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Topological entropy, sets of periods, and transitivity for circle maps

UDC 517.9 Transitivity, the existence of periodic points, and positive topological entropy can be used to characterize complexity in dynamical systems. It is known that, for every graph that is not a tree and any $\varepsilon>0,$ there exist (complicated) totally transitive m...

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Дата:2024
Автори: Alsedà, Lluís, Bordignon, Liane, Groisman, Jorge
Формат: Стаття
Мова:Англійська
Опубліковано: Institute of Mathematics, NAS of Ukraine 2024
Онлайн доступ:https://umj.imath.kiev.ua/index.php/umj/article/view/7659
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Назва журналу:Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal
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author Alsedà, Lluís
Bordignon, Liane
Groisman, Jorge
Alsedà, Lluís
Bordignon, Liane
Groisman, Jorge
author_facet Alsedà, Lluís
Bordignon, Liane
Groisman, Jorge
Alsedà, Lluís
Bordignon, Liane
Groisman, Jorge
author_sort Alsedà, Lluís
baseUrl_str https://umj.imath.kiev.ua/index.php/umj/oai
collection OJS
datestamp_date 2024-06-19T00:35:05Z
description UDC 517.9 Transitivity, the existence of periodic points, and positive topological entropy can be used to characterize complexity in dynamical systems. It is known that, for every graph that is not a tree and any $\varepsilon>0,$ there exist (complicated) totally transitive maps (then with cofinite set of periods) such that the topological entropy is smaller than $\varepsilon$ (simplicity). To numerically measure the complexity of the set of periods, we introduce a notion of the boundary of cofiniteness. Larger boundary of cofiniteness corresponds to a simpler set of periods. We show that, for any continuous degree one circle maps, every totally transitive (and, hence, robustly complicated) map with small topological entropy has arbitrarily large (simplicity) boundary of cofiniteness.
doi_str_mv 10.3842/umzh.v76i1.7659
first_indexed 2026-03-24T03:33:00Z
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fulltext Skip to main content Log in Menu Find a journal Publish with us Track your research Search Saved research Cart Home Ukrainian Mathematical Journal Article Topological Entropy, Sets of Periods, and Transitivity for Circle Maps Published: 30 July 2024 Volume 76, pages 31–50, (2024) Cite this article Save article View saved research Ukrainian Mathematical Journal Aims and scope Submit manuscript Lluís Alsedà1, Liane Bordignon2 & Jorge Groisman3  54 Accesses Explore all metrics Transitivity, the existence of periodic points, and positive topological entropy can be used to characterize complexity in dynamical systems. It is known that, for every graph that is not a tree and any ε > 0, there exist (complicated) totally transitive maps (then with cofinite set of periods) such that the topological entropy is smaller than ε (simplicity). To numerically measure the complexity of the set of periods, we introduce a notion of the boundary of cofiniteness. Larger boundary of cofiniteness corresponds to a simpler set of periods. We show that, for any continuous circle maps of degree one, every totally transitive (and, hence, robustly complicated) map with small topological entropy has arbitrarily large (simplicity) boundary of cofiniteness. This is a preview of subscription content, log in via an institution to check access. Access this article Log in via an institution Subscribe and save Springer+ from €37.37 /Month Starting from 10 chapters or articles per month Access and download chapters and articles from more than 300k books and 2,500 journals Cancel anytime View plans Buy Now Buy article PDF 39,95 € Price includes VAT (Ukraine) Instant access to the full article PDF. Institutional subscriptions Similar content being viewed by others On Slow Growth and Entropy-Type Invariants Chapter © 2019 Complete subgraphs of the coprime hypergraph of integers I: introduction and bounds Article 15 March 2017 Geometry and Global Stability of 2D Periodic Monotone Maps Article 07 October 2021 Explore related subjects Discover the latest articles, books and news in related subjects, suggested using machine learning. Algorithmic Complexity Complexity Computational Complexity Ergodic Theory Mathematics of Algorithmic Complexity Topology Dynamical Systems and Ergodic Theory References R. L. Adler, A. G. Konheim, and M. H. McAndrew, “Topological entropy,” Trans. Amer. Math. Soc., 114, 309–319 (1965). Article  MathSciNet  Google Scholar  Ll. Alsedà, L. Bordignon, and J. Groisman, Topological Entropy, Sets of Periods and Transitivity for Graph Maps: a Factory of Examples, Preprint (2023). Ll. Alsedà, M. A. del Río, and J. A. Rodríguez, “A splitting theorem for transitive maps,” J. Math. Anal. Appl., 232, No. 2, 359–375 (1999). Ll. Alsedà, M. A. del Río, and J. A. Rodríguez, “A note on the totally transitive graph maps,” Internat. J. Bifurcat. Chaos Appl. Sci. Eng., 11, No. 3, 841–843 (2001). Ll. Alsedà, M. A. Del Río, and J. A. Rodríguez, “A survey on the relation between transitivity and dense periodicity for graph maps,” J. Difference Equat. Appl., 9, No. 3-4, 281–288 (2003). Ll. Alsedà, M. A. del Río, and J. A. Rodríguez, “Transitivity and dense periodicity for graph maps,” J. Difference Equat. Appl., 9, No. 6, 577–598 (2003). Ll. Alsedà, F. Mañosas, and P. Mumbrú, “Minimizing topological entropy for continuous maps on graphs,” Ergodic Theory Dynam. Syst., 20, No. 6, 1559–1576 (2000). Ll. Alsedà, S. Baldwin, J. Llibre, and M. Misiurewicz, “Entropy of transitive tree maps,” Topology, 36, No. 2, 519–532 (1997). Ll. Alsedà, J. Llibre, and M. Misiurewicz, Combinatorial Dynamics and Entropy in Dimension One, 2nd ed., Adv. Ser. Nonlinear Dynam., Vol. 5, World Scientific Publ. Co., Inc., River Edge, NJ (2000). Ll. Alsedà and S. Ruette, “Periodic orbits of large diameter for circle maps,” Proc. Amer. Math. Soc., 138, No. 9, 3211–3217 (2010). J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey, “On Devaney’s definition of chaos,” Amer. Math. Monthly, 99, No. 4, 332–334 (1992). Article  MathSciNet  Google Scholar  A. M. Blokh, “On transitive mappings of one-dimensional branched manifolds,” Differential-Difference Equations and Problems of Mathematical Physics [in Russian], Akad. Nauk Ukr. SSR, Inst. Mat., Kiev (1984), pp. 3–9. A. M. Blokh, “The connection between entropy and transitivity for one-dimensional mappings,” Usp. Mat. Nauk, 42, No. 5, 209–210 (1987). MathSciNet  Google Scholar  R. Ito, “Rotation sets are closed,” Math. Proc. Cambridge Philos. Soc., 89, No. 1, 107–111 (1981). Article  MathSciNet  Google Scholar  S. 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Article  MathSciNet  Google Scholar  Download references Author information Authors and Affiliations Departament de Matemàtiques and Centre de Recerca Matemàtica, Edifici Cc, Universitat Autònoma de Barcelona, Barcelona, Spain Lluís Alsedà Departamento de Matemática, Universidade Federal de São Carlos, São Paulo, Brasil Liane Bordignon IMERL, Facultad de Ingeniería, Universidad de la República, Montevideo, Uruguay Jorge Groisman Authors Lluís AlsedàView author publications Search author on:PubMed Google Scholar Liane BordignonView author publications Search author on:PubMed Google Scholar Jorge GroismanView author publications Search author on:PubMed Google Scholar Corresponding author Correspondence to Lluís Alsedà. Additional information Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, No. 1, pp. 31–47, January, 2024. Ukrainian DOI: https://doi.org/10.3842/umzh.v76i1.7659. Rights and permissions Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Reprints and permissions About this article Cite this article Alsedà, L., Bordignon, L. & Groisman, J. Topological Entropy, Sets of Periods, and Transitivity for Circle Maps. Ukr Math J 76, 31–50 (2024). https://doi.org/10.1007/s11253-024-02305-y Download citation Received: 02 July 2023 Published: 30 July 2024 Version of record: 30 July 2024 Issue date: June 2024 DOI: https://doi.org/10.1007/s11253-024-02305-y Share this article Anyone you share the following link with will be able to read this content: Get shareable linkSorry, a shareable link is not currently available for this article. 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spelling umjimathkievua-article-76592024-06-19T00:35:05Z Topological entropy, sets of periods, and transitivity for circle maps Topological entropy, sets of periods, and transitivity for circle maps Alsedà, Lluís Bordignon, Liane Groisman, Jorge Alsedà, Lluís Bordignon, Liane Groisman, Jorge Topological entropy, sets of periods, total transitivity, boundary of cofiniteness, rotation sets, circle maps UDC 517.9 Transitivity, the existence of periodic points, and positive topological entropy can be used to characterize complexity in dynamical systems. It is known that, for every graph that is not a tree and any $\varepsilon>0,$ there exist (complicated) totally transitive maps (then with cofinite set of periods) such that the topological entropy is smaller than $\varepsilon$ (simplicity). To numerically measure the complexity of the set of periods, we introduce a notion of the boundary of cofiniteness. Larger boundary of cofiniteness corresponds to a simpler set of periods. We show that, for any continuous degree one circle maps, every totally transitive (and, hence, robustly complicated) map with small topological entropy has arbitrarily large (simplicity) boundary of cofiniteness. УДК 517.9 Топологічна ентропія, набори періодів і транзитивність для відображень кіл  Транзитивність, існування періодичних точок і позитивна топологічна ентропія можуть бути використані, щоб охарактеризувати складність динамічних систем. Відомо, що для кожного графа, який не є деревом, і для кожного $\varepsilon>0$ існують (складні) повністю транзитивні відображення (тобто зі скінченною множиною періодів), для яких топологіч\-на ентропія менша за $\varepsilon$ (простота). Для кількісного визначення складності множини періодів ми вводимо поняття межі коскінченності. Чим більша межа коскінченності, тим простіша множина періодів. Показано, що для довільних неперервних відображень кола першого ступеня кожне повністю транзитивне (а отже, робастно складне) відображення з малою топологічною ентропією має як завгодно велику (за простотою) межу коскінченності. Institute of Mathematics, NAS of Ukraine 2024-02-02 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7659 10.3842/umzh.v76i1.7659 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 1 (2024); 31 - 47 Український математичний журнал; Том 76 № 1 (2024); 31 - 47 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7659/9677 Copyright (c) 2024 Lluis Alseda, Liane Bordignon, Jorge Groisman
spellingShingle Alsedà, Lluís
Bordignon, Liane
Groisman, Jorge
Alsedà, Lluís
Bordignon, Liane
Groisman, Jorge
Topological entropy, sets of periods, and transitivity for circle maps
title Topological entropy, sets of periods, and transitivity for circle maps
title_alt Topological entropy, sets of periods, and transitivity for circle maps
title_full Topological entropy, sets of periods, and transitivity for circle maps
title_fullStr Topological entropy, sets of periods, and transitivity for circle maps
title_full_unstemmed Topological entropy, sets of periods, and transitivity for circle maps
title_short Topological entropy, sets of periods, and transitivity for circle maps
title_sort topological entropy, sets of periods, and transitivity for circle maps
topic_facet Topological entropy
sets of periods
total transitivity
boundary of cofiniteness
rotation sets
circle maps
url https://umj.imath.kiev.ua/index.php/umj/article/view/7659
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AT bordignonliane topologicalentropysetsofperiodsandtransitivityforcirclemaps
AT groismanjorge topologicalentropysetsofperiodsandtransitivityforcirclemaps
AT alsedalluis topologicalentropysetsofperiodsandtransitivityforcirclemaps
AT bordignonliane topologicalentropysetsofperiodsandtransitivityforcirclemaps
AT groismanjorge topologicalentropysetsofperiodsandtransitivityforcirclemaps

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