Differentiability of a two-parameter family of self-affine functions

This paper highlights an unexpected connection between expansions of real numbers to noninteger bases (so-called {\em $\beta$-expansions}) and the infinite derivatives of a class of self-affine functions. Precisely, we extend Okamoto’s function (itself a generalization of the well-known functions of Perkins and Katsuura) to a two-parameter family ${F_{N,a}: N\in$\mathbb{N}$, a\in(0,1)}$\mathrm{.}Wefirstshowthatforeach$x$,$F_{N,a}'(x)$iseither$0$,$\pm\infty$, or undefined. We then extend Okamoto’s theorem by proving that for each $N$, depending on the value of $a$ relative to a pair of thresholds, the set $\{x:F_{N,a}^{\mathrm{\prime }}(x)=0\}$ is either empty, uncountable but Lebesgue null, or of full Lebesgue measure. We compute its Hausdorff dimension in the second case. The second result is a characterization of the set ${\mathcal{D}}_{\mathrm{\infty }}(a):=\{x:F_{N,a}^{\mathrm{\prime }}(x)=\pm \mathrm{\infty }\}$, which enables us to closely relate this set to the set of points which have a unique expansion in the (typically noninteger) base $\beta =1\mathrm{/}a$. Recent advances in the theory of $\beta$-expansions are then used to determine the cardinality and Hausdorff dimension of ${\mathcal{D}}_{\mathrm{\infty }}(a)$, which depends qualitatively on the value of $a$ relative to a second pair of thresholds.