2017
Том 69
№ 9

All Issues

Hausdorff–Besicovitch dimension of the graph of one continuous nowhere-differentiable function

O. B. Panasenko

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Abstract

We investigate fractal properties of the graph of the function $$y = f(x) = ∑^{∞}_{k−1}\frac{β_k}{2^k} ≡ Δ^2_{β_1β_2…β_k…},$$ where $$\beta_1 = \begin{cases} 0 & \mbox{if } \alpha_1(x) = 0,\\ 1 & \mbox{if } \alpha_1(x) \neq 0,\\ \end{cases}$$ $$\beta_k = \begin{cases} β_{k−1} & \mbox{if } \alpha_k(x) = \alpha_{k-1}(x),\\ 1 - β_{k−1} & \mbox{if } \alpha_k(x) \neq \alpha_{k-1}(x),\\ \end{cases}$$ and $‎α_k(x)$ is the kth ternary digit of $x$: In particular, we prove that this graph is a fractal set with Hausdorff–Besicovitch $α_0(Г_f) = \log_2(1 +2^{\log_32}$ dimension and cell dimension $α_K (Г_f) = 2-\log_32$.

English version (Springer): Ukrainian Mathematical Journal 61 (2009), no. 9, pp 1448-1466.

Citation Example: O. B. Panasenko Hausdorff–Besicovitch dimension of the graph of one continuous nowhere-differentiable function // Ukr. Mat. Zh. - 2009. - 61, № 9. - pp. 1225-1239.

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