Exceptional minimal sets of group actions on the circle
by Steven Hurder

ABSTRACT:

Let $\bf K$ be an exceptional minimal set for a $C^1$-action $\varphi \colon \Gamma \times \mS^1 \to \mS^1$ of a finitely generated group $\Gamma$ on the circle. We prove that if there is no invariant Borel probability measure for the action $\varphi$ supported on $\K$, then $\K$ has finite type. If, in addition, the action is $C^{1+\alpha}$ for some $\alpha > 0$, then $\K$ has Lebesgue measure zero. Consequently, an exceptional minimal set for a $C^2$-action is always finite type and has measure zero.