Self-Similarity in the Collection of ω-Limit Sets

Abstract

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