Distal strongly ergodic actions

Eli Glasner, Benjamin Weiss

Research output: Contribution to journalArticlepeer-review


Let η be an arbitrary countable ordinal. Using results of Bourgain, Gamburd, and Sarnak on compact systems with spectral gap, we show the existence of an action of the free group on three generators F3 on a compact metric space X, admitting an invariant probability measure μ, such that the resulting dynamical system .X;μ;F3/ is strongly ergodic and distal of rank η. In particular, this shows that there is an F3 system which is strongly ergodic but not compact. This result answers the open question whether such actions exist.

Original languageEnglish
Pages (from-to)333-340
Number of pages8
JournalGroups, Geometry, and Dynamics
Issue number1
StatePublished - 2022


  • Distal action
  • spectral gap
  • strong ergodicity


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