The infinite derivatives of Okamoto's self-affine functions: an application of $\beta$-expansions

Abstract

Okamoto's one-parameter family of self-affine functions $F_a:[0,1]\to [0,1]$, where $0<a<1$, includes the continuous nowhere differentiable functions of Perkins ($a=5/6$) and Bourbaki/Katsuura ($a=2/3$), as well as the Cantor function ($a=1/2$). The main purpose of this article is to characterize the set of points at which $F_a$ has an infinite derivative. We compute the Hausdorff dimension of this set for the case $a\le 1/2$, and estimate it for $a>1/2$. For all $a$, we determine the Hausdorff dimension of the sets of points where: (i) $F_a^{\prime }=0$; and (ii) $F_a$ has neither a finite nor an infinite derivative. The upper and lower densities of the digit $1$ in the ternary expansion of $x\in [0,1]$ play an important role in the analysis, as does the theory of $\beta$-expansions of real numbers.