From discrete iteration in the unit disc to continuous semigroups of holomorphic functions
Argyrios Christodoulou, Konstantinos Zarvalis
TL;DR
This work provides a bridge between discrete holomorphic iteration in the unit disk and continuous semigroups of holomorphic functions by partially embedding orbits into a semigroup (the semigroup-fication) on a fundamental domain. It extends Bracci–Roth’s embedding to non-univalent maps, establishing that the embedded trajectories share the same dynamic type and asymptotics as the original orbits, and that slopes and quasi-geodesic properties are preserved or reflected. Using this framework, the authors obtain sharp rates for convergence to the Denjoy–Wolff point, coherent slope behavior, and a rich array of consequences for harmonic measure, internal tangency of domains, and operator theory, including precise growth rates for composition operators on Hardy and Bergman spaces. The results unify and sharpen discrete and continuous holomorphic dynamics in the unit disk, with broad implications for rate estimates, geometric function theory, and related operator-theoretic applications.
Abstract
The main goal of this article is to bring together the theories of holomorphic iteration in the unit disc and semigroups of holomorphic functions. We develop a technique that allows us to partially embed the orbit of a holomorphic self-map $f$ of the disc, into a semigroup which captures the asymptotic behaviour of the orbit. This extends the semigroup-fication procedure introduced by Bracci and Roth to non-univalent functions. We use our technique in order to obtain sharp estimates for the rate with which the orbits of $f$ converge to the attracting fixed point; a fundamental, yet underdeveloped, concept in discrete iteration. Moreover, our semigroup-fication allows us to evaluate the slope of the orbits of $f$, and prove that they behave similarly to quasi-geodesic curves precisely when they converge non-tangentially.