From discrete iteration in the unit disc to continuous semigroups of holomorphic functions

TL;DR

This paper bridges discrete holomorphic iteration and continuous semigroups in the unit disc, providing new techniques to analyze orbit convergence rates and geometric behavior of holomorphic functions.

Contribution

It introduces a novel method to embed discrete orbits into semigroups, extending previous work to non-univalent functions and enabling detailed asymptotic analysis.

Findings

Established sharp convergence rate estimates for orbits

Demonstrated orbits behave like quasi-geodesics under certain conditions

Extended semigroup embedding techniques to broader classes of functions

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.