Topological entropy, sets of periods, and transitivity for circle maps
Alsedà i Soler, Lluís (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Bordignon, Liane (Universidade Federal de São Carlos. Departamento de Matemática)
Groisman, Jorge (Universidad de la República (Uruguai). Instituto de Matemática y Estadística "Rafael Laguardia")

Additional title:	Topological entropy, sets of periods, and transitivity for graph maps
Date:	2024
Abstract:	Transitivity, the existence of periodic points and positive topological entropy can be used to characterize complexity in dynamical systems. It is known that for graphs that are not trees, for every ε > 0, there exist (complicate) totally transitive maps (then with cofinite set of periods) such that the topological entropy is smaller than ε (simplicity). We show by means of three examples that for any graph that is not a tree, relatively simple maps (with small entropy) which are totally transitive (and hence robustly complicate) can be constructed so that the set of periods is also relatively simple. To numerically measure the complexity of the set of periods we introduce a notion of a boundary of cofiniteness. Larger boundary of cofiniteness means simpler set of periods. With the help of the notion of boundary of cofiniteness we can state precisely what do we mean by extending the entropy simplicity result to the set of periods: there exist relatively simple maps such that the boundary of cofiniteness is arbitrarily large (simplicity) which are totally transitive (and hence robustly complicate). Moreover, we will show that, the above statement holds for arbitrary continuous degree one circle maps.
Grants:	Ministerio de Economía y Competitividad MTM2014-52209-C2-1-P Agencia Estatal de Investigación MTM2017-86795-C3-1-P Agencia Estatal de Investigación PID2020-118281GB-C31 Ministerio de Economía y Competitividad MDM-2014-0445 Agencia Estatal de Investigación CEX2020-001084-M
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Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Topological entropy ; Sets of periods ; Total transitivity ; Boundary of cofiniteness ; Rotation sets ; Circle maps
Published in:	Ukrainian Mathematical Journal, Vol. 76, Issue 1 (June 2024) , p. 31-50, ISSN 1573-9376

Adreça alternativa: https://umj.imath.kiev.ua/index.php/umj/article/view/7659
DOI: 10.1007/s11253-024-02305-y

Postprint 82 p, 1.0 MB

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Research literature > UAB research groups literature > Research Centres and Groups (research output) > Experimental sciences > GSD (Dynamical systems)
Articles > Research articles
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