Flows with minimal subdynamics

Abstract

Let $\hat{I}“$ be a countably infinite discrete group. A $\hat{I}“$-flow $X$ (i.e., a nonempty compact Hausdorff space equipped with a continuous action of $\hat{I}“$) is called $S$-minimal for a subset $S\subseteq \hat{I}“$ if the partial orbit $S\cdot x$ is dense for every point $x\in X$. We show that for any countable family $(S_n{)}_{n\in \mathbb{N}}$ of infinite subsets of $\hat{I}“$, there exists a free $\hat{I}“$-flow $X$ that is $S_n$-minimal for all $n\in \mathbb{N}$; additionally, $X$ can be taken to be a subflow of $2^{\hat{I}}“$. This vastly generalizes a result of Frisch, Seward, and Zucker, in which each $S_n$ is required to be a normal subgroup of $\hat{I}“$. As a corollary, we show that for a given Polish $\hat{I}“$-flow $X$, there exists a free $\hat{I}“$-flow $Y$ disjoint from $X$ in the sense of Furstenberg if and only if $X$ has no wandering points. This completes a line of inquiry started by Glasner, Tsankov, Weiss, and Zucker. As another application, we strengthen some of the results of Gao, Jackson, Krohne, and Seward on the structure of Borel complete sections. For example, we show that if $B$ is a Borel complete section in the free part of $2^{\hat{I}}“$, then every union of sufficiently many shifts of $B$ contains an orbit (previously, this was only known for open sets $B$). Although our main results are purely dynamical, their proofs rely on recently developed machinery from descriptive set-theoretic combinatorics, namely the asymptotic separation index introduced by Conley, Jackson, Marks, Seward, and Tucker-Drob and its links to the Lovász Local Lemma.

28 pp., 2 figures