Characterization of ω -limit sets of continuous maps of the circle

David Pokluda

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 3, page 575-581
  • ISSN: 0010-2628

Abstract

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In this paper we extend results of Blokh, Bruckner, Humke and Sm’ıtal [Trans. Amer. Math. Soc. 348 (1996), 1357–1372] about characterization of ω -limit sets from the class 𝒞 ( I , I ) of continuous maps of the interval to the class 𝒞 ( 𝕊 , 𝕊 ) of continuous maps of the circle. Among others we give geometric characterization of ω -limit sets and then we prove that the family of ω -limit sets is closed with respect to the Hausdorff metric.

How to cite

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Pokluda, David. "Characterization of $\omega $-limit sets of continuous maps of the circle." Commentationes Mathematicae Universitatis Carolinae 43.3 (2002): 575-581. <http://eudml.org/doc/248990>.

@article{Pokluda2002,
abstract = {In this paper we extend results of Blokh, Bruckner, Humke and Sm’ıtal [Trans. Amer. Math. Soc. 348 (1996), 1357–1372] about characterization of $\omega $-limit sets from the class $\mathcal \{C\}(I,I)$ of continuous maps of the interval to the class $\mathcal \{C\}(\mathbb \{S\},\mathbb \{S\})$ of continuous maps of the circle. Among others we give geometric characterization of $\omega $-limit sets and then we prove that the family of $\omega $-limit sets is closed with respect to the Hausdorff metric.},
author = {Pokluda, David},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {dynamical system; circle map; $\omega $-limit set; dynamical system; circle map; -limit set},
language = {eng},
number = {3},
pages = {575-581},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Characterization of $\omega $-limit sets of continuous maps of the circle},
url = {http://eudml.org/doc/248990},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Pokluda, David
TI - Characterization of $\omega $-limit sets of continuous maps of the circle
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 3
SP - 575
EP - 581
AB - In this paper we extend results of Blokh, Bruckner, Humke and Sm’ıtal [Trans. Amer. Math. Soc. 348 (1996), 1357–1372] about characterization of $\omega $-limit sets from the class $\mathcal {C}(I,I)$ of continuous maps of the interval to the class $\mathcal {C}(\mathbb {S},\mathbb {S})$ of continuous maps of the circle. Among others we give geometric characterization of $\omega $-limit sets and then we prove that the family of $\omega $-limit sets is closed with respect to the Hausdorff metric.
LA - eng
KW - dynamical system; circle map; $\omega $-limit set; dynamical system; circle map; -limit set
UR - http://eudml.org/doc/248990
ER -

References

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  2. Block L.S., Coppel W.A., Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer, Berlin, 1992. Zbl0746.58007MR1176513
  3. Blokh A., Bruckner A.M., Humke P.D., Smítal J., The space of ø m e g a -limit sets of a continuous map of the interval, Trans. Amer. Math. Soc. 348 (1996), 1357-1372. (1996) MR1348857
  4. Blokh A.M., On transitive mappings of one-dimensional ramified manifolds, in Differential-difference Equations and Problems of Mathematical Physics, Inst. Mat. Acad. Sci., Kiev, 1984, pp. 3-9 (Russian). Zbl0605.58007MR0884346
  5. Hric R., Topological sequence entropy for maps of the circle, Comment. Math. Univ. Carolinae 41 (2000), 53-59. (2000) Zbl1039.37007MR1756926
  6. Pokluda D., On the transitive and ø m e g a -limit points of the continuous mappings of the circle, Archivum Mathematicum, accepted for publication. Zbl1087.37033
  7. Sharkovsky A.N., The partially ordered system of attracting sets, Soviet Math. Dokl. 7 5 (1966), 1384-1386. (1966) 

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