Conceptions of topological transitivity
type: Item
instance of: scholarly article
title: Conceptions of topological transitivity
zbMATH Open document ID: 1248.37015
author: Ethan Akin
author: Jeffrey D. Carlson
published in: Topology and its Applications
publication date: 2012-07-31T00:00:00Z
full work available at URL: https://arxiv.org/abs/1108.4710
review text: The authors discuss the concept of topological transitivity for a discrete dynamical system generated by a continuous map \(f\) defined on a (nonempty) topological space \(X\). They say that \((X,f)\) is topologically transitive if for every pair of nonempty open sets \(U,V\subset X\) the set \(\{k\in\mathbb{Z}|\;A\cap f^{-k}(B)\not=\emptyset\}\) is not empty. The authors list six further properties related to transitivity and they show that all of them coincide in the case when \(X\) is perfect, Hausdorff, second countable, and non-meager; each of these concepts is traced back in the literature.
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 6062385
zbMATH Keywords: topological transitivity
zbMATH Keywords: transitive point
zbMATH Keywords: minimal set
zbMATH Keywords: isolated point
reviewed by: Christian Fenske
MaRDI profile type: Publication
OpenAlex ID: W2963566784
arXiv ID: 1108.4710
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DOI: 10.1016/J.TOPOL.2012.04.016
description: scientific article; zbMATH DE number 6062385