The Hausdorff dimension reference article from the English Wikipedia on 24-Jul-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Hausdorff dimension

The Hausdorff dimension (also: Hausdorff-Besicovitch dimension, capacity dimension and fractal dimension), introduced by Felix Hausdorff, gives a way to accurately measure the dimension of complicated sets such as fractals. The Hausdorff dimension agrees with the ordinary (topological) dimension on "well-behaved sets", but it is applicable to many more sets and is not always a natural number. The Hausdorff dimension should not be confused with the (similar) box-counting dimension.

Suppose (X,d) is a metric space. We define a family of metric outer measuress on X using the Method II construction of outer measures due to Munroe and described in the article outer measure. Let C be the class of all subsets of X; for each positive real number s, let p_s be the function A → diam(A)^s on C. Hausdorff outer measure of dimension s, denoted H^s is the outer measure corresponding to the function p_s on C.

Thus for any subset E of X

where the infimum is taken over sequences {A_i}_i which cover E by sets each with diameter ≤ δ. Then

The value H^s(E) can be described directly as the infimum of all h > 0 such that for all δ > 0, E can be covered by countably many closed sets of diameter ≤ δ and the sum of the s-th powers of these diameters is less than or equal to h.

Theorem. H^s is a metric outer measure. Thus all Borel subsets of X are measurable and H^s is a countably additive measure on the σ-algebra of Borel sets.

Hausdorff measure is a Lipschitz invariant in the following sense: If d and d₁ are metrics on X such that for some 0< C < ∞ and all x, y in X,

then the corresponding Hausdorff measures H^s, H₁^s satisfy

for any Borel set E.

The function s → H^s(E) is non-increasing. In fact, it turns out that for all values of s, except possibly one H^s(E) is either 0 or ∞. We say E has positive finite Hausdorff dimension iff there is a real number 0<d< ∞ such that if s < d then H^s(E) = ∞ and if s > d, then H^s(E) = 0. If H^s(E)=0 for all positive s, then E has Hausdorff dimension 0. Finally, if H^s(E)=∞ for all positive s, then E has Hausdorff dimension ∞

The Hausdorff dimension is a well-defined extended real number for any set E and we always have 0 ≤ d(E) ≤ ∞. It follows from the Lipschitz property of Hausdorff measure that Hausdorff dimension is a Lipschitz invariant. Its relation to topological properties is outlined below.

Note that if m is a positive integer, the m dimensional Hausdorff measure of R^m is a rescaling of usual m-dimensional Borel measure λ_m which is normalized so that the Borel measure of the m-dimensional unit cube [0,1]^m is 1. In fact, for any Borel set E,

Remark. Some authors adopt a slightly different definition of Hausdorff measure than the one chosen here, the difference being that it is normalized in such a way that Hausdorff m-dimensional measure in the case of Euclidean space coincides exactly with Borel measure λ.

See the Federer reference below for additional material on other fractal measures.

Table of contents

1 Examples 2 Hausdorff dimension and topological dimension 3 References

Examples

• The Euclidean space Rⁿ has Hausdorff dimension n.
• The circle S¹ has Hausdorff dimension 1.
• Countable sets have Hausdorff dimension 0.
• Fractals are defined to be sets whose Hausdorff dimension strictly exceeds the topological dimension. For example, the Cantor set (a zerodimensional topological space) is a union of two copies of itself, each copy shrunk by a factor 1/3; this fact can be used to prove that its Hausdorff dimension is ln(2)/ln(3), which is approximately 0.63 (see natural logarithm). The Sierpinski triangle is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln(3)/ln(2), which is approximately 1.58.
• The trajectory of Brownian motion in dimension 2 and above has Hausdorff dimension 2 almost surely.

Hausdorff dimension and topological dimension

Let X be an arbitrary separable metric space. There is a notion of topological dimension for X which is defined recursively. It is always an integer (or +infinity) and is denoted dim_top(X).

Theorem. Suppose X is non-empty. Then

Moreover

where Y ranges over metric spaces homeomorphic to X. In other words, X and Y have the same underlying set of points and the metric d_Y of Y is topologically equivalent to d_X.

These results were originally established by E. Szpilrajn. The treatment in Chapter VIII of the Hurewicz and Wallman reference is particularly recommended.

References

• K. J. Falconer, The Geometry of Fractal Sets, Cambridge University Press, 1985

• H. Federer, Geometric Measure Theory, Springer-Verlag, 1969.

• W. Hurewicz and H. Wallman, Dimension Theory, Princeton University Press, 1948.

• L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992

• Frank Morgan, Geometric Measure Theory, Academic Press, 1988. Good introductory presentation with lots of illustrations.

• E. Szpilrajn, La dimension et la mesure, Fundamenta Mathematica 28, 1937, pp 81-89.