Finitarily Splitting Skew Products

Abstract

Skew products with ergodic automorphisms of compact abelian groups arise naturally in several contexts. For example, suppose S is an automorphism of the compact group G, and H is an S-invariant closed subgroup. By taking a measurable cross section to the quotient map G → G/H, the transformation S can be regarded as a skew product of the quotient automorphism SG/H with the restriction SH of S to H. We can study S by studying the simpler components, SG/H and SH, and how they are joined in a skew product. This method was used in proving that ergodic automorphisms of compact groups are measure theoretically isomorphic to Bernoulli shifts [3]. Crucial to this method is the result that if SH is ergodic, then the skew product S measure theoretically splits into the direct product SG/H × SH.