On separation properties for iterated function systems of similitudes

Let $E$ be the attractor of an iterated function system ${\lbrace \phi_i (x) = \rho R_i x + a_i \rbrace}^N_{i=1}$ on $\mathbb{R}^d$, where $0 \lt \rho \lt 1$, $a_i \in \mathbb{R}^d$ and $R_i$ are orthogonal transformations on $\mathbb{R}^d$. Suppose that ${\lbrace \phi_i \rbrace}^N_{i=1}$ 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.

Keywords

iterated function systems, self-similar sets, open set condition, strong separation condition

2010 Mathematics Subject Classification

28A78, 28A80

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Feng was partially supported by the General Research Funds (CUHK14301017, CUHK14303021) from the Hong Kong Research Grant Council.

Ruan was partially by NSFC grant 11771391, ZJNSF grant LY22A010023 and the Fundamental Research Funds for the Central Universities of China (grant 2021FZZX001-01).

Xiong was partially supported by NSFC grant 12271175 and 11871227.

Received 17 August 2022

Accepted 16 February 2023

Published 21 July 2023