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Non‐differentiability points of Cantor functions
Author(s) -
Li Wenxia
Publication year - 2007
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200410470
Subject(s) - mathematics , cantor function , cantor set , disjoint sets , probability measure , combinatorics , differentiable function , borel set , hausdorff measure , hausdorff dimension , measure (data warehouse) , probability distribution , hausdorff space , disjoint union (topology) , packing dimension , discrete mathematics , pure mathematics , mathematical analysis , minkowski–bouligand dimension , fractal dimension , fractal , statistics , computer science , database
Let the Cantor set C in ℝ be defined by C = ∪ rj =0 h j ( C ) with a disjoint union, where the h j 's are similitude mappings with ratios 0 < a j < 1. Let μ be the self‐similar Borel probability measure on C corresponding to the probability vector ( p 0 , p 1 , …, p r ). Let S be the set of points at which the probability distribution function F ( x ) of μ has no derivative, finite or infinite. For the case where p i > a i we determine the packing and box dimensions of S and give an approach to calculate the Hausdorff dimension of S . (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)