Necessary conditions for stability of diffeomorphisms

S. Smale has recently given sufficient conditions for a diffeomorphism to be $\Omega$-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 $\Omega$-stable diffeomorphism $f$ has only hyperbolic periodic points and moreover that if $p$ is a periodic point of period $k$ then the $k$th roots of the eigenvalues of $df_p^k$ are bounded away from the unit circle. Other results concern the necessity of transversal intersection of stable and unstable manifolds for an $\Omega$-stable diffeomorphism.