Hausdorff dimension

The Hausdorff dimension was introduced by Felix Hausdorff and offers the possibility of assigning a dimension to any metric room . For simple geometric objects such as lines , polygons , cuboids and the like, their value agrees with that of the usual dimension concept. In general, however, its numerical value is not necessarily a natural number , but can also be a rational or an irrational number , such as when used as a fractal dimension .

Simplified definition

The following representation is a simplified definition of the Hausdorff dimension for a set of points of finite extent in a three-dimensional space . To do this, consider the number of spheres with the radius that is at least required to cover the set of points. This minimum number is a function of the radius . The smaller the radius, the larger it is . From the potency of , with the limit increases to zero, the Hausdorff dimension is calculated according namely ${\displaystyle N}$ ${\displaystyle R}$ ${\displaystyle N(R)}$ ${\displaystyle R}$ ${\displaystyle N}$ ${\displaystyle R}$ ${\displaystyle N(R)}$ ${\displaystyle R}$ ${\displaystyle D}$

${\displaystyle N(R)\sim {\frac {1}{R^{D}}}}$

and thus

${\displaystyle D=-\lim _{R\rightarrow 0}{\frac {\log N}{\log R}}}$.

Instead of balls, cubes or comparable objects can just as easily be used. For sets of points in the plane, circles can also be used for covering. For point sets in more than three dimensions, correspondingly higher-dimensional balls must be used.

For an ordinary finite curve , the number of spheres required increases in inverse proportion to the spherical radius. A curve therefore has the Hausdorff dimension . For an ordinary finite surface such as a rectangle, however , the number of spheres required increases proportionally . It is therefore true . ${\displaystyle D=1}$ ${\displaystyle 1/R^{2}}$ ${\displaystyle D=2}$

For the special case of a geometric object, which consists of disjoint sub-objects, which represent reduced- scale copies of the entire object, the Hausdorff dimension results . If the sub-objects have different sizes, it is through ${\displaystyle n}$ ${\displaystyle 1:m}$ ${\displaystyle D={\tfrac {\log n}{\log m}}}$ ${\displaystyle n}$ ${\displaystyle D}$

${\displaystyle {\frac {1}{m_{1}^{D}}}+{\frac {1}{m_{2}^{D}}}+\dotsb +{\frac {1}{m_{n}^{D}}}=1}$

defined, where the individual measures are ( ). In these cases, one speaks of the similarity dimension . Examples for the similarity dimension: ${\displaystyle 1/m_{i}}$ ${\displaystyle i=1,\dotsc ,n}$

1. A square is made up of 9 squares of 1/3 side, its Hausdorff dimension is ${\displaystyle D={\tfrac {\log {9}}{\log {3}}}=2}$
2. The Koch curve , a fractal , consists of 4 copies of the total curve , each scaled down to 1: 3. There is a non-integer dimension. ${\displaystyle D={\tfrac {\log {4}}{\log {3}}}=1{,}2618595\dotso }$

It should be noted, however, that this simplified definition does not generally coincide with the exact definition (see below). For example, in the case of a cooking curve with spatially varying iteration depth or the like, the dimension defined in this way can deviate from the actual Hausdorff dimension.

The so-called box counting algorithm can be used to numerically determine the Hausdorff dimension of a given set of points . But here too this only applies as long as the Hausdorff dimension matches the box counting dimension, which is not the case in special cases. When embedded in a two-dimensional space, the set is covered with a regular grid of squares without any gaps and the number of squares that contain points from the set is determined depending on the length of the edge. A numerical extrapolation of the above definition equation for the edge length towards zero provides approximately the Hausdorff dimension.

Definition of the Hausdorff measure

A mathematically exact definition of the Hausdorff dimension of a restricted subset is made using the Hausdorff dimension , which is assigned to this set for each dimension . According to this, the Hausdorff dimension of is defined as the infimum of all for which is, or equivalently as the supremum of all for which applies, that is ${\displaystyle \dim X}$ ${\displaystyle X\subset \mathbb {R} ^{n}}$ ${\displaystyle H^{s}}$ ${\displaystyle s\geq 0}$ ${\displaystyle X}$ ${\displaystyle s}$ ${\displaystyle H^{s}(X)=0}$ ${\displaystyle s}$ ${\displaystyle H^{s}(X)=\infty }$

${\displaystyle \dim X=\inf\{s\mid H^{s}(X)=0\}=\sup\{s\mid H^{s}(X)=\infty \}.}$

For solid , sets whose Hausdorff dimension is smaller than have the -dimensional measure zero, while sets of larger dimensions have infinite -dimensional measure. This corresponds to the fact that, for example, a line segment as a subset of the plane has the two-dimensional Lebesgue measure zero. ${\displaystyle s}$ ${\displaystyle s}$ ${\displaystyle s}$ ${\displaystyle s}$

To define the Hausdorff measure, consider the size

${\displaystyle H_{\varepsilon }^{s}(X)=\inf {\Bigg \{}\sum _{i=1}^{\infty }d(A_{i})^{s}\;{\Bigg |}\;X\subseteq \bigcup _{i=1}^{\infty }A_{i};\;d(A_{i})<\varepsilon {\Bigg \}}}$

for any and , with all overlaps of through countable many sets whose respective diameters are smaller than . The - dimensional Hausdorff measure of is now defined as ${\displaystyle s\geq 0}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle (A_{i})}$ ${\displaystyle X}$ ${\displaystyle A_{1},A_{2},\dotsc }$ ${\displaystyle d(A_{i})}$ ${\displaystyle \varepsilon }$ ${\displaystyle s}$ ${\displaystyle X}$

${\displaystyle H^{s}(X)=\lim _{\varepsilon \to 0}H_{\varepsilon }^{s}(X).}$

example

The Hausdorff dimension of a one-dimensional route is determined using the quantity as follows: ${\displaystyle X=[0,1]\subset \mathbb {R} }$

1. The Hausdorff measure for : ${\displaystyle s>1}$

The natural number for is chosen in such a way that applies. With the special cover ${\displaystyle \varepsilon >0}$ ${\displaystyle N_{\varepsilon }}$ ${\displaystyle 1/N_{\varepsilon }<\varepsilon }$

${\displaystyle A_{i}=\left[{\frac {i-1}{N_{\varepsilon }}},{\frac {i}{N_{\varepsilon }}}\right]}$for , for ${\displaystyle 1\leq i\leq N_{\varepsilon }}$ ${\displaystyle A_{i}=\{1\}}$ ${\displaystyle i>N_{\varepsilon }}$

follows

${\displaystyle H_{\varepsilon }^{s}(X)\leq N_{\varepsilon }\cdot \left({\frac {1}{N_{\varepsilon }}}\right)^{s}=\left({\frac {1}{N_{\varepsilon }}}\right)^{s-1}<\varepsilon ^{s-1},}$

so

${\displaystyle H^{s}(X)=0.}$

2. The Hausdorff measure for : ${\displaystyle s<1}$

Because is ${\displaystyle d(A_{i})<\varepsilon }$

${\displaystyle \sum d(A_{i})^{s}=\sum {\frac {d(A_{i})}{d(A_{i})^{1-s}}}>\sum {\frac {d(A_{i})}{\varepsilon ^{1-s}}}.}$

Since they cover the unit interval , the sum of their diameters is at least 1: ${\displaystyle A_{i}}$ ${\displaystyle X}$

${\displaystyle {}\geq {\frac {1}{\varepsilon ^{1-s}}}.}$

So follows

${\displaystyle H_{\varepsilon }^{s}(X)\geq {\frac {1}{\varepsilon ^{1-s}}},}$

so

${\displaystyle H^{s}(X)=\infty .}$

3. The Hausdorff measure for : ${\displaystyle s=1}$

If you put the two arguments together from the first and second case, you get ${\displaystyle H^{1}(X)=1.}$

So it is . ${\displaystyle \dim X=1}$

literature

• Egbert Brieskorn (ed.): Felix Hausdorff on memory , Vieweg Verlag 1996, ISBN 3-528-06493-5 , u. a. Pages 185 ff.