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Topological dynamics of piecewise $\unicode[STIX]{x1D706}$-affine maps

Published online by Cambridge University Press:  10 November 2016

ARNALDO NOGUEIRA ,
BENITO PIRES Open the ORCID record for BENITO PIRES [Opens in a new window]  and
RAFAEL A. ROSALES
ARNALDO NOGUEIRA
Affiliation:
Aix Marseille University, CNRS, Centrale Marseille, I2M, 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, rrosales@usp.br
RAFAEL A. ROSALES
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, rrosales@usp.br
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Abstract

Let $-1<\unicode[STIX]{x1D706}<1$ and let $f:[0,1)\rightarrow \mathbb{R}$ be a piecewise $\unicode[STIX]{x1D706}$-affine contraction: that is, let there exist points $0=c_{0}<c_{1}<\cdots <c_{n-1}<c_{n}=1$ and real numbers $b_{1},\ldots ,b_{n}$ such that $f(x)=\unicode[STIX]{x1D706}x+b_{i}$ for every $x\in [c_{i-1},c_{i})$. We prove that, for Lebesgue almost every $\unicode[STIX]{x1D6FF}\in \mathbb{R}$, the map $f_{\unicode[STIX]{x1D6FF}}=f+\unicode[STIX]{x1D6FF}\,(\text{mod}\,1)$ is asymptotically periodic. More precisely, $f_{\unicode[STIX]{x1D6FF}}$ has at most $n+1$ periodic orbits and the $\unicode[STIX]{x1D714}$-limit set of every $x\in [0,1)$ is a periodic orbit.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 5 , August 2018 , pp. 1876 - 1893
Copyright
© Cambridge University Press, 2016 

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References

Alsedà, L. and Misiurewicz, M.. Semiconjugacy to a map of a constant slope. Discrete Contin. Dyn. Syst. Ser. B 20(10) (2015), 3403–3413.Google Scholar
Bruin, H. and Deane, J. H. B.. Piecewise contractions are asymptotically periodic. Proc. Amer. Math. Soc. 137(4) (2009), 1389–1395.Google Scholar
Bugeaud, Y.. Dynamique de certaines applications contractantes, linéaires par morceaux, sur [0, 1). C. R. Acad. Sci. Paris Sér. I Math. 317(6) (1993), 575–578.Google Scholar
Bugeaud, Y.. Linear mod one transformations and the distribution of fractional parts {𝜉(p/q) n }. Acta Arith. 114(4) (2004), 301–311.Google Scholar
Bugeaud, Y. and Conze, J.-P.. Calcul de la dynamique de transformations linéaires contractantes mod 1 et arbre de Farey. Acta Arith. 88(3) (1999), 201–218.Google Scholar
Catsigeras, E. and Budelli, R.. Topological dynamics of generic piecewise contractive maps in n dimensions. Int. J. Pure Appl. Math. 68(1) (2011), 61–83.Google Scholar
Catsigeras, E., Guiraud, P., Meyroneinc, A. and Ugalde, E.. On the asymptotic properties of piecewise contracting maps. Dyn. Syst. 31(2) (2016), 107–135.CrossRefGoogle Scholar
Coutinho, R.. Dinámica Simbólica Linear. PhD Thesis, Universidade Técnica de Lisboa, Instituto Superior Técnico, 1999.Google Scholar
Gambaudo, J.-M. and Tresser, C.. On the dynamics of quasi-contractions. Bol. Soc. Brasil. Mat. 19(1) (1988), 61–114.Google Scholar
Milnor, J. and Thurston, W.. On iterated maps of the interval. Dynamical Systems (College Park, MD, 1986–87) (Lecture Notes in Mathematics, 1342) . Springer, Berlin, 1988, pp. 465–563.Google Scholar
Misiurewicz, M. and Roth, S.. No semiconjugacy to a map of constant slope. Ergod. Th. & Dynam. Sys. 36(3) (2016), 875–889.Google Scholar
Nogueira, A. and Pires, B.. Dynamics of piecewise contractions of the interval. Ergod. Th. & Dynam. Sys. 35(7) (2015), 2198–2215.Google Scholar
Nogueira, A., Pires, B. and Rosales, R. A.. Asymptotically periodic piecewise contractions of the interval. Nonlinearity 27(7) (2014), 1603–1610.Google Scholar
Veerman, P.. Symbolic dynamics of order-preserving orbits. Phys. D 29(1–2) (1987), 191–201.CrossRefGoogle Scholar
Wells, J. C.. A note on a corollary of Sard’s theorem. Proc. Amer. Math. Soc. 48 (1975), 513–514.Google Scholar