Some Results towards the C1 Density Conjecture of Palis

Let M be a d-dimensional compact manifold without boundary. Denote by Diffr(M) the set of diffeomorphisms endowed with the Cr topology. A main problem in differentiable dynamics is the following famous conjecture of Palis [P, PT]: The Cr Density Conjecture (Palis). Cr diffeomorphisms of M exhibiting either a homoclinic tangency or a heterodimensional cycle are Cr dense in the complement of the Cr closure of hyperbolic systems. Here a homoclinic tangency is a point of non-transverse intersection of the stable and unstable manifolds of some hyperbolic periodic orbit, a heterodimensional cycle is a cycle of hyperbolic periodic orbits with different indices, and a hyperbolic system is one with hyperbolic limit set L(f) (the closure of the union of ω-limit sets and α-limit sets of all points). In dimension 2 the conjecture is proved recently in the C1 topology by Pujals and Sambarino in a remarkable paper [PS]. Note that in dimension 2, a priori, there can be no heterodimensional cycles. Hence in dimension 2, what was conjectured by Palis, and now proved by Pujals and Sambarino, is that any diffeomorphism can be C1 approximated ei-ther by a hyperbolic diffeomorphism, or by one with a homoclinic tangency. In dimension higher than 2, examples show that there are systems that can be C1 approximated neither by hyperbolic systems, nor by systems with homoclinic tangencies [Sh1] [M1] [BD], and Palis conjectures that such a system must be Cr approximated by one with a heterodi-mensional cycle. In this abstract we present some results towards the conjecture in the C1 topology for higher dimensions. §1. A principle of Liao and the C1 preperiodic set The main idea of our approach will be to rule out the existence of minimally non-hyperbolic sets contained in the so called C1 preperiodic set, under the assumptions given in question. A compact invariant set Λ is called minimally non-hyperbolic if Λ is non-hyperbolic but any nonempty compact invariant proper subset of Λ is hyperbolic. This notion plays an important role in Pujals and Sambarino [PS]. It resembles the notion of minimally rambling sets, studied intensively by Liao [L3]. Proposition 1.1. ([PS]) Any non-empty compact non-hyperbolic invariant set contains a minimally non-hyperbolic set