Fractal dimension

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A, possibly non-integer valued, dimension concept. Let $M$ be a metric space and $X \subset M$ a bounded subset. For each $\epsilon$ let $N_\epsilon(X)$ be the minimal number of balls of radius $\epsilon$ necessary to cover $X$. Then $$ d_F(X) = \limsup_{\epsilon \rightarrow 0} \frac{\log N_\epsilon(X)}{\log \epsilon^{-1}} $$ is the fractal dimension of $X$. It has also been called the capacity, the Mandelbrot dimension or the Shnirel'man–Kolmogorov dimension of $X$.

One has $$ d_F(X) = \inf\left\lbrace{ d>0 : \limsup_{\epsilon \rightarrow 0} \epsilon^d N_\epsilon(X)=0 }\right\rbrace $$

If $d_H(X)$ denotes the Hausdorff dimension of $X$, then $d_H(X) \le d_F(X)$.


[a1] B.B. Mandelbrot, "Form, chance and dimension" , Freeman (1977)
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