# 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

## Abstract

Let $ω$ be the map which takes $(x,f)$ in $I×C(I×I)$ to the $ω$-limit set $ω(x,f)$ with $L$ the map taking $f$ in $C(I,I)$ to the family of $ω$-limit sets ${ω(x,f):x∈I}$. We study $R(ω)={ω(x,f):(x,f)∈I×C(I,I)}$, the range of $ω$, and $R(L)={L(f):f∈C(I,I)}$, the range of $L$. In particular, $R(ω)$ and its complement are both dense, $R(ω)$ is path-connected, and $R(ω)$ is the disjoint union of a dense $G_{δ}$ set and a first category $F_{σ}$ set. We see that $R(L)$ is porous and path-connected, and its closure contains $K={F⊆[0,1]:Fis closed}$. Moreover, each of the sets $R(ω)$ and $R(L)$ 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