Cyclicity and invariant subspaces in the Dirichlet spaces

Let $\mu$ be a positive finite measure on the unit circle and $$\mathcal{D}$ (\mu)$ the associated Dirichlet space. The generalized Brown-Shields conjecture asserts that an outer function $f\in \mathcal{D}(\mu )$ is cyclic if and only if $c\mathrm{\_}\mu (Z(f))=0$, where $c\mathrm{\_}\mu$ is the capacity associated with $\mathcal{D}(\mu )$ and $Z(f)$ is the zero set of $f$. In this paper we prove that this conjecture is true for measures with countable support. We also give in this case a complete and explicit characterization of invariant subspaces.