## Missouri Journal of Mathematical Sciences

- Missouri J. Math. Sci.
- Volume 21, Issue 1 (2009), 13-20.

### 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.

#### Article information

**Source**

Missouri J. Math. Sci., Volume 21, Issue 1 (2009), 13-20.

**Dates**

First available in Project Euclid: 14 September 2011

**Permanent link to this document**

https://projecteuclid.org/euclid.mjms/1316032676

**Digital Object Identifier**

doi:10.35834/mjms/1316032676

**Mathematical Reviews number (MathSciNet)**

MR2503170

**Zentralblatt MATH identifier**

1175.37026

**Subjects**

Primary: 54H20: Topological dynamics [See also 28Dxx, 37Bxx]

Secondary: 37B45: Continua theory in dynamics

#### Citation

Sanders, Manuel J. An $n$-Cell in $\mathbb{R}^{n+1}$ that is not the Attractor of any IFS on $\mathbb{R}^{n+1}$. Missouri J. Math. Sci. 21 (2009), no. 1, 13--20. doi:10.35834/mjms/1316032676. https://projecteuclid.org/euclid.mjms/1316032676