Modular orbits on the representation spaces of compact abelian Lie groups

Abstract

Let $S$ be a closed surface of genus $g$ greater than zero. In the present paper, we study the topological-dynamical action of the mapping class group on the ${\mathbb{T}}^n$-character variety giving necessary and sufficient conditions for $\mathrm{Mod}(S)$-orbits to be dense. As an application, such a characterisation provides a dynamical proof of the Kronecker's theorem concerning inhomogeneous Diophantine approximation.