Structure of the class of iterated function systems that generate the same self-similar set
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AbstractLet HK 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 HK isotopic, and give the isotopic class HK a multiplication operation defined by composition. The finitely generated property of HK was first studied by Feng and Wang on R [FW], and by the authors on Rd under the strong separation condition [DL]. In this paper, we continue the investigation of the isotopic class on Rd. By using a new technique with the OSC, we prove that HK is finitely generated if either (i) K is totally disconnected, or (ii) the convex hull Co(K) is a polytope, and there exists a line L passing through a vertex of Co(K) such that L∩K is a totally disconnected infinite set. The conditions are easy to check and are satisfied by many standard self-similar sets.
All Author(s) ListQi-Rong Deng, Ka-Sing Lau
Journal nameJournal of Fractal Geometry
Volume Number4
Issue Number1
Pages43 - 71
LanguagesEnglish-United Kingdom

Last updated on 2021-16-01 at 01:25