Dimensions of non-differentiability points of Cantor functions

Yuanyuan Yao¹ ; Yunxiu Zhang¹ ; Wenxia Li¹

¹ Department of Mathematics East China Normal University Shanghai 200241, P.R. China

Studia Mathematica, Tome 195 (2009) no. 2, pp. 113-125

Résumé

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 ${\mathbb R}$ generated by a contractive similitude $h_i(x)=a_ix+b_i$, $i=0,1$. Let $S$ denote the set of points of $C$ at which the probability distribution function $F(x)$ of $\mu$ 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 $p_0 a_0$.

Détail

DOI : 10.4064/sm195-2-2

Keywords: probability vector there exists corresponding self similar borel probability measure supported cantor set strong separation property mathbb generated contractive similitude denote set points which probability distribution function has derivative finite infinite hausdorff packing dimensions have found several authors i however equivalently structure changes significantly previous approaches fail effective present paper devoted determining hausdorff packing dimensions