Collinear Fractals and Bandt’s Conjecture

For a complex parameter c outside the unit disk and an integer n≥2, we examine the n-ary collinear fractal E(c,n), defined as the attractor of the iterated function system {fk:C-C}nk=1, where fk(z):=1+n-2k+c1z. We investigate some topological features of the connectedness locus Mn defined as the set...

Full description

Bibliographic Details
Authors: Espigulé, Bernat, Juher, David, Saldaña Meca, Joan
Format: article
Status:Published version
Publication Date:2024
Country:España
Institution:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repository:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:10256/25833
Online Access:http://hdl.handle.net/10256/25833
Access Level:Open access
Keyword:Fractals
Bandt, Conjetura de
Bandt conjecture
Conjunts de Mandelbrot
Mandelbrot sets
Polinomis
Polynomials
id ES_a8d34b59b2be9bbb29e73a539da4d656
oai_identifier_str oai:recercat.cat:10256/25833
network_acronym_str ES
network_name_str España
repository_id_str
spelling Collinear Fractals and Bandt’s ConjectureEspigulé, BernatJuher, DavidSaldaña Meca, JoanFractalsBandt, Conjetura deBandt conjectureConjunts de MandelbrotMandelbrot setsPolinomisPolynomialsFor a complex parameter c outside the unit disk and an integer n≥2, we examine the n-ary collinear fractal E(c,n), defined as the attractor of the iterated function system {fk:C-C}nk=1, where fk(z):=1+n-2k+c1z. We investigate some topological features of the connectedness locus Mn defined as the set of those c for which E(c,n) is connected. In particular, we provide a detailed answer to an open question posed by Calegri, Koch, and Walker in 2017. We also extend and refine the technique of the “covering property” by Solomyak and Xu to any n≥2. We use it to show that a nontrivial portion of Mn is regular closed. When n≥21, we enhance this result by showing that in fact, the whole Mn\R lies within the closure of its interior, thus proving that the generalized Bandt’s conjecture is truehis work has been funded by grants PID2023-146424NB-I00 of Ministerio de Ciencia, Innovación y Universidades and 2021 SGR 00113 of Generalitat de Catalunya. The first author has been supported by the research grant from the University of Girona (UdG) in collaboration with Banco Santander, through UdG Grant Programme for Researchers in Training (IFUdG 2022–2024)MDPI (Multidisciplinary Digital Publishing Institute)Agencia Estatal de Investigación2024info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionpeer-reviewedapplication/pdfhttp://hdl.handle.net/10256/25833Fractal and Fractional, 2024, vol. 8, núm. 12, p. 725Articles publicats (D-IMA)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)Inglésinfo:eu-repo/semantics/altIdentifier/doi/10.3390/fractalfract8120725info:eu-repo/semantics/altIdentifier/eissn/2504-3110PID2023-146424NB-I00info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023/PID2023-146424NB-I00Attribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:10256/258332026-05-29T05:05:01Z
dc.title.none.fl_str_mv Collinear Fractals and Bandt’s Conjecture
title Collinear Fractals and Bandt’s Conjecture
spellingShingle Collinear Fractals and Bandt’s Conjecture
Espigulé, Bernat
Fractals
Bandt, Conjetura de
Bandt conjecture
Conjunts de Mandelbrot
Mandelbrot sets
Polinomis
Polynomials
title_short Collinear Fractals and Bandt’s Conjecture
title_full Collinear Fractals and Bandt’s Conjecture
title_fullStr Collinear Fractals and Bandt’s Conjecture
title_full_unstemmed Collinear Fractals and Bandt’s Conjecture
title_sort Collinear Fractals and Bandt’s Conjecture
dc.creator.none.fl_str_mv Espigulé, Bernat
Juher, David
Saldaña Meca, Joan
author Espigulé, Bernat
author_facet Espigulé, Bernat
Juher, David
Saldaña Meca, Joan
author_role author
author2 Juher, David
Saldaña Meca, Joan
author2_role author
author
dc.contributor.none.fl_str_mv Agencia Estatal de Investigación
dc.subject.none.fl_str_mv Fractals
Bandt, Conjetura de
Bandt conjecture
Conjunts de Mandelbrot
Mandelbrot sets
Polinomis
Polynomials
topic Fractals
Bandt, Conjetura de
Bandt conjecture
Conjunts de Mandelbrot
Mandelbrot sets
Polinomis
Polynomials
description For a complex parameter c outside the unit disk and an integer n≥2, we examine the n-ary collinear fractal E(c,n), defined as the attractor of the iterated function system {fk:C-C}nk=1, where fk(z):=1+n-2k+c1z. We investigate some topological features of the connectedness locus Mn defined as the set of those c for which E(c,n) is connected. In particular, we provide a detailed answer to an open question posed by Calegri, Koch, and Walker in 2017. We also extend and refine the technique of the “covering property” by Solomyak and Xu to any n≥2. We use it to show that a nontrivial portion of Mn is regular closed. When n≥21, we enhance this result by showing that in fact, the whole Mn\R lies within the closure of its interior, thus proving that the generalized Bandt’s conjecture is true
publishDate 2024
dc.date.none.fl_str_mv 2024
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
peer-reviewed
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/10256/25833
url http://hdl.handle.net/10256/25833
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.3390/fractalfract8120725
info:eu-repo/semantics/altIdentifier/eissn/2504-3110
PID2023-146424NB-I00
info:eu-repo/grantAgreement/AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023/PID2023-146424NB-I00
dc.rights.none.fl_str_mv Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv MDPI (Multidisciplinary Digital Publishing Institute)
publisher.none.fl_str_mv MDPI (Multidisciplinary Digital Publishing Institute)
dc.source.none.fl_str_mv Fractal and Fractional, 2024, vol. 8, núm. 12, p. 725
Articles publicats (D-IMA)
reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869415931513405440
score 15.811543

Similar Items