Let be a univalent self-map of the unit disc. We introduce a technique, that we call semigroup-fication, which allows to construct a continuous semigroup of holomorphic self-maps of the unit disc whose time one map is, in a sense, very close to . The semigroup-fication of is of the same type as (elliptic, hyperbolic, parabolic of positive step or parabolic of zero step) and there is a one-to-one correspondence between the set of boundary regular fixed points of with a given multiplier and the corresponding set for . Moreover, in case (and hence ) has no interior fixed points, the slope of the orbits converging to the Denjoy–Wolff point is the same. The construction is based on holomorphic models, localization techniques and Gromov hyperbolicity. As an application of this construction, we prove that in the non-elliptic case, the orbits of converge non-tangentially to the Denjoy–Wolff point if and only if the Koenigs domain of is “almost symmetric” with respect to vertical lines.
Soit une application univalente du disque unité dans lui-même. On introduit une technique, appelée semigroupe-fication, qui nous permet de construire un semigroupe continu d’applications holomorphes du disque unité dans lui-même tel que l’application au temps , , est très proche de . La “ semigroupe-fication ” de est du même type de (elliptique, hyperbolique, parabolique d’étape positive, parabolique d’étape zéro) et il existe une correspondance entre l’ensemble des points fixes de qui sont réguliers, au bord et avec un multiplicateur donné, et le même ensemble pour . De plus, si (et donc ) n’a pas de points fixes à l’intérieur, la pente des orbites qui convergent au point de Denjoy–Wolff est la même. La construction repose sur les modèles holomorphes, les techniques de localisation et l’hyperbolicité de Gromov. Comme application, on démontre que dans le cas non-elliptique, les orbites de convergent au point de Denjoy–Wolff de façon non-tangentielle si et seulement si le domaine de Koenigs de est “presque symétrique” par rapport aux lignes verticales.
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Keywords: Semigroups of holomorphics maps, univalent functions, asymptotic behavior of orbits, iteration theory.
Mot clés : Semigroupes des applications holomorphes, fonctions univalentes, comportement asymptotique des orbites, théorie de l’itération.
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@article{AIF_2023__73_1_251_0,
author = {Bracci, Filippo and Roth, Oliver},
title = {Semigroup-fication of univalent self-maps of the unit disc},
journal = {Annales de l'Institut Fourier},
pages = {251--277},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {73},
number = {1},
year = {2023},
doi = {10.5802/aif.3517},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3517/}
}
TY - JOUR AU - Bracci, Filippo AU - Roth, Oliver TI - Semigroup-fication of univalent self-maps of the unit disc JO - Annales de l'Institut Fourier PY - 2023 SP - 251 EP - 277 VL - 73 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3517/ DO - 10.5802/aif.3517 LA - en ID - AIF_2023__73_1_251_0 ER -
%0 Journal Article %A Bracci, Filippo %A Roth, Oliver %T Semigroup-fication of univalent self-maps of the unit disc %J Annales de l'Institut Fourier %D 2023 %P 251-277 %V 73 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3517/ %R 10.5802/aif.3517 %G en %F AIF_2023__73_1_251_0
Bracci, Filippo; Roth, Oliver. Semigroup-fication of univalent self-maps of the unit disc. Annales de l'Institut Fourier, Volume 73 (2023) no. 1, pp. 251-277. doi : 10.5802/aif.3517. https://aif.centre-mersenne.org/articles/10.5802/aif.3517/
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