Separation Conditions for Iterated Function Systems with a Common Attractor

Abstract

Let $K$ be the non-singleton attractor of a finite iterated function system $\Phi =\{{\varphi }_i(x)=a_i{O}_{i}x+t_i{\}}_{i=1}^m$ on ${\mathbb{R}}^d$, where $0<a_i<1$, $t_i\in {\mathbb{R}}^d$ and the $O_i$ are orthogonal transformations on ${\mathbb{R}}^d$. Suppose that $\Phi$ satisfies the open set condition, but not the strong separation condition. We show that $K$ cannot be generated by any finite iterated function system of similitudes satisfying the strong separation condition. This removes the homogeneity assumption from a result of Feng, Ruan and Xiong and gives a negative answer to a folklore question about the separation conditions on the generating iterated function systems of self-similar sets.

11 pages, no figures