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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Non-tangential limits and the slope of trajectories of holomorphic semigroups of the unit disc
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by Filippo Bracci, Manuel D. Contreras, Santiago Díaz-Madrigal and Hervé Gaussier PDF
Trans. Amer. Math. Soc. 373 (2020), 939-969 Request permission

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
  • 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
  • © Copyright 2019 American Mathematical Society
  • 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
  • MathSciNet review: 4068255