A condition for mixing of skew products (Q1577675)

Let \(\theta\) be an automorphism of a Lebesgue space \((\Omega,{\mathcal F},P)\), and let \(\varphi:= \{\varphi(\omega): \omega\in\Omega\}\) be a family of measurable transformations of some measurable space \(({\mathcal M},\beta)\) such that \((\omega,x) \mapsto \varphi(\omega)x\) is \(({\mathcal F}\otimes \beta,\beta)\) measurable. Hence \(\theta(\omega,x):= (\theta\omega, \varphi(\omega)x)\) defines a skew product transformation on \(\Omega\times {\mathcal M}\) with base transformation \(\theta\) and fibre maps \(\varphi(\omega)\). This paper deals with conditions for a \(\theta\)-invariant measure \(\mu\) to be mixing. The authors introduce the notion of fibre-mixing, that yields a sufficient (but not necessary) condition for a skew product with mixing base to be mixing.