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Hausdorff dimension

by Melchior Grutzmann (and Brandon Piercy and Hendra I. Nurdin)

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In mathematics, the Hausdorff dimension is a way of defining a possibly fractional exponent for all figures in a metric space such that the dimension describes partially the amount to that the set fills the space around it. For example, a plane would have a Hausdorff dimension of 2, because it fills a 2-parameter subset. However, it would not make sense to give the Sierpiński triangle fractal a dimension of 2, since it does not fully occupy the 2-dimensional realm. The Hausdorff dimension describes this mathematically by measuring the size of the set. For self-similar sets there is a relationship to the number of self-similar subsets and their scale.

Informal definition

Intuitively, the dimension of a set is the number of independent parameters one has to pick in order to fix a point. This is made rigorously with the notion of d-dimensional (topological) manifold which are particularly regular sets. The problem with the classical notion is that you can easily break up the digits of a real number to map it bijectively to two (or d) real numbers. The example of space filling curves shows that it is even possible to do this in a continuous (but non-bijective) way.

The notion of Hausdorff dimension refines this notion of dimension such that the dimension can be any non-negative number.

Benoît Mandelbrot discovered that many objects in nature are not strictly classical smooth bodies, but best approximated as fractal sets, i.e. subsets of R^N whose Hausdorff dimension is strictly greater than its topological dimension.

Hausdorff measure and dimension

Let d be a non-negative real number and S ⊂ X a subset of a metric space (X,ρ). The d-dimesional Hausdorff measure of scale δ>0 is

$H_{\delta }^{d*}(S):=\inf\{\sum _{i=1}^{\infty }r_{i}^{d}:S\subset \bigcup _{i=1}^{\infty }B_{r_{i}}(x_{i}),r_{i}\leq \delta \}$

where B_{ri(xi) is the open ball around xi ∈ X of radius ri. The d-dimensional Hausdorff measure is now the limit}

$H^{d*}(S):=\lim _{\delta \to 0+}H_{\delta }^{d*}(S)$.

As in the Carathéodory construction a set S ⊂ X is called d-measurable iff

$H^{d*}(T)=H^{d*}(S\cap T)+H^{d*}(T\cap X\setminus S)$ for all T ⊂ X.

A set S ⊂ X is called Hausdorff measurable if it is H^d-measurable for all d≥0.

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