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

  • B. Panasenko O.

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 $α_K (Г_f) = 2-\log_32$.
Published
25.09.2009
How to Cite
O.B. P. “Hausdorff–Besicovitch Dimension of the Graph of One Continuous Nowhere-Differentiable Function”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 61, no. 9, Sept. 2009, pp. 1225-39, https://umj.imath.kiev.ua/index.php/umj/article/view/3094.
Section
Research articles