Fatou components and singularities of meromorphic functions
We prove several results concerning the relative position of points in the postsingular set $P(f)$ of a meromorphic map $f$ and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljevi\'c-Brandt and Rempe-Gillen. For...
| Autores: | , , , |
|---|---|
| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2020 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/130195 |
| Acceso en línea: | https://hdl.handle.net/2445/130195 |
| Access Level: | acceso abierto |
| Palabra clave: | Equacions funcionals Funcions analítiques Sistemes dinàmics complexos Polinomis Funcions enteres Funcions meromorfes Functional equations Analytic functions Complex dynamical systems Polynomials Entire functions Meromorphic functions |
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Fatou components and singularities of meromorphic functionsBaranski, KrzysztofFagella Rabionet, NúriaJarque i Ribera, XavierKarpinska, BoguslawaEquacions funcionalsFuncions analítiquesSistemes dinàmics complexosPolinomisFuncions enteresFuncions meromorfesFunctional equationsAnalytic functionsComplex dynamical systemsPolynomialsEntire functionsMeromorphic functionsWe prove several results concerning the relative position of points in the postsingular set $P(f)$ of a meromorphic map $f$ and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljevi\'c-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates $U_n$ of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values $p_n$ such that $\dist(p_n, U_n)\to 0$ as $n\to \infty$. We also prove that if $U_n \cap P(f)=\emptyset$ and the postsingular set of $f$ lies at a positive distance from the Julia set (in $\C$), then the sequence of iterates of any wandering domain must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values.Cambridge University Press2020info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionapplication/pdfhttps://hdl.handle.net/2445/130195Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaInglésVersió postprint del document publicat a: https://doi.org/10.1017/prm.2018.142Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2020, vol. 150, num. 2, p. 633-654https://doi.org/10.1017/prm.2018.142(c) Royal Society of Edinburgh , 2019info:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/1301952026-05-27T06:46:51Z |
| dc.title.none.fl_str_mv |
Fatou components and singularities of meromorphic functions |
| title |
Fatou components and singularities of meromorphic functions |
| spellingShingle |
Fatou components and singularities of meromorphic functions Baranski, Krzysztof Equacions funcionals Funcions analítiques Sistemes dinàmics complexos Polinomis Funcions enteres Funcions meromorfes Functional equations Analytic functions Complex dynamical systems Polynomials Entire functions Meromorphic functions |
| title_short |
Fatou components and singularities of meromorphic functions |
| title_full |
Fatou components and singularities of meromorphic functions |
| title_fullStr |
Fatou components and singularities of meromorphic functions |
| title_full_unstemmed |
Fatou components and singularities of meromorphic functions |
| title_sort |
Fatou components and singularities of meromorphic functions |
| dc.creator.none.fl_str_mv |
Baranski, Krzysztof Fagella Rabionet, Núria Jarque i Ribera, Xavier Karpinska, Boguslawa |
| author |
Baranski, Krzysztof |
| author_facet |
Baranski, Krzysztof Fagella Rabionet, Núria Jarque i Ribera, Xavier Karpinska, Boguslawa |
| author_role |
author |
| author2 |
Fagella Rabionet, Núria Jarque i Ribera, Xavier Karpinska, Boguslawa |
| author2_role |
author author author |
| dc.subject.none.fl_str_mv |
Equacions funcionals Funcions analítiques Sistemes dinàmics complexos Polinomis Funcions enteres Funcions meromorfes Functional equations Analytic functions Complex dynamical systems Polynomials Entire functions Meromorphic functions |
| topic |
Equacions funcionals Funcions analítiques Sistemes dinàmics complexos Polinomis Funcions enteres Funcions meromorfes Functional equations Analytic functions Complex dynamical systems Polynomials Entire functions Meromorphic functions |
| description |
We prove several results concerning the relative position of points in the postsingular set $P(f)$ of a meromorphic map $f$ and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljevi\'c-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates $U_n$ of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values $p_n$ such that $\dist(p_n, U_n)\to 0$ as $n\to \infty$. We also prove that if $U_n \cap P(f)=\emptyset$ and the postsingular set of $f$ lies at a positive distance from the Julia set (in $\C$), then the sequence of iterates of any wandering domain must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values. |
| publishDate |
2020 |
| dc.date.none.fl_str_mv |
2020 |
| dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/acceptedVersion |
| format |
article |
| status_str |
acceptedVersion |
| dc.identifier.none.fl_str_mv |
https://hdl.handle.net/2445/130195 |
| url |
https://hdl.handle.net/2445/130195 |
| dc.language.none.fl_str_mv |
Inglés |
| language_invalid_str_mv |
Inglés |
| dc.relation.none.fl_str_mv |
Versió postprint del document publicat a: https://doi.org/10.1017/prm.2018.142 Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 2020, vol. 150, num. 2, p. 633-654 https://doi.org/10.1017/prm.2018.142 |
| dc.rights.none.fl_str_mv |
(c) Royal Society of Edinburgh , 2019 info:eu-repo/semantics/openAccess |
| rights_invalid_str_mv |
(c) Royal Society of Edinburgh , 2019 |
| eu_rights_str_mv |
openAccess |
| dc.format.none.fl_str_mv |
application/pdf |
| dc.publisher.none.fl_str_mv |
Cambridge University Press |
| publisher.none.fl_str_mv |
Cambridge University Press |
| dc.source.none.fl_str_mv |
Articles publicats en revistes (Matemàtiques i Informàtica) reponame:Dipòsit Digital de la UB instname:Universidad de Barcelona |
| instname_str |
Universidad de Barcelona |
| reponame_str |
Dipòsit Digital de la UB |
| collection |
Dipòsit Digital de la UB |
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| repository.mail.fl_str_mv |
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1869410042305839104 |
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15,301603 |