In this paper we extend results of Blokh, Bruckner, Humke and Sm'{\i}tal [Trans. Amer. Math. Soc. {\bf 348} (1996), 1357--1372] about characterization of $\omega$-limit sets from the class $\Cal{C}(I,I)$ of continuous maps of the interval to the class $\Cal C(\Bbb S,\Bbb S)$ 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.
@article{119347,
author = {David Pokluda},
title = {Characterization of $\omega$-limit sets of continuous maps of the circle},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {43},
year = {2002},
pages = {575-581},
zbl = {1090.37027},
mrnumber = {1920533},
language = {en},
url = {http://dml.mathdoc.fr/item/119347}
}
Pokluda, David. Characterization of $\omega$-limit sets of continuous maps of the circle. Commentationes Mathematicae Universitatis Carolinae, Tome 43 (2002) pp. 575-581. http://gdmltest.u-ga.fr/item/119347/
Combinatorial Dynamics and Entropy in Dimension One, World Scientific Publ., Singapore, 1993. | MR 1255515
Dynamics in One Dimension, Lecture Notes in Math. 1513, Springer, Berlin, 1992. | MR 1176513 | Zbl 0746.58007
The space of $ømega $-limit sets of a continuous map of the interval, Trans. Amer. Math. Soc. 348 (1996), 1357-1372. (1996) | MR 1348857
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). | MR 0884346 | Zbl 0605.58007
Topological sequence entropy for maps of the circle, Comment. Math. Univ. Carolinae 41 (2000), 53-59. (2000) | MR 1756926 | Zbl 1039.37007
On the transitive and $ømega$-limit points of the continuous mappings of the circle, Archivum Mathematicum, accepted for publication. | Zbl 1087.37033
The partially ordered system of attracting sets, Soviet Math. Dokl. 7 5 (1966), 1384-1386. (1966)