Hölder differentiability of self-similar measures

Abstract

We give a self-similar iterated function system $\{S_i=r_{i}x+b_i{\}}_{i=1}^M$ satisfying open set condition. Let $E$ be the self-similar set and $\mu$ the self-similar measure generated by a probability vector $(p_i{)}_{i=1}^M$. Let $f(x)=\mu ((-\infty ,x])$, and we write $N$ for the collection of $\alpha$-Hölder non-differentiable points of $f$. Suppose that $p_i>r_i^{\alpha }$, $i=1,\dots ,M$. We provide the Hausdorff, packing and box dimension formulas for the non-differentiable set $N$ of $f$ under the open set condition.