Dimensions of non-differentiability points of Cantor functions
Volume 195 / 2009
Abstract
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 $p_0< a_0$.
