Hyperbolicity and Exponential Dichotomy for Dynamical Systems

Abstract

The aim of this paper is to show that many of the interesting topological consequences of hyperbolicity follow from just one lemma about exponential dichotomy. Our main lemma asserts that there are local stable manifolds and local unstable manifolds associated with a sequence of maps which are close to hyperbolic linear maps, and that certain local stable manifolds and local unstable manifolds have unique points of intersection. Our main lemma also includes detailed estimates for the positions of the local stable and unstable manifolds, and for the behavior of orbits in the local stable and unstable manifolds. This is an exponential dichotomy result because the hypotheses guarantee that orbits diverge either in the forward direction or in the backward direction. In applications the maps represent a given dynamical system, or dynamical systems in a neighbor hood of a given dynamical system, in local coordinates.