An example of compact extensions of Kronecker factors and skew products of irrational rotations with finite groups
Let \(S\) and \(T\) be commuting ergodic measure preserving transformations on a probability space \((X, {\mathcal B}, \mu)\). The Kronecker factor \(K_1(T)\) is a \(T\)-invariant sub \(\sigma\)-algebra of \(\mathcal B\), and may be thought of as the maximal discrete spectrum factor of \(T\). It is known that \(ST= TS\) implies \(K_1(T)= K_1(S)\). The purpose of this paper is to study certain compact extensions of the Kronecker factor, called the degree 2 Kronecker factor \(K_2(T)\), and show that they can be different for two commuting ergodic automorphisms. The examples used are finite skew products of rotations. Some new criteria for the ergodicity of such maps are given.
ergodic measure preserving transformations
scientific article; zbMATH DE number 836165