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Transactions of the American Mathematical Society

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Non-tangential limits and the slope of trajectories of holomorphic semigroups of the unit disc


Authors: Filippo Bracci, Manuel D. Contreras, Santiago Díaz-Madrigal and Hervé Gaussier
Journal: Trans. Amer. Math. Soc. 373 (2020), 939-969
MSC (2010): Primary 37C10, 30C35; Secondary 30D05, 30C80, 37F99, 37C25
DOI: https://doi.org/10.1090/tran/7977
Published electronically: November 5, 2019
MathSciNet review: 4068255
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Abstract: Let $\Delta \subsetneq \mathbb {C}$ be a simply connected domain, let $f:\mathbb {D} \to \Delta$ be a Riemann map, and let $\{z_k\}\subset \Delta$ be a compactly divergent sequence. Using Gromov’s hyperbolicity theory, we show that $\{f^{-1}(z_k)\}$ converges non-tangentially to a point of $\partial \mathbb {D}$ if and only if there exists a simply connected domain $U\subsetneq \mathbb {C}$ such that $\Delta \subset U$ and $\Delta$ contains a tubular hyperbolic neighborhood of a geodesic of $U$ and $\{z_k\}$ is eventually contained in a smaller tubular hyperbolic neighborhood of the same geodesic. As a consequence we show that if $(\phi _t)$ is a non-elliptic semigroup of holomorphic self-maps of $\mathbb {D}$ with Koenigs function $h$ and $h(\mathbb {D})$ contains a vertical Euclidean sector, then $\phi _t(z)$ converges to the Denjoy-Wolff point non-tangentially for every $z\in \mathbb {D}$ as $t\to +\infty$. Using new localization results for the hyperbolic distance, we also construct an example of a parabolic semigroup which converges non-tangentially to the Denjoy-Wolff point but is oscillating, in the sense that the slope of the trajectories is not a single point.


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Additional Information

Filippo Bracci
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133, Roma, Italia
MR Author ID: 631111
Email: fbracci@mat.uniroma2.it

Manuel D. Contreras
Affiliation: Departamento de Matemática Aplicada II and IMUS, Universidad de Sevilla, Camino de los Descubrimientos, s/n, Sevilla, 41092, Spain
MR Author ID: 335888
Email: contreras@us.es

Santiago Díaz-Madrigal
Affiliation: Departamento de Matemática Aplicada II and IMUS, Universidad de Sevilla, Camino de los Descubrimientos, s/n, Sevilla, 41092, Spain
MR Author ID: 310764
Email: madrigal@us.es

Hervé Gaussier
Affiliation: Université Grenoble Alpes, CNRS, IF, F-38000 Grenoble, France
Email: herve.gaussier@univ-grenoble-alpes.fr

Keywords: Semigroups of holomorphic functions, Gromov’s hyperbolicity
Received by editor(s): May 22, 2018
Received by editor(s) in revised form: March 19, 2019
Published electronically: November 5, 2019
Additional Notes: The first author was partially supported by the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006
The second and third authors were partially supported by the Ministerio de Economía y Competitividad and the European Union (FEDER) MTM2015-63699-P and by La Consejería de Educación y Ciencia de la Junta de Andalucía
The fourth author was partially supported by ERC ALKAGE
Article copyright: © Copyright 2019 American Mathematical Society