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

Lluís Alsedà; Liane Bordignon; Jorge Groisman

doi:10.3842/umzh.v76i1.7659

Lluís Alsedà Departament de Matemàtiques and Centre de Recerca Matemàtica, Edifici Cc, Universitat Autònoma de Barcelona, Spain
Liane Bordignon Departamento de Matemática, Universidade Federal de São Carlos, São Paulo, Brasil
Jorge Groisman IMERL, Facultad de Ingenierìa, Universidad de la República, Montevideo, Uruguay

DOI: https://doi.org/10.3842/umzh.v76i1.7659

Keywords: Topological entropy, sets of periods, total transitivity, boundary of cofiniteness, rotation sets, circle maps

Abstract

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.

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