Commutativity and non-commutativity of topological sequence entropy

Francisco Balibrea; Jose Salvador Cánovas Peña; Víctor Jiménez López

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 5, page 1693-1709
  • ISSN: 0373-0956

Abstract

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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 → X are continuous maps then h A ( f ∘ g ) = h A ( g ∘ 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 | ∩ n ≥ 0 f n ( X ) ) is true if X = [ 0 , 1 ] but does not necessarily hold if X is an arbitrary compact metric space.

How to cite

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Balibrea, Francisco, Peña, Jose Salvador Cánovas, and López, Víctor Jiménez. "Commutativity and non-commutativity of topological sequence entropy." Annales de l'institut Fourier 49.5 (1999): 1693-1709. <http://eudml.org/doc/75399>.

@article{Balibrea1999,
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.},
author = {Balibrea, Francisco, Peña, Jose Salvador Cánovas, López, Víctor Jiménez},
journal = {Annales de l'institut Fourier},
keywords = {commutativity; topological sequence entropy},
language = {eng},
number = {5},
pages = {1693-1709},
publisher = {Association des Annales de l'Institut Fourier},
title = {Commutativity and non-commutativity of topological sequence entropy},
url = {http://eudml.org/doc/75399},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Balibrea, Francisco
AU - Peña, Jose Salvador Cánovas
AU - López, Víctor Jiménez
TI - Commutativity and non-commutativity of topological sequence entropy
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 5
SP - 1693
EP - 1709
AB - 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.
LA - eng
KW - commutativity; topological sequence entropy
UR - http://eudml.org/doc/75399
ER -

References

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  9. [9] S. KOLYADA and L'. SNOHA, Topological entropy of nonautononous dynamical systems, Random and Comp. Dynamics, 4 (1996), 205-233. Zbl0909.54012MR98f:58126
  10. [10] A. LINERO, Cuestiones sobre dinámica topológica de algunos sistemas bidimensionales y medidas invariantes de sistemas unidimensionales asociados, PhD thesis, Universidad de Murcia, 1998. 
  11. [11] M. MISIUREWICZ and J. SMITAL, Smooth chaotic functions with zero topological entropy, Ergod. Th. and Dynam. Sys., 8 (1988), 421-424. Zbl0689.58028MR90a:58118
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  13. [13] P. WALTERS, An introduction to ergodic theory, Springer-Verlag, Berlin, 1982. Zbl0475.28009MR84e:28017

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