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Differentiability and Hölder spectra of a class of self-affine functions. (English) Zbl 1391.28004

The present paper is concerned with the study of a class of continuous functions \(f:[0,1]\rightarrow {\mathbb R}^d\). This class includes some classical examples as Pólya’s space-filling curves, the Riesz-Nagy singular functions and Okamoto’s functions. More precisely, the author assumes that the range of any function in this class is the attractor of an iterated function system \(\{S_1,\ldots ,S_m\}\) consisting of similitudes.
The main purpose of this paper is to completely classify the differentiability of \(f\) in terms of the contraction ratios of the maps \(S_1,\ldots ,S_m\). The main result establish 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.

MSC:

28A78 Hausdorff and packing measures
26A16 Lipschitz (Hölder) classes
26A27 Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives
26A30 Singular functions, Cantor functions, functions with other special properties
28A80 Fractals
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References:

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