A minimal PI cascade with $2^{\mathfrak{c}}$ minimal ideals

TL;DR

This paper improves a classical result on minimal flows, then constructs various minimal PI flows and cascades with many minimal left ideals, including the maximal possible number $2^{rak{c}}$, demonstrating the limits of previous implications.

Contribution

It extends the understanding of minimal PI flows by constructing examples with many minimal left ideals, including the maximum possible number, and shows the failure of a converse implication.

Findings

Constructed a metric minimal PI flow with $rak{c}$ minimal left ideals.

Built a minimal PI cascade with $rak{c}$ minimal left ideals.

Created a minimal PI flow with $2^{rak{c}}$ minimal left ideals.

Abstract

We first improve an old result of McMahon and show that a metric minimal flow whose enveloping semigroup contains less than $2^{\mathfrak{c}}$ (where ${\mathfrak{c}} ={2^{\aleph_0}}$) minimal left ideals is PI. Then we show the existence of various minimal PI flows with many minimal left ideals, as follows. For the acting group $G=SL_2(\mathbb{R})^\mathbb{N}$, we construct a metric minimal PI $G$-flow with $\mathfrak{c}$ minimal left ideals. We then use this example and results established in \cite{GW-79} to construct a metric minimal PI cascade $(X,T)$ with $\mathfrak{c}$ minimal left ideals. We go on and construct an example of a minimal PI-flow $(Y, \mathcal{G})$ on a compact manifold $Y$ and a suitable path-wise connected group $\mathcal{G}$ of homeomorphism of $Y$, such that the flow $(Y, \mathcal{G})$ is PI and has $2^{\mathfrak{c}}$ minimal left ideals. Finally, we use this latter example and a theorem of Dirb\'{a}k to construct a cascade $(X, T)$ which is PI (of order 3) and has $2^\mathfrak{c}$ minimal left ideals. Thus this final result shows that, even for cascades, the converse of the implication "less than $2^\mathfrak{c}$ minimal left ideals implies PI", fails.