Non-ergodicity on the SU(2)-character varieties

We describe the dynamics of a group Γ generated by Dehn twists along two filling multi-curves or a family of filling curves on the SU(2)-representation variety of closed surfaces. Consequently, we provide explicit Γ-invariant rational functions on the representation variety of the genus two closed surface S₂ for some pair of multi-curves. We establish a similar result for the SU(2)-character variety of genus four non-orientable surfaces N₄ for some family of filling curves.