Differentiability and H\"older spectra of a class of self-affine functions

Abstract

This paper studies a large class of continuous functions $f:[0,1]\to\mathbb{R}^d$ whose range is the attractor of an iterated function system $\{S_1,\dots,S_{m}\}$ consisting of similitudes. This class includes such classical examples as P\'olya's space-filling curves, the Riesz-Nagy singular functions and Okamoto's functions. The differentiability of $f$ is completely classified in terms of the contraction ratios of the maps $S_1,\dots,S_{m}$. Generalizing results of Lax (1973) and Okamoto (2006), it is shown that either (i) $f$ is nowhere differentiable; (ii) $f$ is non-differentiable almost everywhere but with uncountably many exceptions; or (iii) $f$ is differentiable almost everywhere but with uncountably many exceptions. The Hausdorff dimension of the exceptional sets in cases (ii) and (iii) above is calculated, and more generally, the complete multifractal spectrum of $f$ is determined.