On the rates of convergence of orbits in semigroups of holomorphic functions

Abstract

Let $({\varphi }_t)$ be a continuous semigroup of holomorphic self-maps of the unit disk $\mathbb{D}$ with Denjoy-Wolff point $\tau \in \overline{\mathbb{D}}$. We study the rate of convergence of the forward orbits of $({\varphi }_t)$ to the Denjoy-Wolff point by finding explicit bounds for the quantity $|{\varphi }_t(z)-\tau |$, $z\in \overline{\mathbb{D}}$, $t>0$. We further discuss the corresponding rate of convergence for the backward orbits of $({\varphi }_t)$.

39 pages, 4 figures