Hyperbolic limit sets

Many known results for diffeomorphisms satisfying Axiom A are shown to be true with weaker assumptions. It is proved that if the negative limit set L − ( f ) {L^ - }(f) of a diffeomorphism f is hyperbolic, then the periodic points of f are dense in L − ( f ) {L^ - }(f) . A spectral decomposition theorem and a filtration theorem for such diffeomorphisms are obtained and used to prove that if L − ( f ) {L^ - }(f) is hyperbolic and has no cycles, then f satisfies Axiom A, and hence is Ω Ω -stable. Examples are given where L − ( f ) {L^ - }(f) is hyperbolic, there are cycles, and f fails to satisfy Axiom A.