Rigidity of actions with extreme deviation from multiple mixing
type: Item
instance of: scholarly article
title: Rigidity of actions with extreme deviation from multiple mixing
zbMATH Open document ID: 1404.37003
DOI: 10.1134/S0001434618050346
author: S. V. Tikhonov
published in: Mathematical Notes
publication date: 2018-08-31T00:00:00Z
review text: Let \(X\) be a measurable space with a Lebesgue measure \(\mu\), \(G\) be a group. An action of a topological group is given by a collection of transformations defined on the space \(X\), where each transformation corresponds to an element of the group, and the composition of two transformations is given by the multiplication of the corresponding elements of the group, i.e., \(L={L^g}_{g\in G}\) with \(L^gL^h=L^{gh}\), and the mapping \(g\to \mu(L^g(A)\cap B)\) is continuous for any measurable sets \(A\) and \(B\). For any positive integer \(k\) and a sequence of transformations \(\{T^{(1)}_i\},\dots,\{T^{(k)}_i\}\subset L\), \(i\in\mathbb{N}\), if \(\lim_{i\to\infty}\mu(\bigcap_jT^{(j)}_iA_j)=\prod^k_{j=1}\mu(A_j)\) for any measurable sets \(A_1,\dots,A_k\), then this set of sequences is said to be \(k\)-mixing. It is evident that \(k\)-mixing implies \((k-1)\)-mixing. The weakened Rokhlin problem for a group \(G\) asks when it is possible that \((k-1)\)-mixing implies \(k\)-mixing. \textit{F. Ledrappier} [C. R. Acad. Sci., Paris, Sér. A 287, 561--563 (1978; Zbl 0387.60084)] constructed an algebraic counterexample to the weakened Rokhlin problem for the group \(\mathbb{Z}^2\). In Ledrappier's construction, an algebraic relation plays an important role. Ledrappier's example is called three-point example, because his algebraic relation involves the intersection of three sets. In this paper, the author generalizes this algebraic relation to \(n\)-point examples by using \(n\) sets. Here the symmetric differences are used to replace the intersection of sets.
reviewed by: Xu Zhang
Mathematics Subject Classification ID: 37A25
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 6930078
zbMATH Keywords: group action
zbMATH Keywords: measure-preserving map
zbMATH Keywords: mixing
zbMATH Keywords: Rokhlin problem
zbMATH Keywords: symmetric difference
MaRDI profile type: Publication
cites work: Q3362614
cites work: Q5790283
cites work: Q4169512
cites work: On the absence of multiple mixing and on the centralizer of measure-preserving actions
cites work: The generic transformation can be embedded in a flow.
cites work: A generic preserving-measure transformation has an extensive centralizer
cites work: Embedding lattice actions in flows with multidimensional time
cites work: Q5690494
cites work: Three results on mixing shapes
cites work: Entropy geometry and disjointness for zero-dimensional algebraic actions
cites work: On the asymmetry of multiple asymptotic properties of ergodic actions
cites work: Dynamical systems of algebraic origin
full work available at URL: s0001434618050346
OpenAlex ID: W2810429185
description: scientific article; zbMATH DE number 6930078