Weak limits of powers, simple spectrum of symmetric products, and rank-one mixing constructions.

*(English)*Zbl 1161.37011
Sb. Math. 198, No. 5, 733-754 (2007); translation from Mat. Sb. 198, No. 5, 137-159 (2007).

This article aims to construct and study properties of (mixing) automorphisms whose symmetric powers having simple spectrum. The motivation for this study comes from Rokhlin’s problem on existence of automorphisms with homogeneous spectrum of finite multiplicity. The author studied this problem earlier in some cases by constructing automorphisms via cutting and stacking method (see [J. Dyn. Control Syst. 5, No. 1, 145–148 (1999; Zbl 0954.37007)] and [Sel. Russ. Math. 1, 13–24 (1999)], where some of the results in the article under consideration were announced).

Given a measure preserving transformation \(T\) of a Lebesgue space \((X, F, \mu), \) let the \(n\)-th symmetric product \(T^{\bigodot n} \) of \(T\) be the transformation \(T\times \dots \times T \) (\(n\) of them) restricted to the \(\sigma\)-algebra \(F^{\bigodot n}\) of all subsets of \(X \times \dots \times X \) (\(n\) of them) which are invariant under all permutations of \(\tau \in S_n. \) The main result is that there exists a mixing \(T\) such that all \(T^{\bigodot n},\;n\geq 2, \) have simple spectrum. Actually, the author obtains a class of automorphisms \(T\), called \(C(\delta)\)-class, constructed as staircase transformations, which also satisfy that \(\sigma^{* k} \perp \sigma^{* m} \) for \(k>m\geq 1, \) where \(\sigma \) is the maximal spectral type of \(T. \) Similar result was obtained for transformations obtained by almost staircase mixing consruction (which is a slight modification of the staircase consruction method). As one application of these results, the author shows the existence of a mixing transformation \(T\) such that \(T\times T^2 \times T^3 \times \dots\) has a simple spectrum.

Given a measure preserving transformation \(T\) of a Lebesgue space \((X, F, \mu), \) let the \(n\)-th symmetric product \(T^{\bigodot n} \) of \(T\) be the transformation \(T\times \dots \times T \) (\(n\) of them) restricted to the \(\sigma\)-algebra \(F^{\bigodot n}\) of all subsets of \(X \times \dots \times X \) (\(n\) of them) which are invariant under all permutations of \(\tau \in S_n. \) The main result is that there exists a mixing \(T\) such that all \(T^{\bigodot n},\;n\geq 2, \) have simple spectrum. Actually, the author obtains a class of automorphisms \(T\), called \(C(\delta)\)-class, constructed as staircase transformations, which also satisfy that \(\sigma^{* k} \perp \sigma^{* m} \) for \(k>m\geq 1, \) where \(\sigma \) is the maximal spectral type of \(T. \) Similar result was obtained for transformations obtained by almost staircase mixing consruction (which is a slight modification of the staircase consruction method). As one application of these results, the author shows the existence of a mixing transformation \(T\) such that \(T\times T^2 \times T^3 \times \dots\) has a simple spectrum.

Reviewer: Doǧan Çömez (Fargo)

##### MSC:

37A30 | Ergodic theorems, spectral theory, Markov operators |

47A35 | Ergodic theory of linear operators |

28D05 | Measure-preserving transformations |