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Periods of self-maps on $\rm S^2$ via their homology

UDC 517.9 As usual, we denote а $2$-dimensional sphere  by $\rm S^2$. We study the periods of  periodic orbits of the maps $f\colon \rm S^2 \rightarrow \rm S^2$ that are either continuous or $C^1$ with all their periodic orbits being hyperbolic, or transverse,...

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Date:2024
Main Author: Llibre, Jaume
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Language:English
Published: Institute of Mathematics, NAS of Ukraine 2024
Online Access:https://umj.imath.kiev.ua/index.php/umj/article/view/7668
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Journal Title:Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal
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author Llibre, Jaume
Llibre, Jaume
author_facet Llibre, Jaume
Llibre, Jaume
author_sort Llibre, Jaume
baseUrl_str https://umj.imath.kiev.ua/index.php/umj/oai
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datestamp_date 2024-06-19T00:35:06Z
description UDC 517.9 As usual, we denote а $2$-dimensional sphere  by $\rm S^2$. We study the periods of  periodic orbits of the maps $f\colon \rm S^2 \rightarrow \rm S^2$ that are either continuous or $C^1$ with all their periodic orbits being hyperbolic, or transverse, or holomorphic, or transverse holomorphic. For the first time, we summarize all  known results on the periodic orbits of these distinct kinds of self-maps on $\rm S^2$ together. We note that every time when a map $f\colon \rm S^2 \rightarrow \rm S^2$ increases its structure, the number of  periodic orbits provided by its action on the homology increases.
doi_str_mv 10.3842/umzh.v76i1.7668
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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 Periods of Self-Maps on \({\mathbb{S}}^{2}\) Via their Homology Published: 30 July 2024 Volume 76, pages 76–79, (2024) Cite this article Save article View saved research Ukrainian Mathematical Journal Aims and scope Submit manuscript Jaume Llibre1  44 Accesses Explore all metrics As usual, we denote a 2-dimensional sphere by \({\mathbb{S}}^{2}\). We study the periods of periodic orbits of the maps f : \({\mathbb{S}}^{2}\to {\mathbb{S}}^{2}\) that are either continuous or C1 with all their periodic orbits being hyperbolic, or transversal, or holomorphic, or transversal holomorphic. For the first time, we summarize all known results on the periodic orbits of these distinct kinds of self-maps on \({\mathbb{S}}^{2}\) together. We note that every time when a map f : \({\mathbb{S}}^{2}\to {\mathbb{S}}^{2}\) increases its structure, the number of periodic orbits provided by its action on the homology increases. 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 The least number of 2-periodic points of a smooth self-map of \(\varvec{S}^\mathbf{2}\) of degree 2 equals 2 Article Open access 22 December 2018 Periodic Points for Sphere Maps Preserving Monopole Foliations Article Open access 27 November 2018 Self-homeomorphisms and Degree \(\pm \,1\) Self-maps on Lens Spaces Article 22 May 2019 Explore related subjects Discover the latest articles, books and news in related subjects, suggested using machine learning. Algebraic Geometry Algebraic Topology Dynamical Systems Hyperbolic Geometry Topological Groups and Lie Groups Topology Lefschetz Theory in Periodic Dynamics References I. N. Baker, “Fix points of polynomials and rational functions,” J. Lond. Math. Soc. (2), 39, 615–622 (1964). Article  Google Scholar  R. F. Brown, The Lefschetz Fixed Point Theorem, Scott, Foresman & Company, Glenview, IL (1971). N. Fagella and J. Llibre, “Periodic points of holomorphic maps via Lefstchetz numbers,” Trans. Amer. Math. Soc., 352, 4711–4730 (2000). Article  MathSciNet  Google Scholar  J. L. Garcia-Guirao and J. Llibre, “\({\mathcal{C}}^{1}\) self-maps on \({\mathbb{S}}^{n}\), \({\mathbb{S}}^{n}\times {\mathbb{S}}^{m}\), \({{\mathbb{C}}P}^{n}\) and \({{\mathbb{H}}P}^{n}\) with all their periodic orbits hyperbolic,” Taiwanese J. Math., 16, 323–334 (2012). MathSciNet  Google Scholar  J. L. Garcia-Guirao and J. Llibre, “Periodic structure of transversal maps on \({{\mathbb{C}}P}^{n}\), \({{\mathbb{H}}P}^{n}\) and \({\mathbb{S}}^{p}\times {\mathbb{S}}^{q}\),” Qual. Theory Dyn. Syst., 12, 417–425 (2013). MathSciNet  Google Scholar  J. L. Garcia-Guirao and J. Llibre, “Periods of continuous maps on some compact spaces,” J. Difference Equat. Appl., 23, 1–7 (2017). Article  MathSciNet  Google Scholar  T. Y. Li and J. A. Yorke, “Period three implies chaos,” Amer. Math. Monthly, 82, No. 10, 985–992 (1975). Article  MathSciNet  Google Scholar  J. Llibre, “Lefschetz numbers for periodic points,” Contemp. Math., 152, 215–227 (1993). Article  MathSciNet  Google Scholar  J. Llibre, “A note on the set of periods of transversal homological sphere self-maps,” J. Difference Equat. Appl., 9, 417–422 (2003). Article  MathSciNet  Google Scholar  A. N. Sharkovsky, “Coexistence of the cycles of a continuous mapping of the line into itself,” Ukr. Math. Zh., 16, No. 1, 61–71 (1964). Google Scholar  J. W. Vick, Homology Theory, 2nd ed., Springer-Verlag, New York (1994). Book  Google Scholar  Download references Author information Authors and Affiliations Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, Spain Jaume Llibre Authors Jaume LlibreView author publications Search author on:PubMed Google Scholar Corresponding author Correspondence to Jaume Llibre. Additional information Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, No. 1, pp. 72–74, January, 2024. Ukrainian DOI: https://doi.org/10.3842/umzh.v76i1.7668. 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 Llibre, J. Periods of Self-Maps on \({\mathbb{S}}^{2}\) Via their Homology. Ukr Math J 76, 76–79 (2024). https://doi.org/10.1007/s11253-024-02308-9 Download citation Received: 05 July 2023 Published: 30 July 2024 Version of record: 30 July 2024 Issue date: June 2024 DOI: https://doi.org/10.1007/s11253-024-02308-9 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-76682024-06-19T00:35:06Z Periods of self-maps on $\rm S^2$ via their homology Periods of self-maps on $\rm S^2$ via their homology Llibre, Jaume Llibre, Jaume Self-maps of the 2-dimensional sphere set of periods periodic points Lefschetz numbers UDC 517.9 As usual, we denote а $2$-dimensional sphere  by $\rm S^2$. We study the periods of  periodic orbits of the maps $f\colon \rm S^2 \rightarrow \rm S^2$ that are either continuous or $C^1$ with all their periodic orbits being hyperbolic, or transverse, or holomorphic, or transverse holomorphic. For the first time, we summarize all  known results on the periodic orbits of these distinct kinds of self-maps on $\rm S^2$ together. We note that every time when a map $f\colon \rm S^2 \rightarrow \rm S^2$ increases its structure, the number of  periodic orbits provided by its action on the homology increases. УДК 517.9 Періоди відображень $\rm S^2$ на себе в термінах їхньої гомології  Як звичайно,  ми позначаємо $2$-вимірну сферу через $\rm S^2$. Вивчаються періоди періодичних орбіт відображень $f\colon \rm S^2 \rightarrow \rm S^2$, які є неперервними або $C^1$, а всі періодичні орбіти цих відображень є гіперболічними, або трансверсальними, або голоморфними, або трансверсально-голоморфними. Вперше узагальнено всі відомі результати про періодичні орбіти цих різних типів відображень  $\rm S^2$ на себе одночасно. Зауважимо, що кожного разу, коли відображення $f\colon \rm S^2 \rightarrow \rm S^2$ збільшує свою структуру, кількість її періодичних орбіт, зумовлених її дією на гомологію, збільшується.  Institute of Mathematics, NAS of Ukraine 2024-02-02 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7668 10.3842/umzh.v76i1.7668 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 1 (2024); 72 - 74 Український математичний журнал; Том 76 № 1 (2024); 72 - 74 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7668/9679 Copyright (c) 2024 Jaume Llibre
spellingShingle Llibre, Jaume
Llibre, Jaume
Periods of self-maps on $\rm S^2$ via their homology
title Periods of self-maps on $\rm S^2$ via their homology
title_alt Periods of self-maps on $\rm S^2$ via their homology
title_full Periods of self-maps on $\rm S^2$ via their homology
title_fullStr Periods of self-maps on $\rm S^2$ via their homology
title_full_unstemmed Periods of self-maps on $\rm S^2$ via their homology
title_short Periods of self-maps on $\rm S^2$ via their homology
title_sort periods of self-maps on $\rm s^2$ via their homology
topic_facet Self-maps of the 2-dimensional sphere
set of periods
periodic points
Lefschetz numbers
url https://umj.imath.kiev.ua/index.php/umj/article/view/7668
work_keys_str_mv AT llibrejaume periodsofselfmapsonrms2viatheirhomology
AT llibrejaume periodsofselfmapsonrms2viatheirhomology

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