Cyclicity and invariant subspaces in the Dirichlet spaces

Let $μ$ be a positive finite measure on the unit circle and $\mathcal{D} (μ)$ the associated Dirichlet space. The generalized Brown-Shields conjecture asserts that an outer function $f \in \mathcal{D} (μ)$ is cyclic if and only if $c\_μ(Z (f))= 0$, where $c\_μ$ is the capacity associated with $\mathcal{D} (μ)$ 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.

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