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Zeitschrift für Analysis und ihre Anwendungen


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Volume 33, Issue 1, 2014, pp. 87–100
DOI: 10.4171/ZAA/1500

Published online: 2013-12-27

Self-Similarity in the Collection of ω-Limit Sets

Emma D'Aniello[1] and Timothy H. Steele[2]

(1) Università degli Studi di Napoli, Caserta, Italy
(2) Weber State University, Ogden, USA

Let $\omega$ be the map which takes $(x,f)$ in $I \times C(I \times I)$ to the $\omega$-limit set $\omega(x,f)$ with ${\cal L}$ the map taking $f$ in $C(I,I)$ to the family of $\omega$-limit sets $\{\omega(x, f): x \in I\}$. We study ${\cal R}(\omega) = \{\omega(x,f): (x,f) \in I \times C(I,I)\}$, the range of $\omega$, and ${\cal R}({\cal L})= \{{\cal L}(f): f \in C(I,I)\}$, the range of ${\cal L}$. In particular, ${\cal R}(\omega)$ and its complement are both dense, ${\cal R}(\omega)$ is path-connected, and ${\cal R}(\omega)$ is the disjoint union of a dense $G_{\delta}$ set and a first category $F_{\sigma}$ set. We see that ${\cal R}({\cal L})$ is porous and path-connected, and its closure contains ${\cal K} = \{F \subseteq [0,1]: F \text{ is closed}\}$. Moreover, each of the sets ${\cal R}(\omega)$ and ${\cal R}({\cal L})$ demonstrates a self-similar structure.

Keywords: Continuous self-map, ω-limit set, porous set

D'Aniello Emma, Steele Timothy: Self-Similarity in the Collection of ω-Limit Sets. Z. Anal. Anwend. 33 (2014), 87-100. doi: 10.4171/ZAA/1500