Commutativity and non-commutativity of topological sequence entropy
type: Item
instance of: scholarly article
title: Commutativity and non-commutativity of topological sequence entropy
zbMATH Open document ID: 0990.37010
author: Francisco Balibrea
author: José Salvador Cánovas Peña
author: Víctor Jiménez López
publication date: 1999-11-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75399
review text: Topological sequence entropy [as defined by \textit{T. N. T. Goodman}, Proc. Lond. Math. Soc. (3) 29, 331-350 (1974; Zbl 0293.54043)] is a natural extension of topological entropy [introduced by \textit{R. L. Adler, A. G. Konheim} and \textit{M. H. McAndrew}, Trans. Am. Math. Soc. 114, 309-319 (1965; Zbl 0127.13102) and in case of metrizable space by \textit{R. Bowen}, Trans. Am. Math. Soc. 153, 401-414 (1971; Zbl 0212.29201), \textit{E. I. Dinaburg}, Izv. Akad. Nauk SSSR, Ser. Mat. 35, 324-366 (1971; Zbl 0216.44704)]. One should reasonably expect that the commutativity property also holds for it. This paper is devoted to commutativity and noncommutativity properties of topological sequence entropy. First, the authors prove that if \(X\) is a compact metric space and \(f,g:X\to X\) are continuous maps then \(h_A(f\circ g)= h_A(g\circ f)\) for every increasing sequence \(A\) if \(X= [0,1]\). Then they construct a counterexample to the commutativity formula in the setting of an appropriate compact metric space using so-called weakly unimodal maps of type \(2^\infty\). In the interim, they also show that the equality \(h_A(f)= h_A(f|_{\bigcap_{n\geq 0}f^n(X)})\) is true if \(X= [0,1]\) but does not necessarily hold if \(X\) is an arbitrary compact metric space. Commutativity for continuous selfmaps of a compact space was first given by \textit{S. F. Kolyada} and \textit{L. Snoha} [Random Comput. Dyn. 4, 205-233 (1996; Zbl 0909.54012)].
Mathematics Subject Classification ID: 37B40
Mathematics Subject Classification ID: 37E05
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 54C70
zbMATH DE Number: 1355997
zbMATH Keywords: commutativity
zbMATH Keywords: topological sequence entropy
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
OpenAlex ID: W2020720501
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DOI: 10.5802/AIF.1735
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 1355997