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'Hausdorff dimension'
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| Title of object: |
Hausdorff dimension |
| Canonical Name: |
HausdorffDimension |
| Type: |
Definition |
| Created on: |
2002-05-31 23:59:28 |
| Modified on: |
2004-07-28 13:03:23 |
| Classification: |
msc:28A80 |
Revision comment (for changes between this and next version):
| Changes for correction #8472 ('clarify'). |
Preamble:
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Content:
Let $\Theta$ be a bounded subset of $\R^n$
let $N_{\Theta}(\epsilon)$ be the minimum number of balls of radius $\epsilon$ required to cover $\Theta$. Then define the Hausdorff dimension
$d_H$ of $\Theta$ to be
\[ d_H (\Theta):= - \lim_{\epsilon \rightarrow 0} \frac{\log N_{\Theta}(\epsilon)}{\log \epsilon}.\]
Hausdorff dimension is easy to calculate for simple objects like the Sierpinski gasket or a Koch curve. Each of these may be covered with a collection of scaled-down copies of itself. In fact, in the case of the Sierpinski gasket, one can take the individual triangles in each approximation as balls in the covering. At stage $n$, there are $3^n$ triangles of radius $\frac{1}{2^n}$, and so the Hausdorff dimension of the Sierpinski triangle is at most $- \frac{n \log 3}{n \log 1/2} = \frac{\log 3}{\log 2}$, and it can be shown that it is equal to $\frac{\log 3}{\log 2}$.
\paragraph{From some notes from Koro} This definition can be extended to a general metric space $X$ with distance function $d$.
Define the \emph{diameter} $|C|$ of a bounded subset $C$ of $X$ to be $\sup_{x,y\in C} d(x,y)$, and define a countable $r$-cover
of $X$ to be a collection of subsets $C_i$ of $X$ indexed by some countable set $I$, such that $X = \cup_{i\in I} C_i$.
We also define the handy function $$H^D_r (X) = \inf \sum_{i\in I} |C_i|^D$$ where the infimum is over all countable $r$-covers of $X$.
The Hausdorff dimension of $X$ may then be defined as $$d_H (X)=\inf \{ D\mid \lim_{r\rightarrow 0} H^D_r(X)=0 \}.$$
When $X$ is a subset of $\R^n$ with any restricted norm-induced metric, then this definition reduces to that given above.
%Alternatively, with $N_{\Theta}(\epsilon)$
%as above, the Hausdorff dimension may
%be defined as the unique real number
%$d$ (if it exists) such that the limit
%\[ \lim_{\epsilon \rightarrow 0} N(\epsilon) \epsilon^d \]
%is neither zero nor infinity. Let $x$ be a
%point in $\Theta$, and let $B(x,r)$ be a
%ball of radius $r$ cetered at $x$. Then
%the limit
%$d_H (\Theta; x) \lim_{r\rightarrow 0} d_H (\Theta \cap B(x,r))$
%(when it exists) can be thought of as the
%dimension of $\Theta$ at$x$. Then in this
%sense, the dimension of $\Theta$ is a kind
%of essential supremum over $\Theta$ of
%$d_H (\Theta;x)............. |
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