From discrete iteration in the unit disc to continuous semigroups of holomorphic functions
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909920696205312 |
|---|---|
| author | Christodoulou, Argyrios Zarvalis, Konstantinos |
| author_facet | Christodoulou, Argyrios Zarvalis, Konstantinos |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19022 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | From discrete iteration in the unit disc to continuous semigroups of holomorphic functions Christodoulou, Argyrios Zarvalis, Konstantinos Complex Variables Dynamical Systems Primary: 30D05, 37F44, Secondary: 30C45, 30D40, 30F45 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. |
| title | From discrete iteration in the unit disc to continuous semigroups of holomorphic functions |
| topic | Complex Variables Dynamical Systems Primary: 30D05, 37F44, Secondary: 30C45, 30D40, 30F45 |
| url | https://arxiv.org/abs/2511.19022 |