For a closed surface $\Sigma$, the mapping class group $Mod(\Sigma)$ has a natural action by pre-composition on the SU(2)-character variety. When the surface is orientable, Goldman showed that the action is ergodic. Palesi on the other hand proved the ergodicity for non-orientable surfaces.
In this talk, we consider a large subgroup $\Gamma$ generated by Dehn-twists along a filling pair of multi-curves or a family of filling curves. Using a description based on square-tiled surfaces, we provide non-ergodic examples on the $SU(2)$-character variety of the non-orientable surface $N_4$ as well as examples on the representation variety of the orientable surface $S_2$, by showing that such a $\Gamma$ admits explicit rational invariant functions.