tailieunhanh - Đề tài " Axiom A maps are dense in the space of unimodal maps in the Ck topology "

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. Introduction . 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. | Annals of Mathematics Axiom A maps are dense in the space of unimodal maps in the Ck topology By O. S. Kozlovski Annals of Mathematics 157 2003 1-43 Axiom A maps are dense in the space of unimodal maps in the Ck topology By O. S. Kozlovski Abstract In this paper we prove Ck structural stability conjecture for unimodal maps. In other words we shall prove that Axiom A maps are dense in the space of Ck unimodal maps in the Ck topology. Here k can be 1 2 . Xi ư. 1. Introduction . 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 . Let X be an interval. We say that a Ck map f X satisfies the Axiom A conditions if f has finitely many hyperbolic periodic attractors the set E f X 23 f is hyperbolic where 23 f is a union of the basins of attracting periodic points. This is more or less a classical definition of the Axiom A maps however in the case of C2 one-dimensional maps Mane has proved that a C2 map satisfies Axiom A if and only if all its periodic points are hyperbolic and the forward iterates of all its critical points converge to some periodic attracting points. It was proved many years ago that Axiom A maps are C2 structurally stable if the critical points are nondegenerate and the no-cycle condition is fulfilled see for example dMvS . However the opposite question Does 2 O. S. KOZLOVSKI structural stability imply Axiom A appeared to be much harder. It was conjectured that the answer to this question is affirmative and it was assigned the name .