Hausdorff–Besicovitch dimension of the graph of one continuous nowhere-differentiable function
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 $$\beta_1 = \begin{cases} 0 & \mbox{if } \alpha_1(x) = 0,\\ 1 & \mbox{if } \alpha_1(x) \neq 0,\\ \end{cases}$$ $$\beta_k = \begin{cases} β_{k−1} &...
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| Datum: | 2009 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2009
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/3094 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | We investigate fractal properties of the graph of the function
$$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 $α_K (Г_f) = 2-\log_32$. |
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