DOI: 10.1017/etds.2026.10329 ISSN: 0143-3857
Dynamics on Markov surfaces: classification of stationary measures
SERGE CANTAT, CHRISTOPHE DUPONT, FLORESTAN MARTIN-BAILLON 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.