Asymptotic stability with rate conditions for dynamical systems

Abstract

We announce existence and smoothness theorems for invariant manifolds characterized by asymptotic stability at specified rates. Our theorems require roughly half of a hyperbolic structure, so various known results about stable and unstable manifolds are included as special cases. Even under hyperbolicity assumptions, however, our results give new information about metric properties, invariant foliations, and asymptotic stability with asymptotic phase. Proofs will appear in