Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-06-08T13:14:11.511Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Anslander, J. and Glasner, S., Distal and highly proximal extensions of minimal flows, Preprint, U. of Maryland, Technical Report (1976).Google Scholar
2. Furstenberg, H. and Glasner, S., On the existence of isometric extensions, preprint.Google Scholar
3. Glasner, S., Proximal flows, Lecture Notes in Math. 517 (Springer-Verlag, New York, 1976).Google Scholar
4. Glasner, S., Relatively invariant measures, Pacific J. Math. 58 (1975), 393–410.Google Scholar
5. Glasner, S., A metric minimal flow whose enveloping semigroup contains finitely many minimal ideals is PI, Israel J. of Math. 22 (1975), 87–92.Google Scholar
6. McMahon, D., Relativized weak disjointness and relatively invariant measures, Trans. AM. 236 (1978), 225–237.Google Scholar
7. Veech, W. A., Point-distal flows, Amer. J. Math. 92 (1970), 205–242.Google Scholar