TY - JOUR

T1 - A FAMILY OF DISTAL FUNCTIONS AND MULTIPLIERS FOR STRICT ERGODICITY

AU - Glasner, Eli

N1 - Publisher Copyright:
© 2023 Juliusz Schauder Centre for Nonlinear Studies Nicolaus Copernicus University in Toruń.

PY - 2023/6

Y1 - 2023/6

N2 - 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 Sd 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|.

AB - 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 Sd 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|.

KW - and phrases. Fistal functions

KW - strict ergodicity

KW - the Weyl algebra

UR - http://www.scopus.com/inward/record.url?scp=85168635010&partnerID=8YFLogxK

U2 - 10.12775/TMNA.2022.030

DO - 10.12775/TMNA.2022.030

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AN - SCOPUS:85168635010

SN - 1230-3429

VL - 61

SP - 661

EP - 680

JO - Topological Methods in Nonlinear Analysis

JF - Topological Methods in Nonlinear Analysis

IS - 2

ER -