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

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.