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Mixing transformations with homogeneous spectrum. (English. Russian original) Zbl 1247.37008
Sb. Math. 202, No. 8, 1231-1252 (2011); translation from Mat. Sb. 202, No. 8, 139-160 (2011).
Let $$(X,{\mathcal B},\mu)$$ be a Lebesgue probability space and $$T: X\to X$$ an ergodic measure preserving transformation. Rokhlin’s problem concerns the existence of an ergodic transformation $$T$$ having a homogeneous spectrum of multiplicity $$n\in\mathbb{Z}^+$$ for $$n> 1$$. This problem was solved in the case $$n= 2$$ independently by O. N. Ageev [“On ergodic transformations with homogeneous spectrum”, J. Dyn. Control Syst. 5, No. 1, 149–152 (1999; Zbl 0943.37005)] and and V. V. Ryzhikov [“Homogeneous spectrum, disjointness of convolutions, and mixing properties of dynamical systems”, Selected Russian Math. 1, No. 1, 13–24 (1999), arXiv:1206.6093]; subsequently it was solved for $$n> 2$$ by O. N. Ageev [Invent. Math. 160, No. 2, 417–446 (2005; Zbl 1064.37003)]. Both proofs for $$n= 2$$ used a Cartesian square, which Ryzhikov showed could be chosen to be mixing. The question of whether or not Rokhlin’s problem could be solved for $$T$$ mixing remained open, and the purpose of this paper is to show that for each $$n> 1$$ there is a mixing transformation $$T$$ having a homogeneous spectrum of multiplicity equal to $$n$$. The difficulty in generalizing the earlier work lay in the fact that the $$T\times T$$ argument for $$n= 2$$ did not seem to generalize since, for example, $$T\times T\times T$$ always has a non-homogenous spectrum.
In addition, Ageev, for $$n> 2$$, used a genericity type of argument, but the mixing transformations are non-generic. However, a recent paper of the author [Sb. Math. 198, No. 4, 575–596 (2007); translation from Mat. Sb. 198, No. 4, 135–158 (2007; Zbl 1140.37005)] shows that the mixing transformations form a complete metric space with respect to some metric. This property enables the author to solve the problem using a genericity approach.

##### MSC:
 37A25 Ergodicity, mixing, rates of mixing 37A30 Ergodic theorems, spectral theory, Markov operators 37A05 Dynamical aspects of measure-preserving transformations 28D05 Measure-preserving transformations
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