Introduction

In this chapter we determine necessary and sufficient conditions for a measure preserving homeomorphism h of a sigma compact manifold X to be the limit of ergodic homeomorphisms, in the compact-open topology. We have already shown in Prasad's Theorem 12.4 that such an approximation is always possible (for any h) when X is Euclidean space Rn with Lebesgue measure. In Examples 13.1 and 13.2 we showed on the contrary that when h is the unit translation on either the Manhattan manifold or the strip manifold, such an approximation is not possible. The obstruction to an ergodic approximation for these systems was explained in terms of the ends E of the manifold X in Chapter 14: The unit translation on the Manhattan manifold induces a homeomorphism on the ends which is compressible, and hence ergodic approximation is precluded by Lemma 14.15; the unit translation on the strip manifold is incompressible, but since it induces a nonzero charge on the ends, an ergodic approximation is ruled out by Theorem 14.23. Conditions on the ends will be used in this chapter to obtain positive results on ergodic approximation. The results in this chapter come from [20] and [22].

We will obtain complete answers to a number of simple questions regarding ergodic approximation in M[X, μ] with respect to the compactopen topology.