Hausdorff dimension

In mathematics, the Hausdorff dimension is an extended non-negative real number (that is a number in the closed infinite interval [0, ∞]) associated to any metric space . It was introduced in 1918 by the mathematician Felix Hausdorff. Many of the technical developments used to compute the Hausdorff dimension for highly irregular sets were obtained by Abram Samoilovitch Besicovitch. For this reason, Hausdorff dimension is sometimes referred to as Hausdorff-Besicovitch dimension. It is also less frequently called the capacity dimension or fractal dimension.

Intuitively, the dimension of a set (for example, a subset of Euclidean space) is the number of independent parameters needed to describe a point in the set. One mathematical concept which closely models this naïve idea is that of topological dimension of a set. For example a point in the plane is described by two independent parameters (the cartesian coordinates of the point), so in this sense, the plane is two-dimensional. As one would expect, topological dimension is always a natural number.

However, topological dimension behaves in quite unexpected ways on certain highly irregular sets such as fractals. For example the Cantor set has topological dimension zero, but in some sense it behaves as a higher dimensional space. Hausdorff dimension gives another way to define dimension, which takes the metric into account.

To define the Hausdorff dimension for X, we first consider the number N(r) of balls of radius at most r required to cover X completely. Clearly, as r gets smaller N(r) gets larger. Very roughly, if N(r) grows in the same way as 1/r^d as r is squeezed down towards zero, then we say X has dimension d. In fact the rigorous definition of Hausdorff dimension is somewhat roundabout, since it first defines an entire family of covering measures for X. It turns out that Hausdorff dimension refines the concept of topological dimension and also relates it to other properties of the space such as area or volume.

It should be noted that there are various closely 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.

Contents

1 Formal definition 2 Results 2.1 Examples 3 Hausdorff dimension and topological dimension 4 Self-similar sets 5 Historical references 6 References

Formal definition

The Hausdorff dimension gives an accurate way to measure the dimension of an arbitrary metric space; this includes complicated sets such as fractals.

Suppose (X,d) is a metric space. As mentioned in the introduction, we are interested in counting the number of balls of some radius necessary to cover a given set. It is possible to try to do this directly for many sets (leading to so-called box counting dimension), but Hausdorff's insight was to approach the problem indirectly using the theory of measure developed earlier in the century by Henri Lebesgue and Constantin Caratheodory. In order to deal with the technical details of this approach, Hausdorff defined an entire family of measures on subsets of X, one for each possible dimension s ∈ [0,∞). For example, if X= R³, this construction assigns an s dimensional measure H^s to all subsets of R³ including the unit segment along the x-axis [0,1] × {0} × {0}, the unit square on the x — y plane [0,1] × [0,1] × {0} and the unit cube [0,1] × [0,1] × [0,1]. For s=2, one would expect

• <math> H^2([0,1] \times \{0\} \times \{0\}) = 0 <math>

• <math> 0 < H^2([0,1] \times [0,1] \times \{0\}) < +\infty <math>

• <math> H^2([0,1] \times [0,1] \times [0,1]) = +\infty <math>

The above example suggests that we can define a set A to have Hausdorff dimension s if its s-dimensional Hausdorff measure is positive and finite; in fact we have to modify this slightly. The Hausdorff dimension of A is the cutoff value s where below s the s-dimensional Hausdorff measure is ∞ and above s it is 0. It is possible for the s dimensional Hausdorff measure of an s dimensional set to be 0 or ∞. For instance R has dimension 1 and its 1-dimensional Hausdorff measure is infinite.

To carry this construction of this measure, we use a theory of measure which is appropriate for metric spaces. 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

<math> H^s_\delta(E) = \inf\Bigg\{\sum_{i=1}^\infty \operatorname{diam}(A_i)^s\Bigg\}<math>

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

<math> H^s(E) = \lim_{\delta \rightarrow 0} H^s_\delta(E). <math>

We can succinctly (though not in a very useful way) describe the value H^s(E) 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

The Hausdorff outer measure H^s is defined for all subsets of X. However, we can in general assert additivity properties, that is

<math> H^s(A \cup B) = H^s(A) + H^s(B) <math>

for disjoint A, B, only when A and B are both Borel sets. From the perspective of assigning measure and dimension to sets with unusual metric properties such as fractals, however, this is not a restriction.

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.

Clearly, if (X, d) and (Y, e) are isomorphic metric spaces, then the corresponding Hausdorff measure spaces are also isomorphic. It is more useful to note however that Hausdorff measure even behaves well under certain bounded modifications of the underlying metric. 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,

<math> C^{-1} d_1(x,y) \leq d(x,y) \leq C d_1(x,y) <math>

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

<math> C^{-s} H^s_1(E) \leq H^s(E) \leq C^s H^s_1(E)<math>

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,

<math> \lambda_m(E) = 2^{-m} \frac{\pi^{m/2}}{\Gamma(\frac{m}{2}+1)} H^m(E). <math>

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 zero-dimensional 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

<math> \operatorname{dim}_{\mathrm{Haus}}(X) \geq \operatorname{dim}_{\mathrm{top}}(X) <math>

Moreover

<math> \inf_Y \operatorname{dim}_{\mathrm{Haus}}(Y) =\operatorname{dim}_{\mathrm{top}}(X) <math>

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.

Self-similar sets

Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set E is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ(E) = E, although the exact definition is given below. The following is Theorem 8.3 of the Falconer reference below:

Theorem. Suppose

<math> \psi_i: \mathbb{R}^n \rightarrow \mathbb{R}^n, \quad i=1, \ldots , m <math>

are contractive mappings on Rⁿ with contraction constant r_j < 1. Then there is a unique non-empty compact set A such that

<math> A = \bigcup_{i=1}^m \psi_i (A). <math>

This follows from Banach's contractive mapping fixed point theorem applied to the complete metric space of non-empty compact subsets of Rⁿ with the Hausdorff distance.

To determine the dimension of the self-similar set A (in certain cases), we need a technical condition called the open set condition on the sequence of contractions ψ_i which is stated as follows: There is a relatively compact open set V such that

<math> \bigcup_{i=1}^m\psi_i (V) \subseteq V <math>

where the sets in union on the left are pairwise disjoint.

Theorem. Suppose the open set condition holds and each ψ_i is a similitude, that is a composition of an isometry and a dilation around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is s where s is the unique solution of

<math> \sum_{i=1}^m r_i^s = 1. <math>

Note that the contraction coefficient of a similitude is the magnitude of the dilation.

We can use this theorem to compute the Hausdorff dimension of the Sierpinski triangle (or sometimes called Sierpinski gasket). Consider three non-collinear points a₁, a₂, a₃ in the plane R² and let ψ_i be the dilation of ratio 1/2 around a_i. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket and the dimension s is the unique solution of

<math> \left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s+\left(\frac{1}{2}\right)^s = 3 \left(\frac{1}{2}\right)^s =1. <math>

Taking natural logarithms of both sides of the above equation, we can solve for s, that is:

<math> s = \frac{\ln 3}{\ln 2}. <math>

The Sierpinski gasket is self-similar. In general a set E which is a fixed point of a mapping

<math> A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) <math>

is self-similar iff the intersections

<math> H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0 <math>

where s is the Hausdorff dimension of E. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally:

Theorem. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar.

Historical references

• A. S. Besicovitch, On Linear Sets of Points of Fractional Dimensions, Mathematische Annalen 101 (1929).
• A. S. Besicovitch and H. D. Ursell, Sets of Fractional Dimensions, Journal of the London Mathematical Society, v12 (1937). Several selections from this volume are reprinted in Classics on Fractals,ed. Gerald A. Edgar, Addison-Wesley (1993) ISBN 0-201-58701-7 See chapters 9,10,11.
• F. Hausdorff, Dimension und äusseres Mass, Mathematische Annalen 79 (1919).

References

• M. Maurice Dodson and Simon Kristensen, Hausdorff Dimension and Diophantine Approximation (http://arxiv.org/abs/math/0305399) (June 12, 2003).
• L. Evans and R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992
• 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.
• 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.de:Hausdorff-Dimension

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