Hausdorff dimension

In mathematics, the dimension of a set, for example a subset of Euclidean space is naively, the number of independent parameters needed to describe a point in the set. The mathematical concept which which most closely models this naive idea is that of topological dimension of a set. For example a point in the plane is described by two independent coordinates, so in this sense, the plane is two-dimensional. One would expect that dimension is always an integer. Accordingly, certain highly irregular sets such as fractals do not have a dimension. An example of such a set is Cantor's middle thirds set.

The Hausdorff dimension is a number that can be attached to many more sets X, by close scrutiny of the number of small balls required to cover X completely, and the way that this number grows as the allowed radius of the ball is squeezed down towards zero. It is one choice of a dimension that may be assigned to a fractal set, though its uses are not restricted to the theory of fractals.

It should be noted that there are various related notions of possibly fractional dimension. For example box-counting dimension, generalises the idea of counting the squares of graph paper in which a point of X can be found, as the size of the squares is made smaller and smaller. In many cases these notions coincide, but the relation between them is highly technical.

Formal definition

The Hausdorff dimension (also: Hausdorff-Besicovitch dimension, capacity dimension and fractal dimension) was introduced by Felix Hausdorff. It gives an accurate way to 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.

Suppose (X,d) is a metric space. We define a family of metric outer measures 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

$H_{\delta }^{s}(E)=\inf {\Bigg \{}\sum _{i=1}^{\infty }\operatorname {diam} (A_{i})^{s}{\Bigg \}}$

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

$H^{s}(E)=\lim _{\delta \rightarrow 0}H_{\delta }^{s}(E).$

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.

Results

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,

$C^{-1}d_{1}(x,y)\leq d(x,y)\leq Cd_{1}(x,y)$

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

$C^{-s}H_{1}^{s}(E)\leq H^{s}(E)\leq C^{s}H_{1}^{s}(E)$

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,

$\lambda _{m}(E)=2^{-m}{\frac {\pi ^{m/2}}{\Gamma ({\frac {m}{2}}+1)}}H^{m}(E).$

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.

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

$\operatorname {dim} _{\mathrm {Haus} }(X)\geq \operatorname {dim} _{\mathrm {top} }(X)$

Moreover

$\inf _{Y}\operatorname {dim} _{\mathrm {Haus} }(Y)=\operatorname {dim} _{\mathrm {top} }(X)$

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 Edward Szpilrajn (1907-1976). 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.