Chaos in countable dynamical system
type: Item
instance of: scholarly article
title: Chaos in countable dynamical system
zbMATH Open document ID: 1034.37005
DOI: 10.1016/S0166-8641(02)00079-2
author: Jozef Bobok
published in: Topology and its Applications
publication date: 2003-04-02T00:00:00Z
review text: Let \((X, \tau )\) be a countable compact Hausdorff space. For any \(Y\subset X\) let \(L(Y)\) be the set obtained from \(Y\) by removing the isolated points. The main result is the following characterization of spaces, for which there exists a distributionally chaotic continuous map \(F\) of \(X\) (for the notion of distributional chaos, cf. \textit{B. Schweizer} and the reviewer [Trans. Am. Math. Soc. 344, 737--754, (1994; Zbl 0812.58062)]): There is a \(\tau\)-metric \(\rho\) on \(X\) and a distributionally chaotic continuous map \(F: X\rightarrow X\) if and only if \(\#L^2(X)>1\). Here distributional chaos means existence of a distributionally chaotic pair of points. Moreover, if \(\#L^2(X)>1\) then there is a homeomorphism of \(X\) which is distributionally chaotic with respect to a suitable \(\tau\)-metric. The paper contains other similar interesting results.
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 37E99
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 1891208
zbMATH Keywords: countable compact Hausdorff space
zbMATH Keywords: countable dynamics
zbMATH Keywords: chaos
reviewed by: Jaroslav Smítal
MaRDI profile type: Publication
cites work: Q3993502
cites work: Homeomorphisms with the whole compacta being scrambled sets
cites work: Measures of Chaos and a Spectral Decomposition of Dynamical Systems on the Interval
description: scientific article; zbMATH DE number 1891208