Generating iterated function systems for self-similar sets with a separation condition
Volume 237 / 2017
Fundamenta Mathematicae 237 (2017), 127-133
MSC: Primary 28A80; Secondary 28A78.
DOI: 10.4064/fm88-10-2016
Published online: 9 January 2017
Abstract
Let $\{f_i\}_{i=1}^m$ be an iterated function system (IFS) for a self-similar set $E\subseteq {\mathbb R}^d$ (which is not a singleton) with the smallest integer $m\ge 2$. Suppose the distance of any two sets of the form $f_{i_1}(E)$ and $f_{i_2}(E)$ is strictly larger than the diameter of any $f_i(E)$. Then the semigroup of all generating IFSs for $E$, equipped with the composition as product, is finitely generated. This partially answers a question posed by Elekes, Keleti and Máthé.
