Dynamics on Markov surfaces: classification of stationary measures

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 ${\mathsf{S}\mathsf{L}}_2(\mathbb{C})$. We classify the stationary probability measures for the action of the mapping class group $\mathsf{M}\mathsf{o}\mathsf{d}({\mathbb{S}}_4^2)$ on these character varieties.