Multiple disjointness and invariant measures on minimal distal flows
Volume 228 / 2015
Abstract
We examine multiple disjointness of minimal flows, that is, we find conditions under which the product of a collection of minimal flows is itself minimal. Our main theorem states that, for a collection $\{X_i\}_{i \in I}$ of minimal flows with a common phase group, assuming each flow satisfies certain structural and algebraic conditions, the product $\prod_{i \in I} X_i$ is minimal if and only if $\prod_{i \in I} X_i^{\rm eq}$ is minimal, where $X_i^{\rm eq}$ is the maximal equicontinuous factor of $X_i$. Most importantly, this result holds when each $X_i$ is distal. When the phase group $T$ is $\mathbb Z$ or $\mathbb R$, we can apply this idea to construct large minimal distal product flows with many ergodic measures. We determine the exact cardinality of (ergodic) invariant measures on the universal minimal distal $T$-flow. Equivalently, we determine the cardinality of (extreme) invariant means on $\mathcal D(T)$, the space of distal functions on $T$. This cardinality is $2^{\mathfrak{c}}$ for both ergodic and invariant measures. The size of the quotient of $\mathcal D(T)$ by a closed subspace with a unique invariant mean is found to be non-separable by using the same techniques.
