Lifting mixing properties by Rokhlin cocycles (Q2908157)

The authors study the problem of lifting various mixing properties from a base automorphism \(T\) on a Lebesque space \((X,\mathcal{B},\mu)\) to skew products of the form \(T_{\phi, (S_g)_{g \in G}}\), where \(\phi~:~X\to G\) is a cocycle with values in locally compact abelian group \(G\) and \((S_g)_{g \in G}\) is a family of automorphisms of a Lebesgue space \((Y,\mathcal{C},\nu)\). We recall that \(T_{\phi, (S_g)_{g \in G}}\) acts on the product space \((X \times Y,\mathcal{B} \otimes \mathcal{C},\mu \otimes \nu)\) by NEWLINE\[NEWLINET_{\phi, (S_g)_{g \in G}}(x,y)=(Tx,S_{\phi(x)}(y)).NEWLINE\]NEWLINE The authors prove that whenever \(T\) is ergodic (midly mixing, mixing) but \(T_{\phi, (S_g)_{g \in G}}\) is not ergodic (is not mildly mixing, not mixing), then on a non-trivial factor \(\mathcal{A} \subset \mathcal{C}\) of \((S_g)_{g \in G}\), the corresponding Rokhlin cocycle \(x \mapsto S_{\phi(x)}|{\mathcal{A}}\) is a coboundary (a quasi-coboundary).