Skip to main content
×
Home
    • Aa
    • Aa

On ergodic actions whose self-joinings are graphs

  • A. del Junco (a1) and D. Rudolph (a2)
Abstract
Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On ergodic actions whose self-joinings are graphs
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      On ergodic actions whose self-joinings are graphs
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      On ergodic actions whose self-joinings are graphs
      Available formats
      ×
Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[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.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax