## 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

### Citations:

Zbl 0552.28016; Zbl 0387.60084
Full Text:

### References:

 [1] V. A. Rokhlin, ?Endomorphisms of compact commutative groups,? Izv. Akad. Nauk SSSR, Ser. Mat.,13, 323-340 (1949). [2] S. A. Kalikov, ?Twofold mixing implies threefold mixing for rank one transformations,? Ergodic Theory Dynamical Systems,4, 237-259 (1984). · Zbl 0552.28016 [3] F. Ledrappier, ?Un champ marcovien peut être d’entropie null et mélangeant,? C. R. Acad. Sci. Paris Ser. A,287, 561-563 (1978). · Zbl 0387.60084 [4] V. V. Ryzhikov, ?Mixing, rank, and minimal self-joinings of measure-preserving maps,? Preprint VINITI, 1-68 (1991). [5] D. Rudolph, ?An example of a measure-preserving map with minimal self-joinings, and applications,? J. Analyse Math.,35, 97-122 (1979). · Zbl 0446.28018 [6] A. del Junco and D. Rudolph, ?On ergodic actions whose self-joinings are graphs,? Ergodic Theory Dynamical Systems,7, 531-557 (1987). · Zbl 0646.60010 [7] A. M. Vershik, ?Multivalued mappings with invariant measure (polymorphisms) and Markov operators,? Zap. Nauchn. Sem. LOMI,72, 26-62 (1977). · Zbl 0408.28014 [8] A. M. Vershik and A. L. Fedorov, Trajectory Theory [in Russian], Current Problems in Mathematics. Newest Results. Itogi Nauki i Tekhniki, Akad. Nauk SSSR, VINITI,26, 171-212 (1985). [9] V. V. Ryzhikov, ?A remark on multiple mixing,? Usp. Mat. Nauk,44, No. 1, 205-206 (1989). [10] V. V. Ryzhikov, ?Joinings of dynamical systems, approximations, and mixing,? Usp. Mat. Nauk,46, No. 5, 177-178 (1991). · Zbl 0791.28013 [11] M. Ratner, ?Horocycle flows, joinings and rigidity of products,? Ann. Math.,118, 277-313 (1983). · Zbl 0556.28020 [12] Ya. G. Sinai, ?A weak isomorphism of transformations with invariant measure,? Mat. Sb.,63, No. 1, 23-42 (1964). [13] A. M. Stepin, ?Spectral properties of generic dynamical systems,? Izv. Akad. Nauk SSSR, Ser. Mat.,50, No. 4, 801-834 (1986). [14] R. Graham, Rudiments of Ramsey Theory, Providence, Rhode Island (1981). [15] J. R. Blum and D. L. Hanson, ?On the mean ergodic theorem for subsequences,? Bull. Am. Math. Soc.,66, 308-311 (1960). · Zbl 0096.09005
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