Hyperbolic Birkhoff centers

The purpose of this paper is to show that if f is a diffeomorphism of a compact manifold whose Birkhoff center, $c(f)$, is hyperbolic and has no cycles, then f satisfies Axiom A and is $\Omega$-stable. To obtain a filtration for $c(f)$, 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 $c(f)$ is hyperbolic and f is $\Omega$-stable, then $c(f)$ has the no cycle property. A characterization of $\Omega$-stability when $c(f)$ is finite is also given.