  Let M be a Riemann surface with boundary $\partial M$ and genus greater than
zero. Let $\Gamma$ be the mapping class group of M fixing $\partial M$. The
group $\Gamma$ acts on ${\mathcal M}_{\mathcal C} = \Hom_{\mathcal
C}(\pi_1(M),SU(2)/SU(2)$ which is the space of SU(2)-gauge equivalence classes
of flat SU(2)-connections on M with fixed holonomy on $\partial M$. We study
the topological dynamics of the $\Gamma$-action and give conditions for the
individual $\Gamma$-orbits to be dense in ${\mathcal M}_{\mathcal C}$.
