Multiple disjointness and invariant measures on minimal distal flows

Juho Rautio

Studia Mathematica, Tome 231 (2015), p. 153-175 / Harvested from The Polish Digital Mathematics Library

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 ${X_{i}}_{i∈I}$ of minimal flows with a common phase group, assuming each flow satisfies certain structural and algebraic conditions, the product $∏_{i∈I} X_{i}$ is minimal if and only if $∏_{i∈I} X_{i}^{eq}$ is minimal, where $X_{i}^{eq}$ 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 ℤ or 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 (T), the space of distal functions on T. This cardinality is $2^{}$ for both ergodic and invariant measures. The size of the quotient of (T) by a closed subspace with a unique invariant mean is found to be non-separable by using the same techniques.

Détail

Publié le : 2015-01-01

Zbl 06504768

EUDML-ID : urn:eudml:doc:285847