Hausdorff dimension

Let Θ be a boundedPlanetmathPlanetmathPlanetmath subset of n let NΘ(ϵ) be the minimum number of balls of radius ϵ required to cover Θ. Then define the Hausdorff dimensionMathworldPlanetmath dH of Θ to be


Hausdorff dimension is easy to calculate for simple objects like the Sierpinski gasket or a Koch curveMathworldPlanetmath. Each of these may be covered with a collectionMathworldPlanetmath of scaled-down copies of itself. In fact, in the case of the Sierpinski gasket, one can take the individual trianglesMathworldPlanetmath in each approximation as balls in the covering. At stage n, there are 3n triangles of radius 12n, and so the Hausdorff dimension of the Sierpinski triangle is at most -nlog3nlog1/2=log3log2, and it can be shown that it is equal to log3log2.

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 supx,yCd(x,y).

Define a r-cover of X to be a collection of subsets Ci of X indexed by some countable set I, such that |Ci|<r and X=iICi.

We also define the function


where the infimumMathworldPlanetmath is over all countableMathworldPlanetmath r-covers of X. The Hausdorff dimension of X may then be defined as


When X is a subset of 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