Properties of topologically transitive maps on the real line
type: Item
instance of: scholarly article
title: Properties of topologically transitive maps on the real line
zbMATH Open document ID: 1015.37030
author: Anima Nagar
author: V. Kannan
author: S. P. Sesha Sai
published in: Real Analysis Exchange
publication date: 2003-01-05T00:00:00Z
review text: Topologically transitive maps defined on intervals of the real line are studied. Let \(I\) be any interval of the real line \(\mathbb{R}\). \(f:I\to I\) is topologically transitive in \(I\) if for any nonempty open subsets \(U, W\) of \(I\) there is a positive integer \(n\) such that \(f^n (U) \cap W\) is nonempty. A point \(x\in I\) is critical point of \(f\) if in every neighborhood \(V\) of \(x\) there are \(y,z \in V\) such that \(f(y)=f(z)\). Let \(C\) be the set of all critical points of \(f\). It is shown that for any topologically transitive map on \(\mathbb{R}\) the sets \(C\) and \(f(C)\) are unbounded, there are at most three open and invariant sets of \(f\) and with a single possible exception for any \(x\in \mathbb{R}\) the backward orbit of \(x\) is dense in \(\mathbb{R}\).
Mathematics Subject Classification ID: 37E05
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 1848923
zbMATH Keywords: topologically transitive maps
zbMATH Keywords: critical points
zbMATH Keywords: invariant set
reviewed by: Katarina Janková
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1848923