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

TL;DR

This paper classifies the differentiability of a broad class of self-affine functions, including classical examples, and computes the Hausdorff dimension of their exceptional sets, revealing complex multifractal structures.

Contribution

It provides a complete classification of differentiability for self-affine functions based on contraction ratios and extends previous results by calculating the multifractal spectra.

Findings

Functions are either nowhere differentiable, differentiable almost everywhere with exceptions, or non-differentiable almost everywhere.

Hausdorff dimensions of exceptional sets are explicitly calculated.

Multifractal spectra of these functions are fully determined.

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.