Real extensions of distal minimal flows and continuous topological ergodic decompositions

TL;DR

This paper establishes a structure theorem for real skew product extensions of distal minimal flows, showing they can be represented by perturbations of Rokhlin skew products and analyzing their topological ergodic decompositions.

Contribution

It introduces a new representation for recurrent real skew product extensions and proves the continuity and compactness of their ergodic decompositions in the topological setting.

Findings

Extensions are representable by perturbations of Rokhlin skew products

Topological ergodic decompositions are continuous and compact

The minimality of the Mackey action is established

Abstract

We prove a structure theorem for topologically recurrent real skew product extensions of distal minimal compact metric flows with a compactly generated Abelian acting group (e.g. $\Z^d$-flows and $\R^d$-flows). The main result states that every such extension apart from a coboundary can be represented by a perturbation of a so-called Rokhlin skew product. We obtain as a corollary that the topological ergodic decomposition of the skew product extension into prolongations is continuous and compact with respect to the Fell topology on the hyperspace. The right translation acts minimally on this decomposition, therefore providing a minimal compact metric analogue to the Mackey action. This topological Mackey action is a distal (possibly trivial) extension of a weakly mixing factor (possibly trivial), and it is distal if and only if perturbation of the Rokhlin skew product is defined by a topological coboundary.