHISMS WITH ROBUSTLY TRANSITIVE SETS

Let Λ be a robustly transitive set of a diffeomorphism f ona closed C ∞ manifold. In this paper, we characterize hyperbolicity of Λin C 1 -generic sense. 1. IntroductionA fundamental problem in differentiable dynamical systems is to understandhow a robust dynamic property (that is, a property that holds for a system andall C 1 nearby ones) on the underlying manifold would influences the behaviorof the tangent map on the tangent bundle. In this paper, we study the robustdynamic property for a transitive set. Let M be a closed C ∞ manifold, and letDiff(M) be the space of diffeomorphisms of M endowed with the C 1 -topology.Denote by d the distance on M induced from a Riemannian metric k·k on thetangent bundle TM. Let f ∈ Diff(M) and Λ be a closed f-invariant set. Theset Λ is transitive if there is a point x ∈ Λ such that ω(x) = Λ. Here ω(x) isthe forward limit set of x. Denote by f| Λ the restriction of f to the set Λ. Amaximal invariant set of f in an open set U, denoted by Λ f (U), is the set ofpoints whose whole orbit contained in U, that is,Λ