JournalszaaVol. 33, No. 1pp. 87–100

Self-Similarity in the Collection of ω-Limit Sets

  • Emma D'Aniello

    Università degli Studi di Napoli, Caserta, Italy
  • Timothy H. Steele

    Weber State University, Ogden, USA
Self-Similarity in the Collection of ω-Limit Sets cover
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Abstract

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

Cite this article

Emma D'Aniello, Timothy H. Steele, Self-Similarity in the Collection of ω-Limit Sets. Z. Anal. Anwend. 33 (2014), no. 1, pp. 87–100

DOI 10.4171/ZAA/1500