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A switched server system semiconjugate to a minimal interval exchange

Part of: Applications - Dynamical systems and ergodic theory Low-dimensional dynamical systems Topological dynamics

Published online by Cambridge University Press:  13 September 2019

FILIPE FERNANDES  and
BENITO PIRES Open the ORCID record for BENITO PIRES [Opens in a new window]
FILIPE FERNANDES
Affiliation:
Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, 13565-905 SP, Brazil email: filipefernandes@dm.ufscar.br
BENITO PIRES
Affiliation:
Departamento de Computação e Matemática, Faculdade de Filosofia, Ciências e Letras, Universidade de São Paulo, Ribeirão Preto, 14040-901 SP, Brazil email: benito@usp.br
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Abstract

Switched server systems are mathematical models of manufacturing, traffic and queueing systems that have being studied since the early 1990s. In particular, it is known that typically the dynamics of such systems is asymptotically periodic: each orbit of the system converges to one of its finitely many limit cycles. In this article, we provide an explicit example of a switched server system with exotic behaviour: each orbit of the system converges to the same Cantor attractor. To accomplish this goal, we bring together recent advances in the understanding of the topological dynamics of piecewise contractions and interval exchange transformations (IETs) with flips. The ultimate result is a switched server system whose Poincaré map is semiconjugate to a minimal and uniquely ergodic IET with flips.

Keywords

Piecewise contractionsymbolic dynamicsswitched server systempseudo billiard

MSC classification

Primary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Secondary: 37B10: Symbolic dynamics 37N35: Dynamical systems in control
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 4 , August 2020 , pp. 682 - 708
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Copyright
© Cambridge University Press 2019

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Footnotes

†

F. Fernandes was financed by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001. B. Pires was partially supported by grant no. 2018/06916-0, São Paulo Research Foundation (FAPESP) and by the National Council for Scientific and Technological Development (CNPq). The authors thank the reviewers for the list of suggestions that contributed to improve the first version of this manuscript.

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