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Joinings and multiple mixing of finite rank actions. (English. Russian original) Zbl 0853.28011

Funct. Anal. Appl. 27, No. 2, 128-140 (1993); translation from Funkts. Anal. Prilozh. 27, No. 2, 63-78 (1993).
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 S. A. Kalikov [Ergodic Theory Dyn. Syst. 4, 237-259 (1984; Zbl 0552.28016)]. As is shown by 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.

MSC:

28D05 Measure-preserving transformations
28D15 General groups of measure-preserving transformations
47D07 Markov semigroups and applications to diffusion processes
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