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arXiv · 2208.04042

A separation property for iterated function systems of similitudes

De-Jun Feng·Huo-Jun Ruan·Ying Xiong
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Abstract

Let $E$ be the attractor of an iterated function system $\{\phi_i(x)=\rho R_ix+a_i\}_{i=1}^N$ on $\Bbb R^d$, where $0<\rho<1$, $a_i\in \Bbb R^d$ and $R_i$ are orthogonal transformations on $\Bbb R^d$. Suppose that $\{\phi_i\}_{i=1}^N$ satisfies the open set condition, but not the strong separation condition. We show that $E$ can not be generated by any iterated function system of similitudes satisfying the strong separation condition. This gives a partial answer to a folklore question about the separation conditions on the generating iterated function systems of self-similar sets.

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De-Jun Feng, Huo-Jun Ruan, Ying Xiong. 2022-08-08. A separation property for iterated function systems of similitudes. https://arxiv.org/abs/2208.04042

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