# 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}| 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 HausdorffDimension 2013-05-18 23:14:26 2013-05-18 23:14:26 Mathprof (13753) unlord (1) 16 Mathprof (1) Definition msc 28A80 Dimension3 HausdorffMeasure countable r-cover diameter