A generalized finite type condition for iterated function systems
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AbstractWe study iterated,function systems (IFSs) of contractive similitudes on R-d with overlaps. We introduce a generalized finite type condition which extends a more restrictive condition in [S.-M. Ngai, Y. Wang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2) 63 (3) (2001) 655-672] and allows us to include some IFSs of contractive similitudes whose contraction ratios are not exponentially commensurable. We show that the generalized finite type condition implies the weak separation property. Under this condition, we can identify the attractor of the IFS with that of a graph-directed IFS, and by modifying a setup of Mauldin and Williams [R.D. Mauldin, S.C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988) 811-829], we can compute the Hausdorff dimension of the attractor in terms of the spectral radius of certain weighted incidence matrix. (c) 2006 Elsevier Inc. All rights reserved.
All Author(s) ListLau KS, Ngai SM
Journal nameAdvances in Mathematics
Volume Number208
Issue Number2
Pages647 - 671
LanguagesEnglish-United Kingdom
Keywordsfinite type condition; generalized finite type condition; Hausdorff dimension; iterated function system; self-similar set
Web of Science Subject CategoriesMathematics; MATHEMATICS

Last updated on 2021-19-01 at 00:37