Necessary conditions for stability of diffeomorphisms

Abstract

S. Smale has recently given sufficient conditions for a diffeomorphism to be Ω \Omega -stable and conjectured the converse of his theorem. The purpose of this paper is to give some limited results in the direction of that converse. We prove that an Ω \Omega -stable diffeomorphism f f has only hyperbolic periodic points and moreover that if p p is a periodic point of period k k then the k k th roots of the eigenvalues of d f p k df_p^k are bounded away from the unit circle. Other results concern the necessity of transversal intersection of stable and unstable manifolds for an Ω \Omega -stable diffeomorphism.