FORWARD TRIPLETS AND TOPOLOGICAL ENTROPY ON TREES

We provide a new and very simple criterion of positive topological entropy for tree maps. We prove that a tree map f has positive entropy if and only if some iterate fk has a periodic orbit with three aligned points consecutive in time, that is, a triplet (a, b, c) such that fk(a) = b, fk(b) = c and...

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Detalles Bibliográficos
Autores: Alsedà, L., Juher, D., Mañosas, F.
Tipo de recurso: artículo
Estado:Versión aceptada para publicación
Fecha de publicación:2022
País:España
Institución:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:recercat.cat:2072/535421
Acceso en línea:http://hdl.handle.net/2072/535421
Access Level:acceso abierto
Palabra clave:Periodic patterns
Topological entropy
Tree maps
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spelling FORWARD TRIPLETS AND TOPOLOGICAL ENTROPY ON TREESAlsedà, L.Juher, D.Mañosas, F.Periodic patternsTopological entropyTree mapsWe provide a new and very simple criterion of positive topological entropy for tree maps. We prove that a tree map f has positive entropy if and only if some iterate fk has a periodic orbit with three aligned points consecutive in time, that is, a triplet (a, b, c) such that fk(a) = b, fk(b) = c and b belongs to the interior of the unique interval connecting a and c (a forward triplet of fk). We also prove a new criterion of entropy zero for simplicial n-periodic patterns P based on the non existence of forward triplets of fk for any 1 ≤ k < n inside P. Finally, we study the set Xn of all n-periodic patterns P that have a forward triplet inside P. For any n, we define a pattern that attains the minimum entropy in Xn and prove that this entropy is the unique real root in (1, ∞) of the polynomial xn − 2x − 1. © 2022 American Institute of Mathematical Sciences. All rights reserved.Ministerio de Economía y Competitividad, MINECO: MDM-2014-0445; Ministerio de Ciencia e Innovación, MICINN: 2017 SGR 1617. This work also acknowledges the CERCA Programme of the Generalitat de Catalunya for institutional support. This work was also supported by the Spanish State Research Agency, through the Severo Ochoa and Maria de Maeztu Program for Centres and Units of Excellence in R&D (CEX2020-001084-M).American Institute of Mathematical Sciences2022info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersion18 p.application/pdfhttp://hdl.handle.net/2072/535421RECERCAT (Dipòsit de la Recerca de Catalunya)reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésL'accés als continguts d'aquest document queda condicionat a l'acceptació de les condicions d'ús establertes per la següent llicència Creative Commons: https://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:recercat.cat:2072/5354212026-05-29T05:05:01Z
dc.title.none.fl_str_mv FORWARD TRIPLETS AND TOPOLOGICAL ENTROPY ON TREES
title FORWARD TRIPLETS AND TOPOLOGICAL ENTROPY ON TREES
spellingShingle FORWARD TRIPLETS AND TOPOLOGICAL ENTROPY ON TREES
Alsedà, L.
Periodic patterns
Topological entropy
Tree maps
title_short FORWARD TRIPLETS AND TOPOLOGICAL ENTROPY ON TREES
title_full FORWARD TRIPLETS AND TOPOLOGICAL ENTROPY ON TREES
title_fullStr FORWARD TRIPLETS AND TOPOLOGICAL ENTROPY ON TREES
title_full_unstemmed FORWARD TRIPLETS AND TOPOLOGICAL ENTROPY ON TREES
title_sort FORWARD TRIPLETS AND TOPOLOGICAL ENTROPY ON TREES
dc.creator.none.fl_str_mv Alsedà, L.
Juher, D.
Mañosas, F.
author Alsedà, L.
author_facet Alsedà, L.
Juher, D.
Mañosas, F.
author_role author
author2 Juher, D.
Mañosas, F.
author2_role author
author
dc.subject.none.fl_str_mv Periodic patterns
Topological entropy
Tree maps
topic Periodic patterns
Topological entropy
Tree maps
description We provide a new and very simple criterion of positive topological entropy for tree maps. We prove that a tree map f has positive entropy if and only if some iterate fk has a periodic orbit with three aligned points consecutive in time, that is, a triplet (a, b, c) such that fk(a) = b, fk(b) = c and b belongs to the interior of the unique interval connecting a and c (a forward triplet of fk). We also prove a new criterion of entropy zero for simplicial n-periodic patterns P based on the non existence of forward triplets of fk for any 1 ≤ k < n inside P. Finally, we study the set Xn of all n-periodic patterns P that have a forward triplet inside P. For any n, we define a pattern that attains the minimum entropy in Xn and prove that this entropy is the unique real root in (1, ∞) of the polynomial xn − 2x − 1. © 2022 American Institute of Mathematical Sciences. All rights reserved.
publishDate 2022
dc.date.none.fl_str_mv 2022
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 http://hdl.handle.net/2072/535421
url http://hdl.handle.net/2072/535421
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv 18 p.
application/pdf
dc.publisher.none.fl_str_mv American Institute of Mathematical Sciences
publisher.none.fl_str_mv American Institute of Mathematical Sciences
dc.source.none.fl_str_mv RECERCAT (Dipòsit de la Recerca de Catalunya)
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
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score 15,812429

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