Axiom A maps are dense in the space of unimodal maps in the Cktopology

Abstract

In this paper we prove C k structural stability conjecture for unimodal maps. In other words, we shall prove that Axiom A maps are dense in the space of C k unimodal maps in the C k topology. Here k can be 1, 2 ,..., ∞ ,ω . 1.1. The structural stability conjecture. The structural stability conjecture was and remains one of the most interesting and important open problems in the theory of dynamical systems. This conjecture states that a dynamical system is structurally stable if and only if it satisfies Axiom A and the transversality condition. In this paper we prove this conjecture in the simplest nontrivial case, in the case of smooth unimodal maps. These are maps of an interval with just one critical turning point. To be more specific let us recall the definition of Axiom A maps: Definition 1.1. Let X be an interval. We say that a C k map f : X ←� satisfies the Axiom A conditions if: • f has finitely many hyperbolic periodic attractors, • the set Σ(f )= X (f )i sh yperbolic, where (f )i sa union of the