On the rate of convergence of semigroups of holomorphic functions at the Denjoy–Wolff point

Abstract

Let $\{{\varphi }_t\}$ be a semigroup of holomorphic self-maps of the unit disc $\mathbb{D}$ with Denjoy–Wolff point $\tau \in \partial \mathbb{D}$. We study the rate of convergence of the semigroup to $\tau$, that is, given $z\in \overline{\mathbb{D}}$, we discuss the behavior of $|{\varphi }_t(z)-\tau |$ as $t$ goes to $+\infty$.