Hausdorff dimension

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

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

Define a $r$ -cover of $X$ to be a collection of subsets $C_{i}$ of $X$ indexed by some countable set $I$ , such that $|C_{i}|<r$ and $X=\cup_{i\in I}C_{i}$ .

We also define the 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 $\mathbb{R}^{n}$ with any norm-induced metric, then this definition reduces to that given above.

Title	Hausdorff dimension
Canonical name	HausdorffDimension
Date of creation	2013-05-18 23:14:26
Last modified on	2013-05-18 23:14:26
Owner	Mathprof (13753)
Last modified by	unlord (1)
Numerical id	16
Author	Mathprof (1)
Entry type	Definition
Classification	msc 28A80
Related topic	Dimension3
Related topic	HausdorffMeasure
Defines	countable r-cover
Defines	diameter