Multiple disjointness and invariant measures on minimal distal flows

Juho Rautio¹

¹ Department of Mathematical Sciences University of Oulu PL 8000, FI-90014 Oulun yliopisto, Finland

Studia Mathematica, Tome 228 (2015) no. 2, pp. 153-175

Résumé

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.

Détail

DOI : 10.4064/sm228-2-4

Keywords: examine multiple disjointness minimal flows conditions under which product collection minimal flows itself minimal main theorem states collection minimal flows common phase group assuming each flow satisfies certain structural algebraic conditions product prod minimal only prod minimal where maximal equicontinuous factor importantly result holds each distal phase group mathbb mathbb apply idea construct large minimal distal product flows many ergodic measures determine exact cardinality ergodic invariant measures universal minimal distal t flow equivalently determine cardinality extreme invariant means mathcal space distal functions cardinality mathfrak ergodic invariant measures size quotient mathcal closed subspace unique invariant mean found non separable using techniques