TYPICAL TRANSITIVITY FOR LIFTS OF ROTATIONLESS ANNULUS OR TORUS HOMEOMORPHISMS

We say that a homeomorphism h of the base space X (which may be either the annulus or n-torus, n greater than or equal to 2) is rotationless if it is area-preserving and has a lift ($) over tilde h to the covering space ($) over tilde X ([0, 1] x R or R(n)) with mean translation zero (integral(Omega)(($) over tilde h(x)-x)dx=0, where Omega is [0, 1] x [O, 1]). We prove (Theorem 1) that in the space of rotationless homeo-morphisms of X with the uniform topology, the subspace consisting of homeomorphisms with transitive lifts to ($) over tilde X contains a dense G(delta) 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.