Relatively weakly mixing models for dynamical systems (Q2813946)

The authors show that the topological model for any factor map of two ergodic systems is either weakly mixing or finite-to-one. The key points of the proof are: (i) the generalization of a well-known lemma of Rohlin, that is, given any nonperiodic ergodic system \((X,\mathcal{X},\mu,T)\), a positive integer \(N\), and \(\epsilon>0\), there is a subset \(B\) such that \(B\),\dots,\(T^{-N+1}(B)\) are pairwise disjoint and \(\mu(\cup^{N-1}_{j=0}T^{-j}(B))>1-\epsilon\), where \((X,\mathcal{X},\mu)\) is a Lebesgue probability space and \(T\) is an invertible measure preserving transformation; (ii) the disintegration of two measures related to the two original measurable systems, and the application of the self-joining; (iii) the application of the partition and its corresponding symbolic representation for an ergodic system.NEWLINENEWLINETwo questions are provided, one is about unique ergodicity of the topological model, another is about a famous result of \textit{R. I. Jewett} [J. Math. Mech. 19, 717--729 (1970; Zbl 0192.40601)] and \textit{W. Krieger} [Proc. 6th Berkeley Sympos. math. Statist. Probab., Univ. Calif. 1970, 2, 327--346 (1972; Zbl 0262.28013)], generalized by \textit{B. Weiss} [Bull. Am. Math. Soc., New Ser. 13, 143--146 (1985; Zbl 0615.28012)].