Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8904905 | Advances in Mathematics | 2018 | 39 Pages |
Abstract
This paper studies a large class of continuous functions f:[0,1]âRd whose range is the attractor of an iterated function system {S1,â¦,Sm} consisting of similitudes. This class includes such classical examples as Pólya'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 S1,â¦,Sm. 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.
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Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Pieter C. Allaart,