Hyperbolic Sets of Diffeomorphisms

Abstract

In the previous chapter, we discussed hyperbolic fixed points. In this chapter we want to consider more general invariant sets consisting of more than one point and perhaps infinitely many points. We give the appropriate definition of hyperbolicity for such sets and show that the continuity of the splitting into stable and unstable bundles follows from the other items in the definition. Then we expound the theory of exponential dichotomies for difference equations and use it to show that hyperbolic sets are expansive and that they are robust under perturbation.