Rigidity of actions with extreme deviation from multiple mixing
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.
scientific article; zbMATH DE number 6930078