Semigroup-fication of univalent self-maps of the unit disc
Annales de l'Institut Fourier, Online first, 27 p.

Let f 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 (ϕ t ) of holomorphic self-maps of the unit disc whose time one map ϕ 1 is, in a sense, very close to f. The semigroup-fication of f is of the same type as f (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 f with a given multiplier and the corresponding set for ϕ 1 . Moreover, in case f (and hence ϕ 1 ) 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 f converge non-tangentially to the Denjoy–Wolff point if and only if the Koenigs domain of f is “almost symmetric” with respect to vertical lines.

Soit f une application univalente du disque unité dans lui-même. On introduit une technique, appelée semigroupe-fication, qui nous permet de construire un semigroupe continu (ϕ t ) d’applications holomorphes du disque unité dans lui-même tel que l’application au temps t=1, (ϕ 1 ), est très proche de f. La “ semigroupe-fication ” de f est du même type de f (elliptique, hyperbolique, parabolique d’étape positive, parabolique d’étape zéro) et il existe une correspondance 1-1 entre l’ensemble des points fixes de f qui sont réguliers, au bord et avec un multiplicateur donné, et le même ensemble pour ϕ 1 . De plus, si f (et donc ϕ 1 ) 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 f convergent au point de Denjoy–Wolff de façon non-tangentielle si et seulement si le domaine de Koenigs de f est “presque symétrique” par rapport aux lignes verticales.

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Accepted:
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DOI: 10.5802/aif.3517
Classification: 37C10,  30C35,  30D05,  30C80,  37F99,  37C25
Keywords: Semigroups of holomorphics maps, univalent functions, asymptotic behavior of orbits, iteration theory.
Bracci, Filippo 1; Roth, Oliver 2

1 Dipartimento di Matematica Università di Roma “Tor Vergata” Via della Ricerca Scientifica 1 00133, Roma (Italia)
2 Department of Mathematics University of Würzburg Emil Fischer Strasse 40 97074, Würzburg (Germany)
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Bracci, Filippo; Roth, Oliver. Semigroup-fication of univalent self-maps of the unit disc. Annales de l'Institut Fourier, Online first, 27 p.

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