Joinings and multiple mixing of finite rank actions
type: Item
instance of: scholarly article
title: Joinings and multiple mixing of finite rank actions
zbMATH Open document ID: 0853.28011
DOI: 10.1007/BF01085983
author: Valery Ryzhikov
publication date: 1994-11-08T00:00:00Z
review text: We prove that the Rokhlin higher-order mixing problem can be solved affirmatively in the class of finite rank transformations. This improves the result of \textit{S. A. Kalikov} [Ergodic Theory Dyn. Syst. 4, 237-259 (1984; Zbl 0552.28016)]. As is shown by \textit{F. Ledrappier} [C. R. Acad. Sci., Paris, Sér. A 287, 561-563 (1978; Zbl 0387.60084)], in the general case, the solution of the Rokhlin problem is negative for actions of the group \(\mathbb{Z}^2\). The results of the present paper can be extended to \(\mathbb{Z}^n\)-actions. Our method is based on the consideration of equations of the form \[ U^*_T {\mathcal J}(x) U_T\equiv {\mathcal J}(Tx),\quad x\in X, \] where \(U_T\) is the operator corresponding to an automorphism \(T\) of a probability measure space, and the solutions are the functions \({\mathcal J}: X\to {\mathcal M}\) with values in a semigroup \(\mathcal M\) of Markov operators. If this equation has no nontrivial solution, then the mixing transformation \(T\) possesses the multiple mixing property.
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 28D15
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 652157
zbMATH Keywords: Rokhlin higher-order mixing problem
zbMATH Keywords: finite rank transformations
zbMATH Keywords: Markov operators
zbMATH Keywords: mixing transformation
zbMATH Keywords: multiple mixing
MaRDI profile type: Publication
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published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 652157