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On ergodic actions whose self-joinings are graphs

Published online by Cambridge University Press:  19 September 2008

A. del Junco  and
D. Rudolph
A. del Junco
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Canada
D. Rudolph
Affiliation:
Department of Mathematics, University of Maryland, College Park, Maryland, USA
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Abstract

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We call an ergodic measure-preserving action of a locally compact group G on a probability space simple if every ergodic joining of it to itself is either product measure or is supported on a graph, and a similar condition holds for multiple self-joinings. This generalizes Rudolph's notion of minimal self-joinings and Veech's property S.

Main results The joinings of a simple action with an arbitrary ergodic action can be explicitly descnbed. A weakly mixing group extension of an action with minimal self-joinings is simple. The action of a closed, normal, co-compact subgroup in a weakly-mixing simple action is again simple. Some corollaries. Two simple actions with no common factors are disjoint. The time-one map of a weakly mixing flow with minimal self-joinings is prime Distinct positive times in a -action with minimal self-joinings are disjoint.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 7 , Issue 4 , December 1987 , pp. 531 - 557
Copyright
Copyright © Cambridge University Press 1987

References

REFERENCES

[Bo]Bourbaki, N.. Elements of Mathematics, General topology, Part 2. Hermann, Paris (1966).Google Scholar
[Fu]Furstenberg, H.. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton (1981).Google Scholar
[Gl]Glasner, S.. Quasi-factors in ergodic theory. Israel J. Math. 45 (1983), 198–208.Google Scholar
[Ju2]del Junco, A.. A family of counterexamples in ergodic theory. Israel J. Math. 44 (1983), 160–188.CrossRefGoogle Scholar
[Ju2]del Junco, A.. Prime Systems in Ergodic Theory and Toplogical Dynamics, Report of the Department of Mathematics. Technische Hogeschool Delft, Delft (1982).Google Scholar
[J, K]del Junco, A. & Keane, M.. On generic points in the cartesian square of Chacón's transformation. To appear in Ergod. Th. & Dynam. Sys.Google Scholar
[J, P]del Junco, A. & Park, K.. An example of a measure-preserving flow with minimal self-joinings. J. d'Analyse Math. 42 (1983), 199–211.CrossRefGoogle Scholar
[J, R]del Junco, A. & Rudolph, D.. An example of a rigid, prime, simple map. Preprint.Google Scholar
[JRS]del Junco, A., Rahe, M. & Swanson, L.. Chacón's automorphism has minimal self-joinings. J. d'Analyse Math. 37 (1980), 276–284.CrossRefGoogle Scholar
[Le]Ledrappier, F.. Un champ markovien peut etre d'ntropie nulle et melangeant. C.R. Acad. Sci. Paris Ser. A 287 (1978), 561–563.Google Scholar
[Ma1]Mackey, G. W.. Point representations of transformation groups. Illinois J. Math. 6 (1962), 327–335.CrossRefGoogle Scholar
[Ma2]Mackey, G. W.. Borel structures in groups and their duals. Trans. Amer. Math. Soc. 85 (1957), 134–165.CrossRefGoogle Scholar
[Ph]Phelps, R. R.. Lectures on Choquet's Theorem. Van Nostrand, Princeton (1966).Google Scholar
[Ram]Ramsay, A.. Virtual groups and group actions. Advances in Math. 6 (1971), 253–322.CrossRefGoogle Scholar
[Ra]Ratner, M.. Joinings of horocycle flows. Preprint.Google Scholar
[Ru1]Rudolph, D.. An example of a measure-preserving map with minimal self-joinings, and applications. J. d'Analyse Math. 35 (1979), 97–122.Google Scholar
[Ru2]Rudolph, D.. The second centralizer of a Bernoulli shift is just its powers. Israel J. Math. 29 (1978), 167–178.Google Scholar
[Va1]Varadarajan, V. S.. Geometry of Quantum Theory, Vol. II. Van Nostrand Reinhold, New York (1970).Google Scholar
[Va2]Varadarajan, V. S.. Groups of automosphisms of Borel spaces. Trans. Amer. Math. Soc. 109 (1963), 191–220.CrossRefGoogle Scholar
[Ve]Veech, W. A.. A criterion for a process to be prime. Monatshefte Math. 94 (1982), 335–341.CrossRefGoogle Scholar
[Zi1]Zimmer, R.. Extensions of ergodic actions. Illinois J. Math. 20 (1976), 373–409.Google Scholar
[Zi2]Zimmer, R.. Ergodic actions with generalized discrete spectrum. Illinois J. Math. 20 (1976), 555–588.CrossRefGoogle Scholar