Abstract
The purpose of this paper is to show that if f is a diffeomorphism of a compact manifold whose Birkhoff center, c ( f ) c(f) , is hyperbolic and has no cycles, then f satisfies Axiom A and is Ω \Omega -stable. To obtain a filtration for c ( f ) c(f) , the concept of an isolated set for a homeomorphism of a compact metric space is introduced. As a partial converse it is proved that if c ( f ) c(f) is hyperbolic and f is Ω \Omega -stable, then c ( f ) c(f) has the no cycle property. A characterization of Ω \Omega -stability when c ( f ) c(f) is finite is also given.
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