On frequencies of parabolic Koenigs domains

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