Dimensions of non-differentiability points of Cantor functions

作者：Yao Yuanyuan; Zhang Yunxiu; Li Wenxia^*

来源：Studia Mathematica, 2009, 195(2): 113-125.

DOI：10.4064/sm195-2-2

• Hausdorff dimension
• Cantor function
• non-differentiability point
• self-similar measure
• distribution function

For a probability vector (p(0),p(1)) there exists a corresponding self-similar Borel probability measure mu supported on the Cantor set C (with the strong separation property) in R generated by a contractive similitude h(i)(x) = a(i)x b(i), i = 0, 1. Let S denote the set of points of C at which the probability distribution function F(x) of p has no derivative, finite or infinite. The Hausdorff and packing dimensions of S have been found by several authors for the case that p(i) > a(i), i = 0, 1. However, when p(0) < a(0) (or equivalently p(1) < a(1)) the structure of S changes significantly and the previous approaches fail to be effective any more. The present paper is devoted to determining the Hausdorff and packing dimensions of S for the case p(0) < a(0).

• 出版日期2009
• 单位华东师范大学