A family of distal functions and multipliers for strict ergodicity

TL;DR

This paper explores the structure of distal functions and multipliers in ergodic theory, proving new relationships among subalgebras, and providing examples that challenge existing assumptions about strict ergodicity.

Contribution

It provides two proofs of Salehi's result on subalgebras, shows that strictly ergodic functions do not form an algebra, and constructs examples of ergodic flows with specific properties.

Findings

Weyl subalgebra is a proper subset of distal functions

Strictly ergodic functions do not form an algebra

Multiplier for strict ergodicity must be constant

Abstract

We give two proofs to an old result of E. Salehi, showing that the Weyl subalgebra $\mathcal{W}$ of $\ell^\infty(\mathbb{Z})$ 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 in particular 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 relatively disjoint over its largest equicontinuous factor from $|\mathcal{W}|$.