Hausdorff dimension

Hausdorff dimension is a concept in mathematics introduced in 1918 by mathematician Felix Hausdorff, and it serves as a measure of the local size of a set of numbers (i.e., a "space"), taking into account the distance between each of its members (i.e., the "points" in the "space"). Applying its mathematical formalisms provides that the Hausdorff dimension of a single point is zero, of a line is 1, and of a square is 2, of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is a counting number (integer) agreeing with a dimension corresponding to its topology. However, formalisms have also been developed that allow calculation of the dimension of other less simple objects, where, based solely on its properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.

The Hausdorff dimension is, more specifically, a further dimensional number associated with a given set of numbers, where the distances between all members of that set are defined, and where the dimension is drawn from the real numbers, ℝ, to which two elements have been added, +∞ and −∞ (read as positive and negative infinity, respectively). The set that provides the Hausdorff dimension is called the extended real numbers, R, and a set of numbers where distances between all members are defined is termed a metric space, so that foregoing can be succinctly stated, saying the Hausdorff dimension is a non-negative extended real number (R ≥ 0) associated with any metric space.

In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an n-dimensional inner product space equals n. This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that irregular sets can have noninteger Hausdorff dimensions. For instance, the Koch curve summarized earlier is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new equilateral triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4. That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original. Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=S^D. This equation is easily solved for D, yielding the ratio of logarithms (or natural logarithms) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects.

The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension.

Intuition

The intuitive concept of dimension of a geometric object X is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the cardinality of the real plane is equal to the cardinality of the real line (this can be seen by an argument involving interweaving the digits of two numbers to yield a single number encoding the same information.) The example of a space-filling curve shows that one can even take one real number into two both surjectively (so all pairs of numbers are covered) and continuously, so that a one-dimensional object completely fills up a higher-dimensional object.

Every space filling curve hits some points multiple times, and does not have a continuous inverse. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. The topological dimension, also called Lebesgue covering dimension, explains why. This dimension is n if, in every covering of X by small open balls, there is at least one point where n + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension n = 1.

But topological dimension is a very crude measure of the local size of a space (size near a point). A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region. A fractal has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space.

The Hausdorff dimension measures the local size of a space taking into account the distance between points, the metric. Consider the number N(r) of balls of radius at most r required to cover X completely. When r is very small, N(r) grows polynomially with 1/r. For a sufficiently well-behaved X, the Hausdorff dimension is the unique number d such that N(r) grows as 1/r^d as r approaches zero. More precisely, this defines the box-counting dimension, which equals the Hausdorff dimension when the value d is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant.

For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the Hausdorff dimension is an integer agreeing with the topological dimension. But Benoît Mandelbrot observed that fractals, sets with noninteger Hausdorff dimensions, are found everywhere in nature. He observed that the proper idealization of most rough shapes you see around you is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes:

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.

For fractals that occur in nature, the Hausdorff and box-counting dimension coincide. The packing dimension is yet another similar notion which gives the same value for many shapes, but there are well documented exceptions where all these dimensions differ.

Hausdorff content

Let X be a metric space. If S ⊂ X and d ∈ [0, ∞), the d-dimensional Hausdorff content of S is defined by

C H d ( S ) := inf { ∑ i r i d : there is a cover of S by balls with radii r i > 0 } .

In other words, C H d ( S ) is the infimum of the set of numbers δ ≥ 0 such that there is some (indexed) collection of balls { B ( x i , r i ) : i ∈ I } covering S with r_i > 0 for each i ∈ I that satisfies ∑ i ∈ I r i d < δ . (Here, we use the standard convention that inf Ø = ∞.)

The Hausdorff dimension of X is defined by

dim H ⁡ ( X ) := inf { d ≥ 0 : C H d ( X ) = 0 } .

Equivalently, dim_H(X) may be defined as the infimum of the set of d ∈ [0, ∞) such that the d-dimensional Hausdorff measure of X is zero. This is the same as the supremum of the set of d ∈ [0, ∞) such that the d-dimensional Hausdorff measure of X is infinite (except that when this latter set of numbers d is empty the Hausdorff dimension is zero).

Examples

Countable sets have Hausdorff dimension 0.

The Euclidean space ℝⁿ has Hausdorff dimension n, and the circle S¹ has Hausdorff dimension 1.

Fractals often are spaces 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; hence, it can be shown that its Hausdorff dimension is ln(2)/ln(3) ≈ 0.63. 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) ≈ 1.58.

Space-filling curves like the Peano and the Sierpiński curve have the same Hausdorff dimension as the space they fill.

The trajectory of Brownian motion in dimension 2 and above has Hausdorff dimension 2 almost surely.

An early paper by Benoit Mandelbrot entitled How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension and subsequent work by other authors have claimed that the Hausdorff dimension of many coastlines can be estimated. Their results have varied from 1.02 for the coastline of South Africa to 1.25 for the west coast of Great Britain. However, 'fractal dimensions' of coastlines and many other natural phenomena are largely heuristic and cannot be regarded rigorously as a Hausdorff dimension. They are based on scaling properties of coastlines at a large range of scales; however, they do not include all arbitrarily small scales, where measurements would depend on atomic and sub-atomic structures, and are not well defined.

Hausdorff dimension and inductive dimension

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

Theorem. Suppose X is non-empty. Then

dim H a u s ⁡ ( X ) ≥ dim ind ⁡ ( X ) .

Moreover,

inf Y dim Haus ⁡ ( Y ) = dim ind ⁡ ( 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), e.g., see Hurewicz and Wallman, Chapter VII.

Hausdorff dimension and Minkowski dimension

The Minkowski dimension is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of rational points in [0, 1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension.

Hausdorff dimensions and Frostman measures

If there is a measure μ defined on Borel subsets of a metric space X such that μ(X) > 0 and μ(B(x, r)) ≤ r^s holds for some constant s > 0 and for every ball B(x, r) in X, then dim_Haus(X) ≥ s. A partial converse is provided by Frostman's lemma.

Behaviour under unions and products

If X = ⋃ i ∈ I X i is a finite or countable union, then

dim Haus ⁡ ( X ) = sup i ∈ I dim Haus ⁡ ( X i ) .

This can be verified directly from the definition.

If X and Y are non-empty metric spaces, then the Hausdorff dimension of their product satisfies

dim Haus ⁡ ( X × Y ) ≥ dim Haus ⁡ ( X ) + dim Haus ⁡ ( Y ) .

This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1. In the opposite direction, it is known that when X and Y are Borel subsets of Rⁿ, the Hausdorff dimension of X × Y is bounded from above by the Hausdorff dimension of X plus the upper packing dimension of Y. These facts are discussed in Mattila (1995).

The Hausdorff Dimension Theorem

Theorem. For any given r > 0 , there are uncountable fractals with Hausdorff dimension r in n-dimensional Euclidean space R n , ( n ≥ ⌈ r ⌉ ) .

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.

Theorem. Suppose

ψ i : R n → R n , i = 1 , … , m

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

A = ⋃ i = 1 m ψ i ( A ) .

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

The open set condition

To determine the dimension of the self-similar set A (in certain cases), we need a technical condition called the open set condition (OSC) on the sequence of contractions ψ_i.

There is a relatively compact open set V such that

⋃ i = 1 m ψ i ( V ) ⊆ V ,

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

The open set condition is a separation condition that ensures the images ψ_i(V) do not overlap "too much".

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

∑ i = 1 m r i s = 1.

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

( 1 2 ) s + ( 1 2 ) s + ( 1 2 ) s = 3 ( 1 2 ) s = 1.

Taking natural logarithms of both sides of the above equation, we can solve for s, that is: s = ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC. In general a set E which is a fixed point of a mapping

A ↦ ψ ( A ) = ⋃ i = 1 m ψ i ( A )

is self-similar if and only if the intersections

H s ( ψ i ( E ) ∩ ψ j ( E ) ) = 0 ,

where s is the Hausdorff dimension of E and H^s denotes Hausdorff measure. 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.

References

Hausdorff dimension Wikipedia

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