Separation conditions for conformal iterated function systems
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AbstractWe extend both the weak separation condition and the finite type condition to include finite iterated function systems (IFSs) of injective C(1) conformal contractions on compact subsets of R(d). For conformal IFSs satisfying the bounded distortion property, we prove that the finite type condition implies the weak separation condition. By assuming the weak separation condition, we prove that the Hausdorff and box dimensions of the attractor are equal and, if the dimension of the attractor is a, then its alpha-dimensional Hausdorff measure is positive and finite. We obtain a necessary and sufficient condition for the associated self-conformal measure mu to be singular. By using these we give a first example of a singular invariant measure mu that is associated with a non-linear IFS with overlaps.
All Author(s) ListLau KS, Ngai SM, Wang XY
Journal nameMonatshefte für Mathematik
Year2009
Month4
Day1
Volume Number156
Issue Number4
PublisherSpringer Verlag (Germany)
Pages325 - 355
ISSN0026-9255
eISSN1436-5081
LanguagesEnglish-United Kingdom
KeywordsAbsolute continuity; Conformal iterated function system; Finite type condition; Hausdorff measure; Self-conformal measure; Singularity; Weak separation condition
Web of Science Subject CategoriesMathematics; MATHEMATICS

Last updated on 2021-13-01 at 00:20