Distal strongly ergodic actions

Abstract

Let $\eta$ 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 $F_3$ on a compact metric space $X$, admitting an invariant probability measure $\mu$, such that the resulting dynamical system $(X,\mu ,F_3)$ is strongly ergodic and distal of rank $\eta$. In particular, this shows that there is an $F_3$ system which is strongly ergodic but not compact. This result answers the open question whether such actions exist.