Most Monothetic Extensions Are Rank-1

摘要

Let T be an ergodic automorphism of a standard probability space and let G be acompact metrizable monothetic group. The space of all measurable G-cocycles is endowed with the L(sup 1) topology. A proof that if T is sufficiently well approximated by cyclic automorphisms then a generic G-extension is rank-1 is given. Moreover, a generic cocycle is weakly mixing. The counterparts of these results are also given in the case of continuous Anzai cocycles of topological degree zero.