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February 2009 An $n$-Cell in $\mathbb{R}^{n+1}$ that is not the Attractor of any IFS on $\mathbb{R}^{n+1}$
Manuel J. Sanders
Missouri J. Math. Sci. 21(1): 13-20 (February 2009). DOI: 10.35834/mjms/1316032676

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.

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Manuel J. Sanders. "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 (1) 13 - 20, February 2009. https://doi.org/10.35834/mjms/1316032676

Information

Published: February 2009
First available in Project Euclid: 14 September 2011

zbMATH: 1175.37026
MathSciNet: MR2503170
Digital Object Identifier: 10.35834/mjms/1316032676

Subjects:
Primary: 54H20‎
Secondary: 37B45

Rights: Copyright © 2009 Central Missouri State University, Department of Mathematics and Computer Science

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Vol.21 • No. 1 • February 2009
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