Dynamics on Markov surfaces: classification of stationary measures

Abstract Consider the four-punctured sphere S 4 2 $\mathbb {S}_4^2$ double struck upper S 4 squared . Each choice of four traces, one for each puncture, determines a relative character variety for the representations of the fundamental group of S 4 2 $\mathbb {S}_4^2$ double struck upper S 4 squared in SL 2 ( C ) ${\textsf {SL}}_2(\mathbb {C})$ sans serif SL Subscript 2 Baseline left parenthesis double struck upper C right parenthesis . We classify the stationary probability measures for the action of the mapping class group Mod ( S 4 2 ) ${\textsf {Mod}}(\mathbb {S}_4^2)$ sans serif Mod left parenthesis double struck upper S 4 squared right parenthesis on these character varieties.