Almost proximal extensions of minimal flows

TL;DR

This paper investigates a class of extensions of minimal flows called almost proximal extensions, characterizing their structure and providing examples of extensions with specific properties related to proximality and fiber size.

Contribution

It introduces the concept of almost proximal extensions, establishes a structure theorem, and constructs explicit examples distinguishing them from proximal and almost finite-to-one extensions.

Findings

Almost proximal extensions are either almost finite to one or contain uncountable scrambled subsets.

A dichotomy theorem classifies these extensions into two distinct types.

Explicit examples of extensions with specific proximality properties are constructed.

Abstract

In this paper we study almost proximal extensions of minimal flows. Let $\pi: (X,T)\rightarrow (Y,T)$ be an extension of minimal flows. $\pi$ is called an almost proximal extension if there is some $N \in \mathbb{N}$ such that the cardinality of any almost periodic subset in each fiber is not greater than $N$. When $N=1$, $\pi$ is proximal. We will give the structure of $\pi$ and give a dichotomy theorem: any almost proximal extension of minimal flows is either almost finite to one, or almost all fibers contain an uncountable strongly scrambled subset. Using category method Glasner and Weiss showed the existence of proximal but not almost one to one extensions [18]. In this paper, we will give explicit such examples, and also examples of almost proximal but not almost finite to one extensions.