An explicit description of all scrambled sets of weakly unimodal functions of type \(2^ \infty \)
type: Item
instance of: scholarly article
title: An explicit description of all scrambled sets of weakly unimodal functions of type \(2^ \infty \)
zbMATH Open document ID: 0879.58044
published in: Real Analysis Exchange
publication date: 1997-12-10T00:00:00Z
review text: Let \(f\) be a weakly unimodal map of type \(2^\infty\) on the interval \(I\). For any \(x,y\in I\) write \(x\sim^sy\) if \(\lim_{n\rightarrow \infty} |f^n(x)-f^n(y)|=0\), and \(x\sim^iy\) if \(\liminf_{n\rightarrow \infty} |f^n(x)-f^n(y)|=0\). Clearly, chaos in the sense of Li and Yorke can be expressed in terms of both relations: there are \(x,y\in I\) such that \(x\sim^iy\), but not \(x\sim^sy\). The main contribution is the study of relations \(\sim^s\) and \(\sim^i\) on the set \(K(f)\) of points with infinite \(\omega\)-limit sets. Using a special kind of coding, the author gives an alternative characterization of chaotic maps of the above mentioned type, and of their maximal scrambled sets. The paper contains some open problems.
Mathematics Subject Classification ID: 37D45
Mathematics Subject Classification ID: 26A18
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 1095859
zbMATH Keywords: chaos
zbMATH Keywords: scrambled set
author: V. Jiménez López
reviewed by: Jaroslav Smítal
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1095859