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

Abstract

Let {\mathscr 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 {\mathscr H}_K isotopic , and give the isotopic class {\mathscr H}_K a multiplication operation defined by composition. The finitely generated property of {\mathscr 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 {\mathscr H}_K is finitely generated if either (i) K is totally disconnected, or (ii) the convex hull {\rm 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.