Let 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 on a compact metric space , admitting an invariant probability measure , such that the resulting dynamical system is strongly ergodic and distal of rank . In particular, this shows that there is an system which is strongly ergodic but not compact. This result answers the open question whether such actions exist.
Cite this article
Eli Glasner, Benjamin Weiss, Distal strongly ergodic actions. Groups Geom. Dyn. 16 (2022), no. 1, pp. 333–340DOI 10.4171/GGD/650