Typical Transitivity for Lifts of Rotationless Annulus or Torus Homeomorphisms

Abstract

We say that a homeomorphism h of the base space X (which may be either the annulus or n-torus, n⩾2) is rotationless if it is area-preserving and has a lift h∼ to the covering space X∼([0,1] × R or Rn) with mean translation zero (∫Ω(h∼(x)–x)dx=0, where Ω is [0,1] × [0,1]). We prove (Theorem 1) that in the space of rotationless homeomorphisms of X with the uniform topology, the subspace consisting of homeo-morphisms with transitive lifts to X ∼ contains a dense Gδ subset. This extends our earlier result, valid only when the base space is the annulus, that typical rotationless homeomorphisms have recurrent lifts. Our result also extends that of Besicovitch, who in 1937 exhibited the first transitive homeomorphism of the plane. In this context we establish such a homeomorphism which is additionally spatially periodic.