Commutativity and non-commutativity of topological sequence entropy

Balibrea, Francisco ; Peña, Jose Salvador Cánovas ; López, Víctor Jiménez

Annales de l'Institut Fourier, Tome 49 (1999), p. 1693-1709 / Harvested from Numdam

Résumé

Dans cet article nous étudions la propriété de commutativité pour l’entropie séquentielle topologique. Nous prouvons que si $X$ est un espace métrique compact et $f,g: X\rightarrow X$ sont deux fonctions continues, alors $h _A(f\circ g)=h_A(g\circ f)$ pour toute suite croissance $A$ où $X=[0,1]$ et nous construisons un contre-exemple dans le cas général. Au passage, nous prouvons aussi que l’égalité $h_A(f)=h_A(f\vert _{\cap _{n\ge 0}f^n(X)})$ est vraie si $X=[0,1]$ mais ne l’est pas nécessairement si $X$ est un espace métrique compact arbitraire.

Abstract

In this paper we study the commutativity property for topological sequence entropy. We prove that if $X$ is a compact metric space and $f,g: X\rightarrow X$ are continuous maps then $h _A(f\circ g)=h_A(g\circ f)$ for every increasing sequence $A$ if $X=[0,1]$, and construct a counterexample for the general case. In the interim, we also show that the equality $h_A(f)=h_A(f\vert _{\cap _{n\ge 0}f^n(X)})$ is true if $X=[0,1]$ but does not necessarily hold if $X$ is an arbitrary compact metric space.

Détail

Publié le : 1999-01-01

MR 2001g:37015 | Zbl 0990.37010 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.1735

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