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Relativized Weak Mixing of Uncountable Order

Published online by Cambridge University Press:  20 November 2018

Douglas McMahon
Douglas McMahon*
Affiliation:
Arizona State University, Tempe, Arizona
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We show that if Y is a metric minimal flow and θ: Y→Z in an open homomorphism that has a section (i.e., a RIM), and if S(θ)= R(θ),then °YΩ contains a dense set of transitive points, where Ω is the first uncountable ordinal

YΩ = П{Y:1 ≦ α < Ω and α not a limit ordinal}, and

°YΩ = {y ∈ YΩ:θ(yα)= θ(yβ)for 1 ≦ α,β < Ω and α, β not limit ordinals},

S(θ) is the relativized equicontinuous structure relation, and

R(θ)= {(y1,y2) ∈ Y X Y:θ(y1) = θ(y2)}.

We use this to generalize a result of Glasner that a metric minimal flow whose enveloping semigroup contains finitely many minimal ideals is PI, [5].

I would like to thank Professor T. S. Wu for making helpful suggestions, and thank the referee for his time and effort.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 3 , 01 June 1980 , pp. 559 - 566
Copyright
Copyright © Canadian Mathematical Society 1980

References

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