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A topological version of a theorem of Veech and almost simple flows

Published online by Cambridge University Press:  19 September 2008

Eli Glasner
Eli Glasner
Affiliation:
School of Mathematical Sciences, The Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel
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Abstract

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Almost simple (AS) minimal flows are defined and it is shown that any factor map of an AS flow is, up to almost 1−1 equivalence, a group factor. An analogous theorem for metric, regular, point distal extensions is proved. In particular a theorem of Gottschalk is strengthened to show that any regular, point distal, metric flow is equicontinuous. When the acting group T is commutative it is shown that every proper minimal joining of an AS flow X and a minimal flow Y, is, up to almost 1−1 extensions, the relative product of X and Y over a common factor which is a group factor of X.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 10 , Issue 3 , September 1990 , pp. 463 - 482
Copyright
Copyright © Cambridge University Press 1990

References

REFERENCES

[A]Auslander, J.. Endomorphisms of minimal sets. Duke Math. J. 30 (1963), 605–614.CrossRefGoogle Scholar
[A1]Auslander, J.. Minimal Flows and Their Extensions. Math. Studies 153, North-Holland: Amsterdam, 1988.Google Scholar
[B]Bronšteǐn, I. U.. Extensions of Minimal Transformation Groups. Sijthoff & Noordhoff: 1979.CrossRefGoogle Scholar
[E1]Ellis, R.. Lectures on Topological Dynamics. Benjamin: New York, 1969.Google Scholar
[E2]Ellis, R.. The Veech structure theorem. Trans. Amer. Math. Soc. 186 (1973), 203–218.CrossRefGoogle Scholar
[E-G]Ellis, R. & Glasner, S.. Pure weak mixing. Trans. Amer. Math. Soc. 243 (1978), 135–146.CrossRefGoogle Scholar
[G1]Glasner, S.. Proximal flows. Lecture Notes in Math. 517. Springer Verlag: New York, 1976.CrossRefGoogle Scholar
[G-W]Glasner, S. & Weiss, B.. A weakly mixing upside down tower of isometric extensions. Ergod. Th. & Dynam. Sys. 1 (1981), 151–157.CrossRefGoogle Scholar
[Go]Gottschalk, W. H.. Transitivity and equicontinuity. Bull. Amer. Math. Soc. 54 (1948), 982–984.CrossRefGoogle Scholar
[J]Junco, A. delOn minimal self-joinings in topological dynamics. Ergod. Th. & Dynam. Sys. 7 (1987), 211–227.CrossRefGoogle Scholar
[J-R]del Junco, A. & Rudolph, D.. On ergodic actions whose self-joinings are graphs. Ergod. Th. & Dynam. Sys. 7 (1987), 531–557.CrossRefGoogle Scholar
[V1]Veech, W. A.. Topological Dynamics, Bull. Amer. Math. Soc. 83 (1977), 775–830.CrossRefGoogle Scholar
[V2]Veech, W. A.. A criterion for a process to be prime. Monat. Math. 94 (1982), 335–341.CrossRefGoogle Scholar
[V3]Veech, W. A.. Point-distal flows. Amer. J. Math. 92 (1970), 205–242.CrossRefGoogle Scholar