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 ($) over tilde h to the cover ing space ($) over tilde X ([0, 1] x R or R(n)) with mean translation zero (integral(Omega)(($) over tilde h(x)-x)dx=0, where Omega is [0, 1 ] x [O, 1]). We prove (Theorem 1) that in the space of rotationless ho meo-morphisms of X with the uniform topology, the subspace consisting of homeomorphisms with transitive lifts to ($) over tilde X contains a dense G(delta) subset. This extends our earlier result, valid only wh en the base space is the annulus, that typical rotationless homeomorph isms have recurrent lifts. Our result also extends that of Besicovitch , who in 1937 exhibited the first transitive homeomorphism of the plan e. In this context we establish such a homeomorphism which is addition ally spatially periodic.