Generalized rings around the McMullen domain

We consider the family of rational maps given by F (z) = z + λ/ z where n, d∈ N with 1 / n+ 1 / d< 1, the variable z∈ C^ and the parameter λ∈ C. It is known that when n= d≥ 3 there are infinitely many rings S with k∈ N, around the McMullen domain. The McMullen domain is a region centered at the o...

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Autores: Garijo, Antoni|||0000-0002-1503-7514, Jang, HyeGyong, Marotta, Sebastian M.|||0000-0001-7286-2222
Tipo de recurso: artículo
Fecha de publicación:2019
País:España
Institución:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:221310
Acceso en línea:https://ddd.uab.cat/record/221310
https://dx.doi.org/urn:doi:10.1007/s12346-018-0287-y
Access Level:acceso abierto
Palabra clave:Singularly perturbed rational maps
McMullen domain
Baby Mandelbrot sets
Sierpinski holes
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spelling Generalized rings around the McMullen domainGarijo, Antoni|||0000-0002-1503-7514Jang, HyeGyongMarotta, Sebastian M.|||0000-0001-7286-2222Singularly perturbed rational mapsMcMullen domainBaby Mandelbrot setsSierpinski holesWe consider the family of rational maps given by F (z) = z + λ/ z where n, d∈ N with 1 / n+ 1 / d< 1, the variable z∈ C^ and the parameter λ∈ C. It is known that when n= d≥ 3 there are infinitely many rings S with k∈ N, around the McMullen domain. The McMullen domain is a region centered at the origin in the parameter λ-plane where the Julia sets of F are Cantor sets of simple closed curves. The rings S converge to the boundary of the McMullen domain as k→ ∞ and contain parameter values that lie at the center of Sierpiński holes, i.e., open simply connected subsets of the parameter space for which the Julia sets of F are Sierpiński curves. The rings also contain the same number of superstable parameter values, i.e., parameter values for which one of the critical points is periodic and correspond to the centers of the main cardioids of copies of Mandelbrot sets. In this paper we generalize the existence of these rings to the case when 1 / n+ 1 / d< 1 where n is not necessarily equal to d. The number of Sierpiński holes and superstable parameters on S is τ1n,d=n-1, and on S for k> 1 is given by τkn,d=dnk-2(n-1)-nk-1+1. 22019-01-0120192019-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/221310https://dx.doi.org/urn:doi:10.1007/s12346-018-0287-yreponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengMinisterio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM-2017-86795-C3-2-Popen accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:2213102026-06-06T12:50:31Z
dc.title.none.fl_str_mv Generalized rings around the McMullen domain
title Generalized rings around the McMullen domain
spellingShingle Generalized rings around the McMullen domain
Garijo, Antoni|||0000-0002-1503-7514
Singularly perturbed rational maps
McMullen domain
Baby Mandelbrot sets
Sierpinski holes
title_short Generalized rings around the McMullen domain
title_full Generalized rings around the McMullen domain
title_fullStr Generalized rings around the McMullen domain
title_full_unstemmed Generalized rings around the McMullen domain
title_sort Generalized rings around the McMullen domain
dc.creator.none.fl_str_mv Garijo, Antoni|||0000-0002-1503-7514
Jang, HyeGyong
Marotta, Sebastian M.|||0000-0001-7286-2222
author Garijo, Antoni|||0000-0002-1503-7514
author_facet Garijo, Antoni|||0000-0002-1503-7514
Jang, HyeGyong
Marotta, Sebastian M.|||0000-0001-7286-2222
author_role author
author2 Jang, HyeGyong
Marotta, Sebastian M.|||0000-0001-7286-2222
author2_role author
author
dc.subject.none.fl_str_mv Singularly perturbed rational maps
McMullen domain
Baby Mandelbrot sets
Sierpinski holes
topic Singularly perturbed rational maps
McMullen domain
Baby Mandelbrot sets
Sierpinski holes
description We consider the family of rational maps given by F (z) = z + λ/ z where n, d∈ N with 1 / n+ 1 / d< 1, the variable z∈ C^ and the parameter λ∈ C. It is known that when n= d≥ 3 there are infinitely many rings S with k∈ N, around the McMullen domain. The McMullen domain is a region centered at the origin in the parameter λ-plane where the Julia sets of F are Cantor sets of simple closed curves. The rings S converge to the boundary of the McMullen domain as k→ ∞ and contain parameter values that lie at the center of Sierpiński holes, i.e., open simply connected subsets of the parameter space for which the Julia sets of F are Sierpiński curves. The rings also contain the same number of superstable parameter values, i.e., parameter values for which one of the critical points is periodic and correspond to the centers of the main cardioids of copies of Mandelbrot sets. In this paper we generalize the existence of these rings to the case when 1 / n+ 1 / d< 1 where n is not necessarily equal to d. The number of Sierpiński holes and superstable parameters on S is τ1n,d=n-1, and on S for k> 1 is given by τkn,d=dnk-2(n-1)-nk-1+1.
publishDate 2019
dc.date.none.fl_str_mv 2
2019-01-01
2019
2019-01-01
dc.type.none.fl_str_mv Article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/221310
https://dx.doi.org/urn:doi:10.1007/s12346-018-0287-y
url https://ddd.uab.cat/record/221310
https://dx.doi.org/urn:doi:10.1007/s12346-018-0287-y
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Economía y Competitividad https://doi.org/10.13039/501100003329 MTM-2017-86795-C3-2-P
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
reponame_str Dipòsit Digital de Documents de la UAB
collection Dipòsit Digital de Documents de la UAB
repository.name.fl_str_mv
repository.mail.fl_str_mv
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