An $n$-Cell in $\mathbb{R}^{n+1}$ that is not the Attractor of any IFS on $\mathbb{R}^{n+1}$

Abstract

Crovisier and Rams [2] recently constructed an embedded Cantor set in $\mathbb{R}$ and showed that it could not be realized as an attractor of any iterated function system (IFS) using measure-theoretic properties. Also, an example of a locally connected continuum in $\mathbb{R} ^2$ which is not the attractor of any IFS on $\mathbb{R}^2$ is constructed in a work of Kwieciński [6]. Kwieciński points out that a variation on his main construction provides an arc in $\mathbb{R} ^2$ which is not the attractor of any IFS either. In this work, for each $n \geq 1$, we construct an $n$-cell in $\mathbb{R}^{n+1}$ and show that this $n$-cell cannot be the attractor of any IFS on $\mathbb{R}^{n+1}$. The $n=1$ case reaffirms the result observed by Kwieciński.