Compositional frequent hypercyclicity on weighted Dirichlet spaces

It is proved that, in most cases, a scalar multiple of a linear-fractional generated composition operator λ C_φ acting on a weighted Dirichlet space S_ν of holomorphic functions in the open unit disk is frequently hypercyclic if and only if it is hypercyclic. In fact, this holds for all triples (ν ,λ , φ ) with the possible exception of those satisfying ν ∈ [1/4,1/2), \, |λ | = 1, \, φ = a parabolic automorphism.