Introduction

In the previous chapter, we defined the space M[X, μ] of all homeomorphisms of a compact connected manifold X which preserve an OU probability measure μ. In addition, we proved the existence of a ‘Brown map’ ø : In → X, and used it to prove (Theorem 9.7) that typical measure theoretic properties V are also typical in the subspace M[X, ø (∂In), μ] of M[X, μ] consisting of homeomorphisms which pointwise fix the singular set K = ø(∂In). In the next section of this chapter we will show (Theorem 10.3) that this genericity result holds for the full space M[X, μ], although it cannot be established by simple bootstrapping arguments involving the Brown map.

The final section of this chapter considers the existence of fixed points for volume preserving homeomorphisms of the open unit n-cube. Recall that we proved earlier (Theorem 5.5) Montgomery's observation that for n = 2 all such homeomorphisms which are orientation preserving have a fixed point. We will negatively answer the question of Bourgin as to whether Montgomery's result can be extended to higher dimensions or to orientation reversing homeomorphisms. The main tool will be the Homeomorphic Measures Theorem (Theorem 9.1), stated in the previous chapter.