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Finite rank transformation and weak closure theorem. (English) Zbl 1198.37009

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)].

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
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