Cyclicity and invariant subspaces in Dirichlet spaces

Let μ   be a positive finite measure on the unit circle and D(μ)D(μ) the associated Dirichlet space. The generalized Brown–Shields conjecture asserts that an outer function f∈D(μ)f∈D(μ) is cyclic if and only if cμ(Z(f))=0cμ(Z(f))=0, where cμcμ is the capacity associated with D(μ)D(μ) and Z(f)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.