On the Spectrum of Cartesian Powers of Classical Automorphisms

Abstract

We prove the following statement: the set of all essential spectral multiplicities of $T^{\left( n \right)} = T \times \cdot \cdot \cdot \times T$ (n times) is $\left\{ {n,\left( {n - 1} \right),...,n!} \right\}$ on $\left\{ {{\text{const}}} \right\}^ \bot$ for Chacon transformations T, or, equivalently, the operator T(ⁿ) has a simple spectrum on the subspace C _Sim of all functions that are invariant with respect to permutations of the coordinates. As an immediate consequence of this fact, we have the disjointness of all convolution powers of the spectral measure for Chacon transformations. If n=2, then T× T has a homogeneous spectrum of multiplicity 2 on $\left\{ {{\text{const}}} \right\}^ \bot$, i.e., this is a solution of Rokhlin's problem for Chacon transformations. Similar statements are considered for other classical automorphisms.