Topological Dynamics on Moduli Spaces, I

Let M be a one-holed torus with boundary $\mathrm{\partial }M$ (a circle) and $\mathrm{\Gamma }$ the mapping class group of M fixing $\mathrm{\partial }M$. The group $\mathrm{\Gamma }$ acts on ${\mathcal{M}}_{\mathcal{C}}(SU(2))$ which is the space of SU(2)-gauge equivalence classes of flat SU(2)-connections on M with fixed holonomy on $\mathrm{\partial }M$. We study the topological dynamics of the $\mathrm{\Gamma }$-action and give conditions for the individual $\mathrm{\Gamma }$-orbits to be dense in ${\mathcal M}_{\mathcal C}(SU(2))$.