Relatively weakly mixing models for dynamical systems

Abstract

A classical result in ergodic theory says that there always exists a topological model for any factor map ${\it\pi}:(X,{\mathcal{X}},{\it\mu},T)\rightarrow (Y,{\mathcal{Y}},{\it\nu},S)$ of ergodic systems. That is, there are some topological factor map $\hat{{\it\pi}}:(\hat{X},\hat{T})\rightarrow ({\hat{Y}},{\hat{S}})$ and invariant measures $\hat{{\it\mu}}$, $\hat{{\it\nu}}$ such that the diagram $$\begin{eqnarray}\displaystyle & & \displaystyle \nonumber\end{eqnarray}$$ is commutative, where ${\it\phi}$ and ${\it\psi}$ are measure theoretical isomorphisms. In this paper, we show that one can require that in the above result $\hat{{\it\pi}}$ is either weakly mixing or finite-to-one. Also, we present some related questions.