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→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 the open set condition (OSC) on the sequence of contractions ψ<sub>i</sub>.
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<sub>i</sub>.
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<sub>''i''</sub>。
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<sub>i</sub>(V) do not overlap "too much".
The open set condition is a separation condition that ensures the images ψ<sub>i</sub>(V) do not overlap "too much".
开集条件是保证图像子ψ<sub>''i''</sub>(''V'') 不重叠“太多”的分离条件。
Theorem. Suppose the open set condition holds and each ψ<sub>i</sub> 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<sub>i</sub> 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<sub>''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'' <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 a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub> in the plane R<sup>2</sup> and let ψ<sub>i</sub> be the dilation of ratio 1/2 around a<sub>i</sub>. 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 a<sub>1</sub>, a<sub>2</sub>, a<sub>3</sub> in the plane R<sup>2</sup> and let ψ<sub>i</sub> be the dilation of ratio 1/2 around a<sub>i</sub>. 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<sup>2</sup> 平面上的三个非共线点，''a''<sub>1</sub>，''a''<sub>2</sub>，''a''<sub>3</sub>，让ψ<sub>''i''</sub>是围绕着''a<sub>i</sub>''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
是自相似的，当且仅当
<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<sup>s</sup> 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<sup>s</sup> 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<sup>s</sup>'' 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.
'''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==
==See also参阅==