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  • Ergodic Theory and Dynamical Systems, Volume 7, Issue 4
  • December 1987, pp. 531-557

On ergodic actions whose self-joinings are graphs

  • A. del Junco (a1) and D. Rudolph (a2)
  • DOI:
  • Published online: 01 September 2008

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.

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[Fu]H. Furstenberg . Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton University Press, Princeton (1981).

[Gl]S. Glasner . Quasi-factors in ergodic theory. Israel J. Math. 45 (1983), 198208.

[Ju2]A. del Junco . A family of counterexamples in ergodic theory. Israel J. Math. 44 (1983), 160188.

[J, P]A. del Junco & K. Park . An example of a measure-preserving flow with minimal self-joinings. J. d'Analyse Math. 42 (1983), 199211.

[JRS]A. del Junco , M. Rahe & L. Swanson . Chacón's automorphism has minimal self-joinings. J. d'Analyse Math. 37 (1980), 276284.

[Ma2]G. W. Mackey . Borel structures in groups and their duals. Trans. Amer. Math. Soc. 85 (1957), 134165.

[Ram]A. Ramsay . Virtual groups and group actions. Advances in Math. 6 (1971), 253322.

[Ru1]D. Rudolph . An example of a measure-preserving map with minimal self-joinings, and applications. J. d'Analyse Math. 35 (1979), 97122.

[Ru2]D. Rudolph . The second centralizer of a Bernoulli shift is just its powers. Israel J. Math. 29 (1978), 167178.

[Va2]V. S. Varadarajan . Groups of automosphisms of Borel spaces. Trans. Amer. Math. Soc. 109 (1963), 191220.

[Ve]W. A. Veech . A criterion for a process to be prime. Monatshefte Math. 94 (1982), 335341.

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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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