$F$-minimal sets

Abstract

An $F$-minimal set is the simplest proximal extension of an equicontinuous minimal set. It has one interesting proximal cell and all the points in this proximal cell are uniformly asymptotic. The Sturmian minimal sets are the best known examples of $F$-minimal sets. Our analysis of them is in terms of their maximal equicontinuous factors. Algebraically speaking $F$-minimal sets are obtained by taking an invariant *-closed algebra of almost periodic functions and adjoining some suitable functions to it. Our point of view is to obtain these functions from the maximal equicontinuous factor. In §3 we consider a subclass of $F$-minimal sets which generalize the classical Sturmian minimal sets, and in §4 we examine the class of minimal sets obtained by taking the minimal right ideal of an $F$-minimal set and factoring by a closed invariant equivalence relation which is smaller than the proximal relation.