# 分形维数

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Example of non-integer dimensions. The first four iterations of the Koch curve, where after each iteration, all original line segments are replaced with four, each a self-similar copy that is 1/3 the length of the original. One formalism of the Hausdorff dimension uses this scale factor (3) and the number of self-similar objects (4) to calculate the dimension, D, after the first iteration to be D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26.^{[1]} That is, while the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3, for fractals such as this, the object can have a non-integer dimension.

Example of non-integer dimensions. The first four iterations of the Koch curve, where after each iteration, all original line segments are replaced with four, each a self-similar copy that is 1/3 the length of the original. One formalism of the Hausdorff dimension uses this scale factor (3) and the number of self-similar objects (4) to calculate the dimension, D, after the first iteration to be D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26. That is, while the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3, for fractals such as this, the object can have a non-integer dimension.

非整数维度示例：前四个Koch 曲线的迭代，在每次迭代后，所有原始线段都被替换为四个，每个自相似的复制是原始线段长度的1 / 3。豪斯多夫维数的一个建模是使用比例因子(3)和自相似对象的数量(4)来计算维度 D，在第一次迭代后为 D = (log N)/(log S) = (log 4)/(log 3) ≈ 1.26.^{[1]} 也就是说，单点的豪斯多夫维数为零，线段为1，正方形为2，立方体为3时，但对于分形，对象可以有一个非整数维度。

In mathematics, **Hausdorff dimension** is a measure of *roughness*, or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff.^{[2]} For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and 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 an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their 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 or "rough" sets, this dimension is also commonly referred to as the *Hausdorff–Besicovitch dimension.*

In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and 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 an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their 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 or "rough" sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension.

在数学中，**豪斯多夫维数Hausdorff dimension**是一种粗糙度的度量单位，或者更确切地说，分形维数，是由数学家 Felix Hausdorff 在1918年首次提出的。例如，单点的豪斯多夫维数为零，线段为1，正方形为2，立方体为3。也就是说，对于定义了一个光滑形状或一个有少数几个角的形状---- 传统几何学和科学的形状---- 的点集来说，豪斯多夫维数是一个整数，符合通常的维度意义，也称为拓扑维度。然而，还有一些公式允许计算其他不太简单的对象的维数，其中仅仅根据它们的标度和自相似特性，就可以得出结论: 特定的对象ー包括分形ー具有非整数的 Hausdorff 维数。由于阿布拉姆·萨莫伊洛维奇·贝西科维奇的重大技术进步，允许计算高度不规则或“粗糙”集的维度，这个维度通常也被称为 Hausdorff-Besicovitch 维度。

The Hausdorff dimension, more specifically, is a further dimensional number associated with a given set, where the distances between all members of that set are defined. Such a set is termed a metric space. The dimension is drawn from the extended real numbers, [math]\displaystyle{ \overline{\mathbb{R}} }[/math], as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.

The Hausdorff dimension, more specifically, is a further dimensional number associated with a given set, where the distances between all members of that set are defined. Such a set is termed a metric space. The dimension is drawn from the extended real numbers, [math]\displaystyle{ \overline{\mathbb{R}} }[/math], as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers.

更具体地说，豪斯多夫维数是一个与给定集合相关联的更进一步的维数，其中定义了该集合所有成员之间的距离。这样的集合称为度量空间。维数是从扩展的实数，[math]\displaystyle{ \overline{\mathbb{R}} }[/math]，而不是更直观的维数概念(它不与一般的度量空间相关联，只取非负整数的值)。

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 snowflake shown at right 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.^{[3]} 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.^{[1]} 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}.^{[4]} 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.

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 snowflake shown at right 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. 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.

用数学术语来说，豪斯多夫维数概括了实向量空间维数的概念。也就是说，n 维内积空间的豪斯多夫维数等于 n。 这就是早期假设的基础，一个点的豪斯多夫维数是零，一条线是一等等，不规则集可以有非整数的豪斯多夫维数。例如，右边所示的 Koch 雪花是由一个正三角形构成的; 在每次迭代中，它的组成线段被分成单位长度的3段，新创建的中间线段被用作一个指向外部的新正三角形的基础，然后人们删除这个基础线段用来保留单位长度4的迭代中的最终对象。也就是说，在第一次迭代之后，每个原始线段都被替换为 N=4，其中每个自相似拷贝的长度是原始线段的1/S = 1/3 。这个方程很容易求解为 D，产生出现在图形中的对数(或自然对数)的比率，并给出——在 Koch 和其他分形情况下——这些对象的非整数维数。

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

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

豪斯多夫维数是更简单但通常等价的**计盒维数box-counting**或**闵可夫斯基维数Minkowski-Bouligand**的继承者。

## Intuition概念

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 map the real line to the real plane surjectively (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and *continuously*, so that a one-dimensional object completely fills up a higher-dimensional object.

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 map the real line to the real plane surjectively (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and continuously, so that a one-dimensional object completely fills up a higher-dimensional object.

几何对象X的尺寸的直观概念是指需要多少个独立参数才能找到一个独特的点。但是，任何由两个参数指定的点都可以由一个参数指定，因为为实平面的基数等于实线的基数(这可以通过交织两个数字以产生一个编码相同信息的单个数字看到)。**皮亚诺曲线**的例子表明，可以将实线完美和连续地映射到实平面(把一个实数转换成一对实数，从而覆盖所有实数对)，由此一维对象完全填充了一个高维对象。

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.

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.

每条空间填充曲线都会多次击中某些点，且不存在连续的逆。将二维以连续和连续可逆的方式映射到一维是不可能的。**拓扑维度 topological dimension**，也被称为Lebesgue覆盖维数，解释了为什么。如果在 x 的每个小开球覆盖中，至少有一个点 n + 1个球重叠，这个维度是 n。例如，当用短的开区间覆盖一条线时，某些点必须被覆盖两次，给出维数*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.

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.

但是，拓扑维度是对空间局部尺寸(点附近的尺寸)的一个非常粗略的度量。一条几乎是空间填充的曲线仍然可以有一维拓扑，即使它填充了一个区域的大部分面积。分形具有整数的拓扑维数，但就其所占的空间量而言，它看起来像一个更高维的空间。

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.

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.

豪斯多夫维数测量一个空间的局部大小时，会考虑到点之间距离（度量）。考虑半径最大为 r 的球数 *N* (r) ，需要完全覆盖 *X*。当 r 很小时，n (r)以1/*r* 的多项式增长。对于一个表现足够好的 *X*，豪斯多夫维数是唯一的数 d，这样当 r 趋近于零时， N(r)增长为1/r^{d} 。更确切地说，这定义了盒子计数维度，当值 d 是不足以覆盖空间的增长率和过度充裕的增长率之间的临界边界时，它等于豪斯多夫维数。

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 Benoit 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:

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 Benoit 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:

对于光滑的形状，或者有少量棱角的形状，传统几何和科学的形状，豪斯多夫维数是一个整数，与拓扑维度一致。但是伯努·曼德布洛特观察到分形---- 具有非整数豪斯多夫维数的集合---- 在自然界中随处可见。他观察到，我们周围大多数粗糙形状的理想化不是光滑的理想化形状，而是分形理想化形状:

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.

^{[5]}

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.

云不是球体，山不是锥体，海岸线不是圆圈，树皮不平滑，闪电也不是直线运动。

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.

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.

对于自然界中出现的分形，豪斯多夫维数和盒计数维数是一致的。封装尺寸是另一个类似的概念，它为许多形状提供相同的值，但是在所有这些尺寸不同的情况，都做了很好的说明。

## Formal definitions形式化定义

### Hausdorff content豪斯多夫集

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

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

设 *X*是度量空间。若*S* ⊂ *X* 和 *d* ∈ [0, ∞) ，则 *S* 的*d*维无限 豪斯多夫集定义为

- [math]\displaystyle{ C_H^d(S):=\inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii }r_i\gt 0\Bigr\}. }[/math]

[math]\displaystyle{ C_H^d(S):=\inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii }r_i\gt 0\Bigr\}. }[/math]

[math]\displaystyle{ C_H^d(S):=\inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii }r_i\gt 0\Bigr\}. }[/math]

In other words, [math]\displaystyle{ C_H^d(S) }[/math] is the infimum of the set of numbers [math]\displaystyle{ \delta \geq 0 }[/math] such that there is some (indexed) collection of balls [math]\displaystyle{ \{B(x_i,r_i):i\in I\} }[/math] covering *S* with *r _{i}* > 0 for each

*i*∈

*I*that satisfies [math]\displaystyle{ \sum_{i\in I} r_i^d\lt \delta }[/math]. (Here, we use the standard convention that inf Ø = ∞.)

In other words, [math]\displaystyle{ C_H^d(S) }[/math] is the infimum of the set of numbers [math]\displaystyle{ \delta \geq 0 }[/math] such that there is some (indexed) collection of balls [math]\displaystyle{ \{B(x_i,r_i):i\in I\} }[/math] covering S with r_{i} > 0 for each i ∈ I that satisfies [math]\displaystyle{ \sum_{i\in I} r_i^d\lt \delta }[/math]. (Here, we use the standard convention that inf Ø = ∞.)

换句话说，[math]\displaystyle{ C_H^d(S) }[/math] 是数字集合[math]\displaystyle{ \delta \geq 0 }[/math]的下确界，使得在 *i* ∈ 中存在一些球集合 [math]\displaystyle{ \{B(x_i,r_i):i\in I\} }[/math] i 包含 s，对于每个 *r _{i}* > 0 满足 i 中的和[math]\displaystyle{ \sum_{i\in I} r_i^d\lt \delta }[/math] (在这里，我们使用inf Ø = ∞ 的标准约定)。

### Hausdorff measurement豪斯多夫分形测量

The Hausdorff outer measure is different from the unbounded Hausdorf content in that rather than considering all possible coverings of *S*, we see what happens when the sizes of the balls go to zero. This is for [math]\displaystyle{ d \geq 0 }[/math], we define the *d*-dimensional Hausdorff outer measure of *S* as

The Hausdorff outer measure is different from the unbounded Hausdorf content in that rather than considering all possible coverings of S, we see what happens when the sizes of the balls go to zero. This is for [math]\displaystyle{ d \geq 0 }[/math], we define the d-dimensional Hausdorff outer measure of S as

豪斯多夫外测度不同于无界的豪斯多夫，因为我们不考虑 s 的所有可能，我们看到当球的大小变为零时会发生什么。对于 [math]\displaystyle{ d \geq 0 }[/math]，我们定义了 *S*的 *d*维豪斯多夫Hausdorff 外测度为

- [math]\displaystyle{ \mathcal{H}^d(S):=\lim_{r \to 0} \inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii } 0 \lt r_i \lt r\Bigr\}. }[/math]

[math]\displaystyle{ \mathcal{H}^d(S):=\lim_{r \to 0} \inf\Bigl\{\sum_i r_i^d:\text{ there is a cover of } S\text{ by balls with radii } 0 \lt r_i \lt r\Bigr\}. }[/math]

[math]\displaystyle{ \mathcal{H}^d(S):=\lim_{r \to 0} \inf\Bigl\{\sum_i r_i^d:\text{有一个面} S\text{ 球 } 0 \lt r_i \lt r\Bigr\}. }[/math]

### Hausdorff dimension豪斯多夫维数

The **Hausdorff dimension** of *X* is defined by

The Hausdorff dimension of X is defined by

X 的豪斯多夫维数定义为

- [math]\displaystyle{ \dim_{\operatorname{H}}(X):=\inf\{d\ge 0: \mathcal{H}^d(X)=0\}. }[/math]

[math]\displaystyle{ \dim_{\operatorname{H}}(X):=\inf\{d\ge 0: \mathcal{H}^d(X)=0\}. }[/math]

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).

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).

等价地，dim _{H}(*X*)可定义为 *d* ∈ [0, ∞) 集的下确界，使得*X* 的*d*-维 豪斯多夫测度 为零。这与 *d* ∈ [0, ∞)的集合的上确界相同，因此*X*的 d 维豪斯多夫测度是无限的(除非后一个集合 *d* 是空的，豪斯多夫维数为零)。

## Examples实例

Dimension of a further fractal example. The Sierpinski triangle, an object with Hausdorff dimension of log(3)/log(2)≈1.58.

- Countable sets have Hausdorff dimension 0.
^{[6]} - 可数集拥有豪斯多夫维数0.
^{[6]} - The Euclidean space ℝ
^{n}has Hausdorff dimension*n*, and the circle**S**^{1}has Hausdorff dimension 1.^{[6]} - 教育空间 ℝ
^{n}有豪斯多夫维数*n*，循环**S**^{1}拥有豪斯多夫维数1.^{[6]} - Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological dimension.
^{[5]}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.^{[7]}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.^{[1]}These Hausdorff dimensions are related to the "critical exponent" of the Master theorem for solving recurrence relations in the analysis of algorithms. - 分形一般是那些豪斯多夫维数直接超过其拓扑维数的空间。例如康托尔集是一个o维拓扑空间，由两个自己复制而成，每一个复制品都是原来的三分之一，因此它的豪斯多夫维数是 ln(2)/ln(3) ≈ 0.63。
^{[8]}一个谢尔宾斯基三角是他自身三个复制的组合。每一个是原来的 1/2，它的豪斯多夫维数ln(3)/ln(2) ≈ 1.58。^{[1]}在递归算法中解决递归关系时，这些豪斯多夫维数与算法分析主定理的临界指标相联系。 - Space-filling curves like the Peano curve have the same Hausdorff dimension as the space they fill.
- 空间填充曲线拥有和他们填充空间同样的豪斯多夫维数，如皮亚诺曲线。
- The trajectory of Brownian motion in dimension 2 and above is conjectured to be Hausdorff dimension 2.
^{[9]}

coast of Great Britain]]

[[大不列颠海岸

- 刘易斯·弗赖伊·理查森Lewis Fry Richardson has performed detailed experiments to measure the approximate Hausdorff dimension for various coastlines. His results have varied from 1.02 for the coastline of South Africa to 1.25 for the west coast of Great Britain.
^{[5]} - 刘易斯·弗赖伊·理查森（Lewis Fry Richardson）已经通过豪斯多夫维数去测量了很多海岸线。它的结果涵盖从1.02的南非海岸线到1.25的大英帝国西海岸模型。]]

## Properties of Hausdorff dimension豪斯多夫维数特性

### Hausdorff dimension and inductive dimension 豪斯多夫维数和归纳维数

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*).

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).

设*X* 是任意可分度量空间。对于*X* 有一个递归定义的归纳维数拓扑概念。它总是一个整数(或 + ∞) ，记为dim_{ind}(*X*)。

**Theorem**. Suppose *X* is non-empty. Then

Theorem. Suppose X is non-empty. Then

**定理**：假设*X* 是非空的, 那么

- [math]\displaystyle{ \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X). }[/math]

[math]\displaystyle{ \dim_{\mathrm{Haus}}(X) \geq \dim_{\operatorname{ind}}(X). }[/math]

Moreover,

Moreover,

此外,

[math]\displaystyle{ \inf_Y \dim_{\operatorname{Haus}}(Y) =\dim_{\operatorname{ind}}(X), }[/math]

[math]\displaystyle{ \inf_Y \dim_{\operatorname{Haus}}(Y) =\dim_{\operatorname{ind}}(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}.

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}.

其中*Y*是度量空间同胚到 *X*的范围。换句话说， *X*和 *Y*具有相同的基本点集，*Y*的度量*d*_{Y} 拓扑等价于*d*_{X}。

These results were originally established by Edward Szpilrajn (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.模板:Full citation needed

These results were originally established by Edward Szpilrajn (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII.

这些结果最初是由 Edward Szpilrajn (1907–1976)建立的, 参见 Hurewicz and Wallman, Chapter VII.模板:Full citation needed第七章。

### Hausdorff dimension and Minkowski dimension 豪斯多夫维数和闵可夫斯基维度

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.

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.

闵可夫斯基维数与豪斯多夫维数相似，至少和它一样大，而且在许多情况下是相等的。然而，[0,1]中有理点集的豪斯多夫维数为0，闵可夫斯基维数为1。还有一些紧集的闵可夫斯基维数严格大于豪斯多夫维数。

### Hausdorff dimensions and Frostman measures 豪斯多夫维度和弗洛斯曼测度

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.

^{[citation needed]}

^{[10]}

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.

如果在度量空间 *X* 的 Borel 子集上定义一个测度 μ，使得*μ*(*X*) > 0 和*μ*(*B*(*x*, *r*)) ≤ *r ^{s}*，对于某个常数

*s*> 0 和

*X*中的每个球

*B*(

*x*,

*r*) 成立，则 dim

_{Haus}(

*X*) ≥

*s*。 部分逆向转换由弗洛斯曼引理定义。

### Behaviour under unions and products

If [math]\displaystyle{ X=\bigcup_{i\in I}X_i }[/math] is a finite or countable union, then

If [math]\displaystyle{ X=\bigcup_{i\in I}X_i }[/math] is a finite or countable union, then

如果 [math]\displaystyle{ X=\bigcup_{i\in I}X_i }[/math] 是一个有限或可数的联合，则

[math]\displaystyle{ \dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i). }[/math]

[math]\displaystyle{ \dim_{\operatorname{Haus}}(X) =\sup_{i\in I} \dim_{\operatorname{Haus}}(X_i). }[/math]

This can be verified directly from the definition.

This can be verified directly from the definition.

这可以直接从定义得到验证。

If *X* and *Y* are non-empty metric spaces, then the Hausdorff dimension of their product satisfies^{[11]}

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

如果 *X* 和*Y*是非空度量空间，那么它们乘积的豪斯多夫维数满足

- [math]\displaystyle{ \dim_{\operatorname{Haus}}(X\times Y)\ge \dim_{\operatorname{Haus}}(X)+ \dim_{\operatorname{Haus}}(Y). }[/math]

[math]\displaystyle{ \dim_{\operatorname{Haus}}(X\times Y)\ge \dim_{\operatorname{Haus}}(X)+ \dim_{\operatorname{Haus}}(Y). }[/math]

This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1.^{[12]} In the opposite direction, it is known that when *X* and *Y* are Borel subsets of **R**^{n}, 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).

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 R^{n}, 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).

这种不平等可以是严格的。有可能找到两个维数为0的集合，其乘积的维数为1。相反，我们知道当*X*和*Y*是 **R**^{n}的 Borel 子集时， *X* × *Y*的豪斯多夫维数从上面以 *X*的豪斯多夫维数加上 *Y*的上填充维数为界。Mattila (1995)曾就这些情况进行了讨论。

## Self-similar sets自相似集合

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.

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.

许多由自相似条件定义的集合具有可以显式确定的维数。粗略地说，如果集合*E*是集值ψ变换的不动点，即ψ(*E*) = *E*， 则它是自相似的，尽管下面给出了确切的定义。

Theorem. SupposeTheorem. Suppose

定理：假设

- [math]\displaystyle{ \psi_i: \mathbf{R}^n \rightarrow \mathbf{R}^n, \quad i=1, \ldots , m }[/math]
[math]\displaystyle{ \psi_i: \mathbf{R}^n \rightarrow \mathbf{R}^n, \quad i=1, \ldots , m }[/math]

are contractive mappings on

R^{n}with contraction constantr< 1. Then there is a unique_{j}non-emptycompact setAsuch thatare contractive mappings on R

^{n}with contraction constant r_{j}< 1. Then there is a unique non-empty compact set A such that是

R^{n}上的压缩常数r_{j}'< 1的压缩映射。则有一个唯一的非空紧集A

- [math]\displaystyle{ A = \bigcup_{i=1}^m \psi_i (A). }[/math]
[math]\displaystyle{ A = \bigcup_{i=1}^m \psi_i (A). }[/math]

The theorem follows from Stefan Banach's contractive mapping fixed point theorem applied to the complete metric space of non-empty compact subsets ofR^{n}with the Hausdorff distance.^{[13]}The theorem follows from Stefan Banach's contractive mapping fixed point theorem applied to the complete metric space of non-empty compact subsets of R

^{n}with the Hausdorff distance.这个定理来源于 Stefan Banach 的压缩映射不动点定理，该定理应用于 具有豪斯多夫距离的

R^{n}的非空紧子集的完整度量空间。

## The open set condition开集条件

To determine the dimension of the self-similar set

A(in certain cases), we need a technical condition called theopen set condition(OSC) on the sequence of contractions ψ_{i}.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}.为了确定自相似集

A的维数(在某些情况下) ，我们需要一个关于收缩序列的称为开集条件(OSC)的技术条件ψ_{i}。

There is a relatively compact open set

Vsuch thatThere is a relatively compact open set V such that

有一个相对紧的开集

V

- [math]\displaystyle{ \bigcup_{i=1}^m\psi_i (V) \subseteq V, }[/math]
[math]\displaystyle{ \bigcup_{i=1}^m\psi_i (V) \subseteq V, }[/math]

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

左边并集的集合成对不相交。

The open set condition is a separation condition that ensures the images ψ

_{i}(V) do not overlap "too much".The open set condition is a separation condition that ensures the images ψ

_{i}(V) do not overlap "too much".开集条件是保证图像ψ

_{i}(V) 不重叠“太多”的分离条件。

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 isswheresis the unique solution of^{[14]}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

定理假设开集条件成立，并且每个ψ_{i}是一个相似度，即等距和某个点周围的膨胀的组合。。那么唯一的不动点是Hausdorff维数为s的集合，其中s是s^{[15]}的唯一解

- [math]\displaystyle{ \sum_{i=1}^m r_i^s = 1. }[/math]
[math]\displaystyle{ \sum_{i=1}^m r_i^s = 1. }[/math]

The contraction coefficient of a similitude is the magnitude of the dilation.

The contraction coefficient of a similitude is the magnitude of the dilation.

相似物的收缩系数就是膨胀的大小。

We can use this theorem to compute the Hausdorff dimension of the Sierpinski triangle (or sometimes called Sierpinski gasket). Consider three non-collinear points

a_{1},a_{2},a_{3}in the planeR^{2}and let ψ_{i}be the dilation of ratio 1/2 arounda. The unique non-empty fixed point of the corresponding mapping ψ is a Sierpinski gasket and the dimension_{i}sis the unique solution ofWe can use this theorem to compute the Hausdorff dimension of the Sierpinski triangle (or sometimes called Sierpinski gasket). Consider three non-collinear points a

_{1}, a_{2}, a_{3}in the plane R^{2}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我们可以使用这个定理来计算谢尔宾斯基三角形的豪斯多夫维数(或者有时候叫做谢尔宾斯基垫圈)。考虑R

^{2}平面上的三个非共线点，a_{1}，a_{2}，a_{3}，让ψ_{i}是围绕着a比率1/2的膨胀。对应映射的唯一非空不动点是一个谢尔宾斯基垫圈，其维数_{i}s是对应映射的唯一解

- [math]\displaystyle{ \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]
[math]\displaystyle{ \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 fors, that is:s= ln(3)/ln(2). The Sierpinski gasket is self-similar and satisfies the OSC. In general a setEwhich is a fixed point of a mappingTaking 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

取上述方程两边的自然对数，我们可以求出

s，即:s= ln(3)/ln(2)。该密封垫具有自相似性，满足 OSC 要求。一般来说，集合E是一个映射的不动点

- [math]\displaystyle{ A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) }[/math]
`[math]\displaystyle{ A \mapsto \psi(A) = \bigcup_{i=1}^m \psi_i(A) }[/math]`

is self-similar if and only if the intersectionsis self-similar if and only if the intersections

是自相似的，当且仅当

- [math]\displaystyle{ H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, }[/math]
[math]\displaystyle{ H^s\left(\psi_i(E) \cap \psi_j(E)\right) =0, }[/math]

where

sis the Hausdorff dimension ofEandHdenotes Hausdorff measure. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally:^{s}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:其中

s是E的豪斯多夫维数，H表示 豪斯多夫测度。对于谢尔宾斯基垫圈(交叉点就是点)来说，这一点很明显，但更普遍的情况是:^{s}

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

定理：在与前一定理相同的条件下，其唯一不动点是自相似的。## See also参阅

- List of fractals by Hausdorff dimension豪斯多夫分形维数列表 Examples of deterministic fractals, random and natural fractals.确定性分形与随机和自然分形的示例。

- Assouad dimension Assouad维数, another variation of fractal dimension that, like Hausdorff dimension, is defined using coverings by balls其他被球覆盖方式定义的分形维数，例如豪斯多夫维数。
## References参考文献

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}^{1.4}MacGregor Campbell, 2013, "5.6 Scaling and the Hausdorff Dimension," atAnnenberg Learner:MATHematics illuminated, see [1], accessed 5 March 2015. 引用错误：无效`<ref>`

标签；name属性“CampbellAnnenberg15”使用不同内容定义了多次- ↑ Gneiting, Tilmann; Ševčíková, Hana; Percival, Donald B. (2012). "Estimators of Fractal Dimension: Assessing the Roughness of Time Series and Spatial Data".
Statistical Science.27(2): 247–277. arXiv:1101.1444. doi:10.1214/11-STS370.- ↑ Larry Riddle, 2014, "Classic Iterated Function Systems: Koch Snowflake", Agnes Scott College e-Academy (online), see [2], accessed 5 March 2015.
- ↑
^{4.0}^{4.1}^{4.2}Keith Clayton, 1996, "Fractals and the Fractal Dimension,"Basic Concepts in Nonlinear Dynamics and Chaos(workshop), Society for Chaos Theory in Psychology and the Life Sciences annual meeting, June 28, 1996, Berkeley, California, see [3], accessed 5 March 2015.- ↑
^{5.0}^{5.1}^{5.2}Mandelbrot, Benoît (1982).The Fractal Geometry of Nature. Lecture notes in mathematics 1358. W. H. Freeman. ISBN 0-7167-1186-9. https://archive.org/details/fractalgeometryo00beno.- ↑
^{6.0}^{6.1}^{6.2}^{6.3}Schleicher, Dierk (June 2007). "Hausdorff Dimension, Its Properties, and Its Surprises".The American Mathematical Monthly(in English).114(6): 509–528. arXiv:math/0505099. doi:10.1080/00029890.2007.11920440. ISSN 0002-9890. 引用错误：无效`<ref>`

标签；name属性“schleicher”使用不同内容定义了多次- ↑ Falconer, Kenneth (2003).
Fractal Geometry: Mathematical Foundations and Applications(2nd ed.). John Wiley and Sons.- ↑ Falconer, Kenneth (2003).
Fractal Geometry: Mathematical Foundations and Applications(2nd ed.). John Wiley and Sons.- ↑ Morters, Peres (2010).
Brownian Motion. Cambridge University Press.- ↑ This Wikipedia article also discusses further useful characterizations of the Hausdorff dimension.模板:Clarify
- ↑ Marstrand, J. M. (1954). "The dimension of Cartesian product sets".
Proc. Cambridge Philos. Soc.50(3): 198–202. Bibcode:1954PCPS...50..198M. doi:10.1017/S0305004100029236.- ↑ Falconer, Kenneth J. (2003).
Fractal geometry. Mathematical foundations and applications. John Wiley & Sons, Inc., Hoboken, New Jersey.- ↑ Falconer, K. J. (1985). "Theorem 8.3".
The Geometry of Fractal Sets. Cambridge, UK: Cambridge University Press. ISBN 0-521-25694-1.- ↑ Hutchinson, John E. (1981). "Fractals and self similarity".
Indiana Univ. Math. J.30(5): 713–747. doi:10.1512/iumj.1981.30.30055.- ↑ Hutchinson, John E. (1981). "Fractals and self similarity".
Indiana Univ. Math. J.30(5): 713–747. doi:10.1512/iumj.1981.30.30055.## Further reading拓展阅读

- Dodson, M. Maurice; Kristensen, Simon (June 12, 2003). "Hausdorff Dimension and Diophantine Approximation".
Fractal Geometry and Applications: A Jubilee of Benoît Mandelbrot. Proceedings of Symposia in Pure Mathematics.72. pp. 305–347. arXiv:math/0305399. Bibcode 2003math......5399D. doi:10.1090/pspum/072.1/2112110. ISBN 9780821836378.

- Hurewicz, Witold; Wallman, Henry (1948).
Dimension Theory. Princeton University Press. https://archive.org/details/in.ernet.dli.2015.84609.

- "La dimension et la mesure".
Fundamenta Mathematicae.28: 81–9. 1937. Unknown parameter`|author作者=`

ignored (help)

- 马斯特兰德, J. M. (1954). "笛卡儿积集的维数".
Proc. Cambridge Philos. Soc.50(3): 198–202. Bibcode:1954PCPS...50..198M. doi:10.1017/S0305004100029236. Text " 1954年 " ignored (help); Text " 第50卷" ignored (help); Text " 第198-202页" ignored (help); Text " 第三期" ignored (help)| doi=10.1017/S0305004100029236|bibcode = 1954PCPS...50..198M }}10.1017 / S0305004100029236 | bibcode 1954PCPS... 50. . 198 m }

- Mattila, Pertti (1995).
Geometry of sets and measures in Euclidean spaces. Cambridge University Press. ISBN 978-0-521-65595-8.| isbn=978-0-521-65595-8 | year=1995}}| isbn 978-0-521-65595-8 | year 1995}

- A. S. Besicovitch (1929). "On Linear Sets of Points of Fractional Dimensions".
Mathematische Annalen.101(1): 161–193. doi:10.1007/BF01454831.

- A. S. Besicovitch; H. D. Ursell (1937). "Sets of Fractional Dimensions".
Journal of the London Mathematical Society.12(1): 18–25. doi:10.1112/jlms/s1-12.45.18.

Several selections from this volume are reprinted in Edgar, Gerald A. (1993).Classics on fractals. Boston: Addison-Wesley. ISBN 0-201-58701-7. See chapters 9,10,11

- F. Hausdorff (March 1919). "Dimension und äußeres Maß" (PDF).
Mathematische Annalen.79(1–2): 157–179. doi:10.1007/BF01457179. hdl:10338.dmlcz/100363.

- Hutchinson, John E. (1981). "Fractals and self similarity".
Indiana Univ. Math. J.30(5): 713–747. doi:10.1512/iumj.1981.30.30055.

- Falconer, Kenneth (2003).
Fractal Geometry: Mathematical Foundations and Applications(2nd ed.). John Wiley and Sons.## External links外部链接

Category分类:Metric geometry度量几何

Category分类:Dimension theory维度理论

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