Expanding Maps and Anosov Diffeomorphisms

Abstract

In Section I.1 we highlighted the fact that some maps have an invariant measure naturally associated with them, but that ergodic theory also comprises, among its aims, the dynamical study of maps which are not born with an associated invariant measure. The first step in this direction was showing that every continuous map of a compact metric space has an invariant measure (I.8). The second was the ergodic decomposition theorem and the concept of regular points (II.6). Not much more can be said about continuous maps in general; in order to develop our theory further, we will restrict our attention to differentiable maps of closed manifolds. Before doing that, however, we will introduce the concept of expanding maps of metric spaces—roughly, maps which locally increase distances. For such maps, it is possible to find, among all invariant measures, one whose properties make it especially interesting. We start with two particular cases of the definition, where the results obtained are particularly relevant.