Difference between revisions of "Fractal dimension"
From Encyclopedia of Mathematics
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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 | |
| − | + | $$ | |
| − | is the fractal dimension of | + | d_F(X) = \limsup_{\epsilon \rightarrow 0} \frac{\log N_\epsilon(X)}{\log \epsilon^{-1}} |
| + | $$ | ||
| + | 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 | 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 $d_H(X) \le d_F(X)$. | |
| − | |||
| − | If | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> B.B. Mandelbrot, "Form, chance and dimension" , Freeman (1977)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> B.B. Mandelbrot, "Form, chance and dimension" , Freeman (1977)</TD></TR> | ||
| + | </table> | ||
Latest revision as of 21:09, 18 December 2014
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 $d_H(X) \le d_F(X)$.
References
| [a1] | B.B. Mandelbrot, "Form, chance and dimension" , Freeman (1977) |
How to Cite This Entry:
Fractal dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fractal_dimension&oldid=11902
Fractal dimension. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fractal_dimension&oldid=11902