On real extensions of distal minimal homeomorphisms

TL;DR

This paper establishes a structure theorem for real skew product extensions of distal minimal flows, showing they can be represented as perturbations of Rokhlin skew products, with counterexamples demonstrating the necessity of all components.

Contribution

It provides a novel structural characterization of conservative real skew product extensions of distal minimal flows, including explicit counterexamples.

Findings

Extensions can be represented as perturbations of Rokhlin skew products.

Counterexamples show all components of the construction are necessary.

The structure theorem applies to topologically conservative extensions.

Abstract

We prove a structure theorem for topologically conservative real skew product extensions of distal minimal compact metric $\Z$-flows. The main result states that every such extension can be represented by a perturbation of a Rokhlin skew product. Moreover, we give certain counterexamples to point out that all components of the construction are in fact inevitable.