Transitive Homeomorphisms

A mapping h of a topological space is called transitive if for any open sets U and V, there is a positive integer k with hkU ∩ V ≠ ø. For manifolds (actually, for any complete, separable metric space without isolated points) this is equivalent to the existence of a dense orbit under h. For some spaces, it is easy to exhibit transitive homeomorphisms (e.g., irrational rotations of the circle), while for other spaces where they exist more subtlety is required. Besicovitch [40] first defined such a transformation for the plane in 1937, partially answering a question of Ulam (see Chapter 12). In the same year, Oxtoby [89] used the Baire Category Theorem to demonstrate that in fact transitivity is typical for volume preserving homeomorphisms of the cube. At that time, the existence of such transformations had not been established. Recently Xu ([108, 109]) has shown how Besicovitch's transitive homeomorphism of the plane can be used to construct an explicit transitive homeomorphism of the closed unit square and Cairns, Jessup and Nicolau [48] have given examples on the 2-sphere and more generally on quotients of tori.

In this chapter we present a simplification of Oxtoby's original proof based on the combinatorial techniques of Chapter 3, and then we extend this method to show the existence of spatially periodic transitive homeomorphisms of Rn, or equivalently, rotationless homeomorphisms of the torus with transitive lifts.