A separation property for iterated function systems of similitudes

Abstract

Let $E$ be the attractor of an iterated function system $\{{\varphi }_i(x)=\rho R_{i}x+a_i{\}}_{i=1}^N$ on ${\mathbb{R}}^d$, where $0<\rho <1$, $a_i\in {\mathbb{R}}^d$ and $R_i$ are orthogonal transformations on ${\mathbb{R}}^d$. Suppose that $\{{\varphi }_i{\}}_{i=1}^N$ 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.