On frequencies of parabolic Koenigs domains

Let $({\phi }_t{)}_{t\ge 0}$ be a parabolic semigroup of analytic functions on $\mathbb{D}$, with Koenigs function $h$ and Koenigs domain $\mathrm{\Omega }=h(\mathbb{D})$. We study the point spectrum ${\sigma }_p(\mathrm{\Delta }{\mid }_{H^p})$ of $\mathrm{\Delta }$, the infinitesimal generator of the $C_0$-semigroup $(C_{{\phi }_t}{)}_{t\ge 0}$ of composition operators on $H^p$. This reduces to characterizing the frequencies of $\mathrm{\Omega }$. That is, those $\lambda \in \mathbb{C}$ such that $e^{\lambda h}\in H^p$. We first derive containment relations for ${\sigma }_p(\mathrm{\Delta }{\mid }_{H^p})$ and provide sufficient conditions for its complete characterization. Our approach relies heavily on the geometric properties of $\mathrm{\Omega }$ and on careful estimates of the harmonic measure of some boundary subsets of $\mathrm{\Omega }$. Furthermore, assuming that $\mathrm{\Omega }$ is convex, we also obtain necessary conditions for $\lambda$ to be a frequency of $\mathrm{\Omega }$. Using these, we are able to completely describe ${\sigma }_p(\mathrm{\Delta }{\mid }_{H^p})$ in a broad range of situations e.g. when $\mathrm{\Omega }$ contains an angular sector. We conclude with some consequences regarding the spectrum of the composition operators $(C_{{\phi }_t}{)}_{t\ge 0}$. These results extend a previous work of Betsakos on hyperbolic semigroups.