Non-differentiability points of Cantor functions

Wenxia Li

doi:10.1002/mana.200410470

Non-differentiability points of Cantor functions

Wenxia Li

School of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

Let the Cantor set C in ℝ be defined by C = U_j=0 ^r h_j (C) 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 (p₀, P₁,...., Pr). Let 5 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.

Original language	English
Pages (from-to)	140-151
Number of pages	12
Journal	Mathematische Nachrichten
Volume	280
Issue number	1-2
DOIs	https://doi.org/10.1002/mana.200410470
State	Published - 2007

Keywords

Cantor function
Hausdorff dimension
Non-differentiability
Packing dimension

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10.1002/mana.200410470

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abstract = "Let the Cantor set C in ℝ be defined by C = Uj=0 r hj (C) disjoint union, where the hj's are similitude mappings with ratios 0 < aj < 1. Let μ be the self-similar Borel probability measure on C corresponding to the probability vector (p0, P1,...., Pr). Let 5 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 pi > ai we determine the packing and box dimensions of S and give an approach to calculate the Hausdorff dimension of S.",

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AB - Let the Cantor set C in ℝ be defined by C = Uj=0 r hj (C) disjoint union, where the hj's are similitude mappings with ratios 0 < aj < 1. Let μ be the self-similar Borel probability measure on C corresponding to the probability vector (p0, P1,...., Pr). Let 5 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 pi > ai we determine the packing and box dimensions of S and give an approach to calculate the Hausdorff dimension of S.

KW - Cantor function

KW - Hausdorff dimension

KW - Non-differentiability

KW - Packing dimension

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IS - 1-2

ER -

Non-differentiability points of Cantor functions

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Keywords

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