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Multiple disjointness and invariant measures on minimal distal flows

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Rautio J.

Studia Mathematica

2015

tom 228

nr 2

153-175

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 𝓓(T), the space of distal functions on T. This cardinality is $2^{𝔠}$ for both ergodic and invariant measures. The size of the quotient of 𝓓(T) by a closed subspace with a unique invariant mean is found to be non-separable by using the same techniques.

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1 Studia Mathematica

1 Rautio J.

1 2015

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Multiple disjointness and invariant measures on minimal distal flows