  Consider the four punctured sphere ${\mathbb{S}}_4^2$. Each choice of four
traces, one for each puncture, determines a relative character variety for the
representations of the fundamental group of ${\mathbb{S}}_4^2$ in
${\sf{SL}}_2(\mathbb{C})$. We classify the stationary probability measures for
the action of the mapping class group ${\sf{Mod}}({\mathbb{S}}_4^2)$ on these
character varieties.
