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arXiv · 2511.19022

Semigroup models for discrete holomorphic iteration in the unit disc

Argyrios Christodoulou·Konstantinos Zarvalis
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Abstract

The main goal of this article is to develop a technique that allows us to partially embed the orbit of any 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 results allow 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.

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BibTeXRIS

Argyrios Christodoulou, Konstantinos Zarvalis. 2026-09-14. Semigroup models for discrete holomorphic iteration in the unit disc. https://doi.org/10.1016/j.aim.2026.111203

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