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

  • 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
Keywords: 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.


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How to Cite
Alsedà, L., L. Bordignon, and J. Groisman. “Topological Entropy, Sets of Periods, and Transitivity for Circle Maps”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, no. 1, Feb. 2024, pp. 31 -47, doi:10.3842/umzh.v76i1.7659.
Research articles