A separation property for iterated function systems of similitudes

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