Multiple disjointness and invariant measures on minimal distal flows

TL;DR

This paper establishes a characterization of multiple disjointness in minimal distal flows and determines the cardinality of invariant means on spaces of distal functions, revealing they are of size continuum.

Contribution

It provides a new criterion for multiple disjointness in minimal distal flows and computes the exact cardinality of invariant means on certain function spaces.

Findings

Multiple disjointness characterized by maximal equicontinuous factors.

Invariant means on $\\mathcal{D}(\mathbb{Z})$ and $\\mathcal{D}(\mathbb{R})$ have cardinality $2^{\mathfrak{c}}$.

Quotients of these spaces by subspaces with unique invariant mean are non-separable.

Abstract

As the main theorem, it is proved that a collection of minimal $PI$-flows with a common phase group and satisfying a certain algebraic condition is multiply disjoint if and only if the collection of the associated maximal equicontinuous factors is multiply disjoint. In particular, this result holds for collections of minimal distal flows. The disjointness techniques are combined with Furstenberg's example of a minimal distal system with multiple invariant measures to find the exact cardinalities of (extreme) invariant means on $\mathcal{D}(\mathbb{Z})$ and $\mathcal{D}(\mathbb{R})$, the spaces of distal functions on $\mathbb{Z}$ and $\mathbb{R}$, respectively. In all cases, this cardinality is $2^{\mathfrak{c}}$. The size of the quotient of $\mathcal{D}(\mathbb{Z})$ or of $\mathcal{D}(\mathbb{R})$ by a closed subspace with a unique invariant mean is observed to be non-separable by applying the same ideas.