Complete frequencies for Koenigs domains
We provide a complete characterization of those non-elliptic semigroups of holomorphic self-maps of the unit disc for which the linear span of exponentials of the eigenvalues of the generator of the corresponding semigroup of composition operators is weak-star dense in $H^\infty$ of the Koenigs domain associated to the semigroup. We also give some sufficient and necessary conditions for completeness in $H^p$. As a tool needed for the results, we introduce and study discontinuities of semigroups.