A FAMILY OF DISTAL FUNCTIONS AND MULTIPLIERS FOR STRICT ERGODICITY

We give two proofs to an old result of E. Salehi, showing that the Weyl subalgebra W of ℓ^∞ (Z) is a proper subalgebra of D, the algebra of distal functions. We also show that the family S^d of strictly ergodic functions in D does not form an algebra and hence in particular does not coincide with W. We then use similar constructions to show that a function which is a multiplier for strict ergodicity, either within 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 selfjoining. 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 the universal Weyl flow |W|.