Separation conditions for conformal iterated function systems
Finite type is a useful tool in the studies of self-similar sets with overlaps, the motivation and applications of finite type were discussed by \textit{H. Rao} and \textit{Z. Wen} first in the studies of \(\lambda\)-Cantor sets [Adv. Appl. Math. 20, No. 1, 50--72 (1998; Zbl 0894.28003)] and \textit{D. Feng} and \textit{Z. Wen} for Pisot numbers [J. Number Theory 97, No. 2, 305--316 (2002; Zbl 1026.11078)]. \textit{S.-M. Ngai} and \textit{Y. Wang} consider a more general case [J. Lond. Math. Soc., II. Ser. 63, No. 3, 655--672 (2001; Zbl 1013.28008)]. In this paper, the authors extend the weak separation and the finite type condition to some iterated function systems (IFS) of injective \(C^1\)-conformal contractions on compact subsets of \(\mathbb R^d\). For conformal IFS with the bounded distortion property, the authors prove that the finite type condition implies the weak separation condition, and that the Hausdorff dimension and the Bouligand dimension coincide, moreover, the Hausdorff measure is finite positive.
separation conditions for iterated function
scientific article; zbMATH DE number 5547506
scientific article