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dynamical system; circle map; $\omega$-limit set
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.
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