Hausdorff dimension
Let be a bounded subset of let be the minimum number of balls of radius required to cover . Then define the Hausdorff dimension of to be
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 , there are triangles of radius , and so the Hausdorff dimension of the Sierpinski triangle is at most , and it can be shown that it is equal to .
From some notes from Koro
This definition can be extended to a general metric space with distance function .
Define the diameter of a bounded subset of to be .
Define a -cover of to be a collection of subsets of indexed by some countable set , such that and .
We also define the function
where the infimum is over all countable -covers of . The Hausdorff dimension of may then be defined as
When is a subset of 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 |