Methods and Applications of Analysis

Volume 30 (2023)

Number 1

On separation properties for iterated function systems of similitudes

Pages: 17 – 26

DOI: https://dx.doi.org/10.4310/MAA.2023.v30.n1.a2

Authors

De-Jun Feng (Department of Mathematics, Chinese University of Hong Kong, djfeng@math.cuhk.edu.hk)

Huo-Jun Ruan (School of Mathematical Sciences, Zhejiang University, Hangzhou, Zhejiang, China)

Ying Xiong (Department of Mathematics, South China University of Technology, Guangzhou, Guangdong, China)

Abstract

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

Received 17 August 2022

Accepted 16 February 2023

Published 21 July 2023