Ergodic continuous skew product actions of amenable groups

Abstract

Given two compact, metric topological dynamical systems (F, 7\ ) and (Z, G, v), where T and G are locally compact separable groups acting continuously on spaces, preserving finite ergodic measures and v respectively, a continuous cocycle a on (Y, 7\ ) defines a skew product T action on Z X Y by (z, y) t -> (z(>\ /), j; /). We prove that for a large class of amenable groups T and, under some very general conditions on spaces Y, Z and <7, residually many continuous cocycles lift various ergodic and mixing properties from Y to Z X Y. Similar results are obtained for non-trivial compact group extensions of (Y, 7\ ).