A separation property for iterated function systems of similitudes

TL;DR

This paper proves that self-similar sets generated by iterated function systems of similitudes satisfying the open set condition cannot be generated by systems satisfying the strong separation condition, addressing a longstanding question in fractal geometry.

Contribution

It establishes a separation property showing that certain self-similar sets cannot be generated by strongly separated systems, clarifying the relationship between separation conditions.

Findings

Self-similar sets under open set condition cannot be generated by strongly separated systems.

Provides a partial answer to a folklore question in fractal geometry.

Highlights limitations of strong separation condition in generating certain self-similar sets.

Abstract

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