Introduction

The study of typical properties of volume preserving homeomorphisms of noncompact manifolds was initiated by Prasad [96], with a proof that ergodicity is generic when the manifold is Euclidean n-space Rn, n ≥ 2. Shortly thereafter Prasad's result for ergodicity was extended by Alpern [12, 14] to all properties generic for automorphisms of an infinite Lebesgue space, but still only on the particular manifold Rn. This chapter is devoted to explaining and proving these results for Rn. Unlike the compact case, where the analysis for the cube In is essentially the same as for all compact manifolds, in the noncompact case the study of Rn is directly applicable only to the special class of noncompact manifolds with a single end. However, some of the ideas used here will be of general use for the noncompact setting, so this chapter will give the reader a gentle introduction to the more varied manifolds to come later.

The interest in dynamics on Euclidean space Rn dates at least to the famous Scottish Book [85] of 1935. This was a record kept by Polish mathematicians including Banach, Steinhaus, and Ulam, of problems discussed at the ‘Scottish Cafe’ in Lwów, Poland. Problem 115 of the Scottish Book, posed by Ulam, asks the following question:

In our terminology, this questions asks whether there is a transitive homeomorphism of Rn.