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Hausdorff dimension (Definition)

Let $ \Theta$ be a bounded subset of $ \mathbb{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

$\displaystyle d_H (\Theta):= - \lim_{\epsilon \rightarrow 0} \frac{\log N_{\Theta}(\epsilon)}{\log \epsilon}.$

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}$.

From some notes from Koro

This definition can be extended to a general metric space $ X$ with distance function $ d$.

Define the diameter $ \vert C\vert$ of a bounded subset $ C$ of $ X$ to be $ \sup_{x,y\in C} d(x,y)$.

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 $ \vert C_i\vert < r$ and $ X = \cup_{i\in I} C_i$.

We also define the function

$\displaystyle H^D_r (X) = \inf \sum_{i\in I} \vert C_i\vert^D$
where the infimum is over all countable $ r$-covers of $ X$. The Hausdorff dimension of $ X$ may then be defined as
$\displaystyle d_H (X)=\inf \{ D\mid \lim_{r\rightarrow 0} H^D_r(X)=0 \}.$
When $ X$ is a subset of $ \mathbb{R}^n$ with any restricted norm-induced metric, then this definition reduces to that given above.



"Hausdorff dimension" is owned by Mathprof. [ full author list (4) | owner history (3) ]
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See Also: dimension, Hausdorff measure

Also defines:  countable r-cover, diameter
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Cross-references: metric, infimum, countable, indexed by, function, distance, metric space, covering, approximation, triangles, collection, Koch curve, Sierpinski gasket, objects, simple, calculate, cover, radius, balls, number, subset, bounded
There are 4 references to this entry.

This is version 12 of Hausdorff dimension, born on 2002-05-31, modified 2006-08-02.
Object id is 2981, canonical name is HausdorffDimension.
Accessed 7650 times total.

Classification:
AMS MSC: 28A80 (Measure and integration :: Classical measure theory :: Fractals)
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