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AN ALMOST EVERYWHERE VERSION OF SMÍTAL’S ORDER–CHAOS DICHOTOMY FOR INTERVAL MAPS

Part of: Low-dimensional dynamical systems Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  01 August 2008

ALEJO BARRIO BLAYA  and
VÍCTOR JIMÉNEZ LÓPEZ
ALEJO BARRIO BLAYA
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain (email: alejobar@um.es)
VÍCTOR JIMÉNEZ LÓPEZ*
Affiliation:
Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain (email: vjimenez@um.es)
*
✉For correspondence; e-mail: vjimenez@um.es
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Abstract

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We prove that if f:I=[0,1]→I is a C3-map with negative Schwarzian derivative, nonflat critical points and without wild attractors, then exactly one of the following alternatives must occur: (i) R(f) has full Lebesgue measure λ; (ii) both S(f) and Scramb(f) have positive measure. Here R(f), S(f), and Scramb(f) respectively stand for the set of approximately periodic points of f, the set of sensitive points to the initial conditions of f, and the two-dimensional set of points (x,y) such that {x,y} is a scrambled set for f. Also, we show that if f is piecewise monotone and has no wandering intervals, then either λ(R(f))=1 or λ(S(f))>0, and provide examples of maps g,h of this type satisfying S(g)=S(h)=I such that, on the one hand, λ(R(g))=0 and λ2 (Scramb (g))=0 , and, on the other hand, λ(R(h))=1 .

Keywords

adding machineapproximate periodicityLi–Yorke chaosnegative Schwarzian derivativepiecewise monotone map scrambled setsensitivity to initial conditionssolenoidwandering intervalwild attractor

MSC classification

Secondary: 37E05: Maps of the interval (piecewise continuous, continuous, smooth) 37D45: Strange attractors, chaotic dynamics 37E15: Combinatorial dynamics (types of periodic orbits)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 85 , Issue 1 , August 2008 , pp. 29 - 50
Copyright
Copyright © 2008 Australian Mathematical Society

Footnotes

This work has been partially supported by MEC (Ministerio de Educación y Ciencia, Spain) and FEDER (Fondo Europeo de Desarrollo Regional), grant MTM2005-03868, and Fundación Séneca (Comunidad Autónoma de la Región de Murcia, Spain), grant 00684/PI/04.

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