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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 [email protected]
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 [email protected], [email protected]
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 [email protected], [email protected]
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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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