Geodesic
Distal strongly ergodic actions
Eli Glasner1 ; Benjamin Weiss2
1Tel Aviv University, Israel
2Hebrew University of Jerusalem, Israel
Groups, geometry, and dynamics, Tome 16 (2022) no. 1, pp. 333-340

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DOI

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.
DOI : 10.4171/ggd/650
Classification : 37-XX, 22-XX
Mots-clés : Distal action, strong ergodicity, spectral gap

Eli Glasner  1   ; Benjamin Weiss  2

1 Tel Aviv University, Israel
2 Hebrew University of Jerusalem, Israel 
Eli Glasner; Benjamin Weiss. Distal strongly ergodic actions. Groups, geometry, and dynamics, Tome 16 (2022) no. 1, pp. 333-340. doi: 10.4171/ggd/650
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