Distal strongly ergodic actions (Q2141715)

A strongly ergodic, distal, non-compact action of the free group on three generators is constructed, thus answering questions raised by \textit{T. Ibarlucía} and \textit{T. Tsankov} [Ann. Sci. Éc. Norm. Supér. (4) 54, No. 3, 751--777 (2021; Zbl 1490.37002)] and others. The result behind this is a construction using results of \textit{J. Bourgain} et al. [Invent. Math. 179, No. 3, 559--644 (2010; Zbl 1239.11103)] on spectral gaps, for any given countable ordinal \(\eta\), of an action of the free group on three generators on a compact metric space with an invariant probability measure which is strongly ergodic and distal of rank \(\eta\).