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Accesses to infinity from Fatou components


Authors: Krzysztof Barański, Núria Fagella, Xavier Jarque and Bogusława Karpińska
Journal: Trans. Amer. Math. Soc. 369 (2017), 1835-1867
MSC (2010): Primary 30D05, 37F10, 30D30
DOI: https://doi.org/10.1090/tran/6739
Published electronically: May 2, 2016
MathSciNet review: 3581221
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Abstract: We study the boundary behaviour of a meromorphic map $ f: \mathbb{C} \to \widehat {\mathbb{C}}$ on its simply connected invariant Fatou component $ U$. To this aim, we develop the theory of accesses to boundary points of $ U$ and their relation to the dynamics of $ f$. In particular, we establish a correspondence between invariant accesses from $ U$ to infinity or weakly repelling fixed points of $ f$ and boundary fixed points of the associated inner function on the unit disc. We apply our results to describe the accesses to infinity from invariant Fatou components of the Newton maps.


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Additional Information

Krzysztof Barański
Affiliation: Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland
Email: baranski@mimuw.edu.pl

Núria Fagella
Affiliation: Barcelona Graduate School of Mathematics and Departament de Matemàtiques i Informàtica, Universitat de Barcelona, 08007 Barcelona, Catalonia, Spain
Email: fagella@maia.ub.es

Xavier Jarque
Affiliation: Barcelona Graduate School of Mathematics and Departament de Matemàtiques i Informàtica, Universitat de Barcelona, 08007 Barcelona, Catalonia, Spain
Email: xavier.jarque@ub.edu

Bogusława Karpińska
Affiliation: Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland
Email: bkarpin@mini.pw.edu.pl

DOI: https://doi.org/10.1090/tran/6739
Received by editor(s): November 25, 2014
Received by editor(s) in revised form: March 9, 2015
Published electronically: May 2, 2016
Additional Notes: The second and third authors were partially supported by the Catalan grant 2009SGR-792, the Spanish grants MTM2011-26995-C02-02 and MTM2014-52209-22-2-P and the María de Maeztu grant MDM-2014-0445.
The four authors were supported by the Polish NCN grant decision DEC-2012/06/M/ ST1/00168
Article copyright: © Copyright 2016 American Mathematical Society