JournalsjfgVol. 3 , No. 1pp. 1–31

The infinite derivatives of Okamoto's self-affine functions: an application of β\beta-expansions

  • Pieter Allaart

    University of North Texas, Denton, USA
The infinite derivatives of Okamoto's self-affine functions: an application of $\beta$-expansions cover
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Abstract

Okamoto's one-parameter family of self-affine functions Fa:[0,1][0,1]F_a: [0,1] \to [0,1], where 0<a<10 < a < 1, includes the continuous nowhere differentiable functions of Perkins (a=5/6a=5/6) and Bourbaki/Katsuura (a=2/3a=2/3), as well as the Cantor function (a=1/2a=1/2). The main purpose of this article is to characterize the set of points at which FaF_a has an infinite derivative. We compute the Hausdorff dimension of this set for the case a1/2a \leq 1/2, and estimate it for a>1/2a > 1/2. For all aa, we determine the Hausdorff dimension of the sets of points where: (i) Fa=0F_a'=0; and (ii) FaF_a has neither a finite nor an infinite derivative. The upper and lower densities of the digit 11 in the ternary expansion of x[0,1]x \in [0,1] play an important role in the analysis, as does the theory of β\beta-expansions of real numbers.

Cite this article

Pieter Allaart, The infinite derivatives of Okamoto's self-affine functions: an application of β\beta-expansions. J. Fractal Geom. 3 (2016), no. 1 pp. 1–31

DOI 10.4171/JFG/28