Every C² minimal action of Thompson group T on the circle is ergodic w.r.t. Lebesgue measure; non-minimal actions have exceptional minimal sets of Lebesgue measure zero.
Navas.Groups of circle diffeomorphisms
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Ergodicity of $C^2$ minimal actions of Thompson group $T$ on the circle
Every C² minimal action of Thompson group T on the circle is ergodic w.r.t. Lebesgue measure; non-minimal actions have exceptional minimal sets of Lebesgue measure zero.