Probabilistic Tits alternative for circle diffeomorphisms

Abstract

Let ${\mu }_1$, ${\mu }_2$ be probability measures on ${\mathrm{Diff}}_{+}^1(S^1)$ satisfying a suitable moment condition and such that their supports generate discrete groups acting proximally on $S^1$. Let $(f_{\omega }^n{)}_{n\ge 0}$, $(f_{{\omega }^{\prime }}^n{)}_{n\ge 0}$ be two independent realizations of the random walk driven by ${\mu }_1$, ${\mu }_2$, respectively. We show that almost surely there is an $N\in \mathbb{N}$ such that for all $n\ge N$ the elements $f_{\omega }^n$, $f_{{\omega }^{\prime }}^n$ generate a non-abelian free group. The proof is inspired by the strategy by R. Aoun for linear groups. A weaker (and easier) statement holds for measures supported on ${\mathrm{Homeo}}_{+}(S^1)$ with no moment conditions.