Open homomorphisms of minimal flows

Abstract

The consideration of homomorphisms (‘‘extensions’’) of minimal flows is central to topological dynamics. Building upon two earlier papers of the first author, with two different collaborators, we study open extensions of minimal flows. These are classified using maximally highly proximal generators, certain closed subsets of the universal minimal flow. The orbit closures of such in the space of closed subsets are the maximally highly proximal minimal flows. Given a maximally highly proximal generator C, the action of the universal minimal flow on a space gives rise to a “C extension” (the inverse images are determined by C). These are open and every open homomorphism is in fact a C extension for some MHP generator C. This generalizes the classical special case of RIC extensions. Moreover, the open extensions can be represented as quasi-factors (minimal sets in the space of closed subsets). We also consider a category whose objects are extensions of minimal flows and define the morphisms of these.