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Difference between revisions of "Fractal dimension"

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A, possibly non-integer valued, dimension concept. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041180/f0411801.png" /> be a metric space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041180/f0411802.png" /> a bounded subset. For each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041180/f0411803.png" /> let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041180/f0411804.png" /> be the minimal number of balls of radius <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041180/f0411805.png" /> necessary to cover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041180/f0411806.png" />. Then
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{{TEX|done}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041180/f0411807.png" /></td> </tr></table>
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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
 
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$$
is the fractal dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041180/f0411808.png" />. It has also been called the capacity, the Mandelbrot dimension or the Shnirel'man–Kolmogorov dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041180/f0411809.png" />.
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d_F(X) = \limsup_{\epsilon \rightarrow 0} \frac{\log N_\epsilon(X)}{\log \epsilon^{-1}}
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$$
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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
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$$
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d_F(X) = \inf\left\lbrace{ d>0 : \limsup_{\epsilon \rightarrow 0} \epsilon^d N_\epsilon(X)=0    }\right\rbrace
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041180/f04118010.png" /></td> </tr></table>
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If $d_H(X)$ denotes the [[Hausdorff dimension]] of $X$, then $d_H(X) \le d_F(X)$.
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041180/f04118011.png" /> denotes the [[Hausdorff dimension|Hausdorff dimension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041180/f04118012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041180/f04118013.png" />.
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B.B. Mandelbrot,  "Form, chance and dimension" , Freeman  (1977)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  B.B. Mandelbrot,  "Form, chance and dimension" , Freeman  (1977)</TD></TR>
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</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