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第74行： − 紧致拓扑空间 x 是自相似的，如果存在一个有限集 s 对 s / math 中的一组非满射同胚数学进行索引+ − − − − :<math>X=\bigcup_{s\in S} f_s(X)</math> − − S } f (x) / math 中的数学 x 大杯 − − − If <math>X\subset Y</math>, we call X self-similar if it is the only non-empty subset of Y such that the equation above holds for <math>\{ f_s : s\in S \} </math>. We call+ − 如果数学 x 子集 y / math，我们称 x 自相似，如果它是 y 的唯一非空子集，使得上面的方程适用于数学 s: s / math。我们打电话+ + + − :<math>\mathfrak{L}=(X,S,\{ f_s : s\in S \} )</math>+ − Math mathfrak { l }(x，s，f s: s in s) / math+ − − − − a self-similar structure. The homeomorphisms may be iterated, resulting in an iterated function system. The composition of functions creates the algebraic structure of a monoid. When the set S has only two elements, the monoid is known as the dyadic monoid. The dyadic monoid can be visualized as an infinite binary tree; more generally, if the set S has p elements, then the monoid may be represented as a p-adic tree. − − 第115行：
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无编辑摘要
A compact topological space X is self-similar if there exists a finite set S indexing a set of non-surjective homeomorphisms <math>\{ f_s : s\in S \}</math> for which
A compact topological space X is self-similar if there exists a finite set S indexing a set of non-surjective homeomorphisms <math>\{ f_s : s\in S \}</math> for which
如果存在一个有限集 ''S'' 对<math>\{ f_s : s\in S \}</math>中的一组非满射同胚集进行索引，则紧致拓扑空间 ''X'' 是自相似的，有：
<math>X=\bigcup_{s\in S} f_s(X)</math>
<math>X=\bigcup_{s\in S} f_s(X)</math>
If <math>X\subset Y</math>, we call ''X'' self-similar if it is the only [[Non-empty set|non-empty]] [[subset]] of ''Y'' such that the equation above holds for <math>\{ f_s : s\in S \} </math>. We call
If <math>X\subset Y</math>, we call ''X'' self-similar if it is the only [[Non-empty set|non-empty]] [[subset]] of ''Y'' such that the equation above holds for <math>\{ f_s : s\in S \} </math>. We call
假设有
<math>X\subset Y</math>，
当且仅当X是Y的唯一非空子集，使得上式对
<math>\{ f_s : s\in S \} </math>
成立，则X是自相似的。而且我们称
<math>\mathfrak{L}=(X,S,\{ f_s : s\in S \} )</math>
<math>\mathfrak{L}=(X,S,\{ f_s : s\in S \} )</math>
是自相似结构。
a ''self-similar structure''. The homeomorphisms may be [[iterated function|iterated]], resulting in an [[iterated function system]]. The composition of functions creates the algebraic structure of a [[monoid]]. When the set ''S'' has only two elements, the monoid is known as the [[dyadic monoid]]. The dyadic monoid can be visualized as an infinite [[binary tree]]; more generally, if the set ''S'' has ''p'' elements, then the monoid may be represented as a [[p-adic number|p-adic]] tree.
a ''self-similar structure''. The homeomorphisms may be [[iterated function|iterated]], resulting in an [[iterated function system]]. The composition of functions creates the algebraic structure of a [[monoid]]. When the set ''S'' has only two elements, the monoid is known as the [[dyadic monoid]]. The dyadic monoid can be visualized as an infinite [[binary tree]]; more generally, if the set ''S'' has ''p'' elements, then the monoid may be represented as a [[p-adic number|p-adic]] tree.
一个自相似的结构。同胚可以迭代，产生迭代函数系统。函数的组合创建了 monoid 的代数结构。当集合 s 只有两个元素时，这个幺半群称为二元幺半群。二元幺半群可以被视为一棵无限的二叉树，更一般地说，如果集合 s 有 p 个元素，那么幺半群可以被表示为一棵 p-adic 树。
一个自相似的结构。同胚可以迭代，产生迭代函数系统。函数的组合创建了 monoid 的代数结构。当集合 s 只有两个元素时，这个幺半群称为二元幺半群。二元幺半群可以被视为一棵无限的二叉树，更一般地说，如果集合 s 有 p 个元素，那么幺半群可以被表示为一棵 p-adic 树。
The [[automorphism]]s of the dyadic monoid is the [[modular group]]; the automorphisms can be pictured as [[Hyperbolic coordinates|hyperbolic rotation]]s of the binary tree.
The [[automorphism]]s of the dyadic monoid is the [[modular group]]; the automorphisms can be pictured as [[Hyperbolic coordinates|hyperbolic rotation]]s of the binary tree.
二元幺半群的自同构是模群，自同构可以描述为二叉树的双曲旋转。
二元幺半群的自同构是模群，自同构可以描述为二叉树的双曲旋转。
A more general notion than self-similarity is [[Self-affinity]].
A more general notion than self-similarity is [[Self-affinity]].