Almost proximal extensions of minimal flows

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.