Non-ergodicity on the $\mathsf{SU}(2)$-character varieties

Abstract

We describe the dynamics of a group $\Gamma$ generated by Dehn twists along two filling multi-curves or a family of filling curves on the $\mathsf{SU}(2)$-representation variety of closed surfaces. Consequently, we provide explicit $\Gamma$-invariant rational functions on the representation variety of the genus two closed surface $S_2$ for some pair of multi-curves. We establish a similar result for the $\mathsf{SU}(2)$-character variety of genus four non-orientable surfaces $N_4$ for some family of filling curves.