Journal of Fractal Geometry


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Volume 3, Issue 1, 2016, pp. 1–31
DOI: 10.4171/JFG/28

Published online: 2016-05-09

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

Pieter Allaart[1]

(1) University of North Texas, Denton, USA

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

Keywords: Continuous nowhere differentiable function, singular function, Cantor function, infinite derivative, ternary expansion, beta-expansion, Komornik–Loreti constant, Thue–Morse sequence, Hausdorff dimension

Allaart Pieter: The infinite derivatives of Okamoto's self-affine functions: an application of $\beta$-expansions. J. Fractal Geom. 3 (2016), 1-31. doi: 10.4171/JFG/28