Topological sequence entropy for maps of the circle

Roman Hric

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 1, page 53-59
  • ISSN: 0010-2628

Abstract

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A continuous map f of the interval is chaotic iff there is an increasing sequence of nonnegative integers T such that the topological sequence entropy of f relative to T , h T ( f ) , is positive ([FS]). On the other hand, for any increasing sequence of nonnegative integers T there is a chaotic map f of the interval such that h T ( f ) = 0 ([H]). We prove that the same results hold for maps of the circle. We also prove some preliminary results concerning topological sequence entropy for maps of general compact metric spaces.

How to cite

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Hric, Roman. "Topological sequence entropy for maps of the circle." Commentationes Mathematicae Universitatis Carolinae 41.1 (2000): 53-59. <http://eudml.org/doc/248584>.

@article{Hric2000,
abstract = {A continuous map $f$ of the interval is chaotic iff there is an increasing sequence of nonnegative integers $T$ such that the topological sequence entropy of $f$ relative to $T$, $h_T(f)$, is positive ([FS]). On the other hand, for any increasing sequence of nonnegative integers $T$ there is a chaotic map $f$ of the interval such that $h_T(f)=0$ ([H]). We prove that the same results hold for maps of the circle. We also prove some preliminary results concerning topological sequence entropy for maps of general compact metric spaces.},
author = {Hric, Roman},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {chaotic map; circle map; topological sequence entropy; chaotic map; circle map; topological sequence; entropy},
language = {eng},
number = {1},
pages = {53-59},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Topological sequence entropy for maps of the circle},
url = {http://eudml.org/doc/248584},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Hric, Roman
TI - Topological sequence entropy for maps of the circle
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 1
SP - 53
EP - 59
AB - A continuous map $f$ of the interval is chaotic iff there is an increasing sequence of nonnegative integers $T$ such that the topological sequence entropy of $f$ relative to $T$, $h_T(f)$, is positive ([FS]). On the other hand, for any increasing sequence of nonnegative integers $T$ there is a chaotic map $f$ of the interval such that $h_T(f)=0$ ([H]). We prove that the same results hold for maps of the circle. We also prove some preliminary results concerning topological sequence entropy for maps of general compact metric spaces.
LA - eng
KW - chaotic map; circle map; topological sequence entropy; chaotic map; circle map; topological sequence; entropy
UR - http://eudml.org/doc/248584
ER -

References

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  11. Kuchta M., Characterization of chaos for continuous maps of the circle, Comment. Math. Univ. Carolinae 31 (1990), 383-390. (1990) Zbl0728.26011MR1077909
  12. Kuchta M., Smítal J., Two point scrambled set implies chaos, European Conference on Iteration Theory ECIT'87 World Sci. Publishing Co. Singapore. MR1085314
  13. Lemańczyk M., The sequence entropy for Morse shifts and some counterexamples, Studia Math. 52 (1985), 221-241. (1985) MR0825480
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