Structure of the class of iterated function systems that generate the same self-similar set

Abstract

Let ${\mathcal{H}}_K$ denote the family of homogenous IFSs that satisfy the open set condition (OSC) and generate the same self-similar set $K$, we call the IFSs in ${\mathcal{H}}_K$ isotopic, and give the isotopic class ${\mathcal{H}}_K$ a multiplication operation defined by composition. The finitely generated property of ${\mathcal{H}}_K$ was first studied by Feng and Wang on $\mathbb{R}$ [FW], and by the authors on ${\mathbb{R}}^d$ under the strong separation condition [DL]. In this paper, we continue the investigation of the isotopic class on ${\mathbb{R}}^d$. By using a new technique with the OSC, we prove that ${\mathcal{H}}_K$ is finitely generated if either (i) $K$ is totally disconnected, or (ii) the convex hull $\mathrm{Co}(K)$ is a polytope, and there exists a line $L$ passing through a vertex of Co$(K)$ such that $L\cap K$ is a totally disconnected infinite set. The conditions are easy to check and are satisfied by many standard self-similar sets.