Separation Conditions for Iterated Function Systems with a Common Attractor

RESUMO

Let $K$ be the non-singleton attractor of a finite iterated function system $\Phi=\{\phi_i(x)=a_i O_i x+t_i\}_{i=1}^m$ 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.