Joinings and multiple mixing of finite rank actions
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.
scientific article; zbMATH DE number 652157