# Hausdorff dimension

Let $\mathrm{\Theta}$ be a bounded^{} subset of ${\mathbb{R}}^{n}$
let ${N}_{\mathrm{\Theta}}(\u03f5)$ be the minimum number of balls of radius $\u03f5$ required to cover $\mathrm{\Theta}$. Then define the *Hausdorff dimension ^{}*
${d}_{H}$ of $\mathrm{\Theta}$ to be

$${d}_{H}(\mathrm{\Theta}):=-\underset{\u03f5\to 0}{lim}\frac{\mathrm{log}{N}_{\mathrm{\Theta}}(\u03f5)}{\mathrm{log}\u03f5}.$$ |

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 $-\frac{n\mathrm{log}3}{n\mathrm{log}1/2}=\frac{\mathrm{log}3}{\mathrm{log}2}$, and it can be shown that it is equal to $\frac{\mathrm{log}3}{\mathrm{log}2}$.

## 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 $$ and $X={\cup}_{i\in I}{C}_{i}$.

We also define the function

$${H}_{r}^{D}(X)=inf\sum _{i\in I}{|{C}_{i}|}^{D}$$ |

where the infimum^{} is over all countable^{} $r$-covers of $X$.
The *Hausdorff dimension* of $X$ may then be defined as

$${d}_{H}(X)=inf\{D\mid \underset{r\to 0}{lim}{H}_{r}^{D}(X)=0\}.$$ |

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 |