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.

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.

Reviewer: Geoffrey R. Goodson (Towson)