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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.

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) ，我们需要一个称为开集条件(OSC)的技术条件。

为了确定自相似集''A'' 的维数(在某些情况下) ，我们需要一个称为''开集条件''(OSC)的技术条件ψ_''i''。

There is a relatively compact open set V such that

There is a relatively compact open set V such that

有一个相对紧的开集 v

有一个相对紧的开集''V''

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

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

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子 i / 子(v)不重叠“太多”的分离条件。

开集条件是保证图像子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 is s where s is the unique solution of

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，并且每个子 i / sub 是一个相似量，这是一个等距和围绕某一点的膨胀的合成。那么唯一的不动点是一个集合，它的豪斯多夫维数是 s，其中 s 是 s 的唯一解

'''定理'''假设开集条件成立，并且每个ψ_''i'' 是一个相似量，这是一个等距和围绕某一点的膨胀的合成。那么唯一的不动点是一个集合，它的豪斯多夫维数是 ''s'' ，其中  ''s'' 是 ''s'' <ref>{{cite journal | last=Hutchinson | first=John E. | title=Fractals and self similarity | journal=Indiana Univ. Math. J. | volume=30 | year=1981 | pages=713–747 | doi=10.1512/iumj.1981.30.30055 | issue=5 | doi-access=free }}</ref>的唯一解

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

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

 

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

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

我们可以使用这个定理来计算豪斯多夫维数谢尔宾斯基三角形的密封圈(或者有时候叫做 Sierpinski 垫圈)。考虑平面上的三个非共线点，一个子1 / 子，一个子2 / 子，一个子3 / 子。对应映射的唯一非空不动点是一个 Sierpinski 垫片，其维数 s 是对应映射的唯一解

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R² 平面上的三个非共线点，''a''₁，''a''₂，''a''₃，让ψ_''i''是围绕着''a_i''膨胀比率的1/2。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''s''是对应映射的唯一解

:<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>

:<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>

<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>

<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: 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

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

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

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

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

is self-similar if and only if the intersections

is self-similar if and only if the intersections

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

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

 

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:

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 sup s / sup 表示 Hausdorff 测度。对于 Sierpinski 垫圈(交叉点就是点)来说，这一点很明显，但更普遍的情况是:

其中 ''s''是''E''的豪斯多夫维数， ''E'' and ''H^s'' 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:

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.

==See also参阅==

==See also参阅==