Lifting mixing properties by Rokhlin cocycles

Abstract

AbstractWe study the problem of lifting various mixing properties from a base automorphismT∈Aut(X,ℬ,μ) to skew products of the formTφ,𝒮, where φ:X→Gis a cocycle with values in a locally compact Abelian groupG, 𝒮=(Sg)g∈Gis a measurable representation ofGinAut(Y,𝒞,ν) andTφ,𝒮acts on the product space (X×Y,ℬ⊗𝒞,μ⊗ν) byIt is also shown that wheneverTis ergodic (mildly mixing, mixing) butTφ,𝒮is not ergodic (is not mildly mixing, not mixing), then, on a non-trivial factor 𝒜⊂𝒞 of 𝒮, the corresponding Rokhlin cocyclex↦Sφ(x)∣𝒜is a coboundary (a quasi-coboundary).