Neighborhoods of hyperbolic sets

Abstract

Inventiones math. 9, 121 - 134 (1970) Neighborhoods of Hyperbolic Sets M. HIRSCH, J. PALIS, C. PUGH, and M. SHUB (Univ. of Warwick) w 1. Introduction In this paper we study the asymptotic behavior of points near a compact hyperbolic set of a C r diffeomorphism (r__>l)f: M - ~ M , M being a compact manifold. The purpose of our study is to complete the proof of Smale's O-stability Theorem by demonstrating (2.1), (2.4) of [6]. O denotes the set of non-wandering points for f Smale's Axiom A requires [5]: (a) O has a hyperbolic structure, (b) the periodic points are dense in O. Hyperbolic structure, the stable manifold of O, and fundamental neighborhoods are discussed in ~j 2 and 5. The result of [-6] proved here is: I f f obeys Axiom A then there exists a proper fundamental neighbor- hood V for the stable manifold of f2 such that the union of the unstable manifold oft2 and the forward orbit of V contains a neighborhood of 0 in M. As a consequence we have: l f f obeys Axiom A then any point whose orbit stays near 0 is asymp- totic with a point of 0. Section 8 of the mimeographed version of [ 1 ] contains a generalization of the above results with an incorrect proof. A correct generalization is: (1.1) Theorem. I f A is a compact hyperbolic set then WU(A)uO+ V contains a neighborhood U of A, where V is any fundamental neighbor- hood for WS(A) and 0+ V= U f ( v ) . i f A has local product structure n>O then a proper fundamental neighborhood may be found and any point whose forward orbit lies in U is asymptotic with some point of A. Theorem (1.1) is proved in w 5, local product structure is discussed in [-5] and in w In w we prove the analogous theorems for flows. Here is an example, due to Bowen, of a compact hyperbolic set A which does not have local product structure, has no proper fundamental neighborhood and for which there are points asymptotic to A without being asymptotic with any point of A.