Kwiatkowski, Jan; Lacroix, Yves Finite rank transformation and weak closure theorem. (English) Zbl 1198.37009 Ann. Inst. Henri Poincaré, Probab. Stat. 38, No. 2, 155-192 (2002). Summary: We introduce a new class of cocycles which provides examples of measure preserving dynamical systems \((X,B,\mu,T)\), such that given positive integers \(r\geq 2\) and \(m\geq 1\), possibly infinite, with \((r,m) \neq (\infty,\infty)\), the rank is \(r\) and the order of the quotient group in the measure-theoretic centralizer, \(\#\frac{C(T )}{\mathrm{wcl}\{T^n,~n\in\mathbb{Z}\}}\), is \(m\). Moreover, \(\mathrm{wcl}\{T^n,~n\in\mathbb{Z}\}\) is uncountable. For the case \((r,m) = (\infty,\infty)\), we produce a mixing \(T\). This completes the weak closure theorem of J. King [Ergodic Theory Dyn. Syst. 6, 363–384 (1986; Zbl 0595.47005)]. Cited in 1 Document MSC: 37A05 Dynamical aspects of measure-preserving transformations 37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations 37A25 Ergodicity, mixing, rates of mixing 28D05 Measure-preserving transformations Citations:Zbl 0595.47005 PDFBibTeX XMLCite \textit{J. Kwiatkowski} and \textit{Y. Lacroix}, Ann. Inst. Henri Poincaré, Probab. Stat. 38, No. 2, 155--192 (2002; Zbl 1198.37009) Full Text: DOI Numdam EuDML