Structural Stability and Hyperbolic Attractors

A necessary condition for structural stability is presented that in the two dimensional case means that the system has a finite number of topological attractors. Introduction. One of the basic questions in Dynamical System Theory is the characterization and the study of the properties of structurally stable systems (vector fields or diffeomorphisms). The standing conjecture about characterization, first formulated by Palis and Smale [2], is that a system is structurally stable if and only if it satisfies Axiom A and the strong transversality condition. That these conditions are sufficient was proved by Robbin [8] and Robinson [9], [10] after a series of important results in this direction. It is also known that in the presence of Axiom A, strong transversality is a necessary condition for stability. Therefore the open question is whether stability implies Axiom A. Relevant partial results were obtained by Manei [3], [4] and Pliss [5], [6]. Pliss [5] showed that C'-stability of a diffeomorphism implies that there is only a finite number of attracting periodic orbits (all hyperbolic). The purpose of this work is to show, generalizing this last result, that C "structurally stability of a diffeomorphism implies that there is only a finite number of hyperbolic attractors. This result is a well known fact for Axiom A diffeomorphisms (Smale [11]). We will now explain some facts about our result. Let M be a C '-manifold without boundary and Diffr(M) be the space of Cr-diffeomorphisms of M with the Cr-topology, r > 1. A diffeomorphism f E Diff`(M) is Cr_structurally stable if there exists a neighborhood U of f in Diffr(M), such that if g E U there exist a homeomorphism h of M satisfying f = hg. Such a diffeomorphism f has the property that all diffeomorphisms sufficiently near to it have the same orbit structure. We will call a point x E M nonwandering for f when V U neighborhood of x in M there exist n E Z {0}, such that f (U) n U #0. We will denote such a set by R(f). A closed subset A of the nonwandering set Q(f) is called an attractor when: (1) A is f-invariant (that is, f(A) = A). Received by the editors April 4, 1978. AMS (MOS) subject classifications (1970). Primary 58F10; Secondary 58F15.