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Dynamics of piecewise contractions of the interval

Published online by Cambridge University Press:  03 July 2014

ARNALDO NOGUEIRA and
BENITO PIRESOpen the ORCID record for BENITO PIRES[Opens in a new window]
ARNALDO NOGUEIRA
Affiliation:
Aix-Marseille Université, CNRS, Centrale Marseille, I2M, UMR 7373, 13453 Marseille, France email arnaldo.nogueira@univ-amu.fr
BENITO PIRES
Affiliation:
Departamento de Computação e Matemática, Faculdade de Filosofia, Ciências e Letras, Universidade de São Paulo, 14040-901, Ribeirão Preto - SP, Brazil email benito@usp.br
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Abstract

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We study the long-term behavior of injective piecewise contractions of the interval. We prove that every injective piecewise contraction with $n-1$ discontinuities has at most $n$ periodic orbits and is topologically conjugate to a piecewise linear contraction.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 7 , October 2015, pp. 2198 - 2215
Copyright
© Cambridge University Press, 2014 

References

Blank, B. and Bunimovich, L.. Switched flow systems: pseudo-billiard dynamics. Dyn. Syst. 19(4) (2004), 359–370.CrossRefGoogle Scholar
Brémont, J.. Dynamics of injective quasi-contractions. Ergod. Th. & Dynam. Sys. 26 (2006), 19–44.CrossRefGoogle Scholar
Bruin, H. and Deane, J. H. B.. Piecewise contractions are asymptotically periodic. Proc. Amer. Math. Soc. 137(4) (2009), 1389–1395.CrossRefGoogle Scholar
Chase, C., Serrano, J. and Ramadge, P. J.. Periodicity and chaos from switched flow systems: contrasting examples of discretely controlled continuous systems. IEEE Trans. Automat. Control. 38(1) (1993), 70–83.CrossRefGoogle Scholar
Gutiérrez, C.. Smoothability of Cherry Flows (Lecture Notes in Mathematics, 1007). Springer, Berlin, 1983, pp. 308–331.Google Scholar
Gutiérrez, C.. On two-dimensional recurrence. Bull. Braz. Math. Soc. 10(1) (1979), 1–16.CrossRefGoogle Scholar
Gambaudo, J.-M. and Tresser, C.. On the dynamics of quasi-contractions. Bull. Braz. Math. Soc. 19(1) (1988), 61–114.CrossRefGoogle Scholar
Jeong, I.-J.. Outer billiards with contraction. Senior Thesis, Brown University, 2012.Google Scholar
MacPhee, I. M., Menshikov, M. V., Popov, S. and Volkov, S.. Periodicity in the transient regime of exhaustive polling systems. Ann. Appl. Probab. 16(4) (2006), 1816–1850.CrossRefGoogle Scholar
Nogueira, A.. Almost all interval exchange transformation with flips are nonergodic. Ergod. Th. & Dynam. Sys. 9 (1989), 515–525.CrossRefGoogle Scholar
Nogueira, A., Pires, B. and Rosales, R.. Asymptotically periodic piecewise contractions of the interval. Preprint, http://arxiv.org/pdf/1310.5784v1.pdf, pp. 1–8.Google Scholar
Nogueira, A., Pires, B. and Troubetzkoy, S.. Orbit structure of interval exchange transformations with flip. Nonlinearity 26(2) (2013), 525–537.CrossRefGoogle Scholar