A family of distal functions and multipliers for strict ergodicity - Glasner, Eli (Topological Methods in Nonlinear Analysis, 2023)
We give two proofs to an old result of E. Salehi, showing that the Weyl subalgebra $\mathcal{W}$ of $\ell^∞(\Z)$ is a proper subalgebra of $\mathcal{D}$,the algebra of distal functions. We also show that the family $\mathcal{S}^d$ of strictly ergodic functions in $\mathcal{D}$ does not form an algebra and hence inparticular does not coincide with $\mathcal{W}$.We then use similar constructions to show that a function which is a multiplier for strict ergodicity,either within $\mathcal{D}$ or in general, is necessarily a constant.An example of a metric, strictly ergodic, distal flow is constructed which admits a non-strictly ergodic $2$-fold minimal self-joining.It then follows that the enveloping group of this flow is not strictly ergodic (as a $T$-flow).Finally we show that the distal, strictly ergodic Heisenberg nil-flow is relativelydisjoint over its largest equicontinuous factor from the universal Weyl flow $|\mathcal{W}|$.