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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

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

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>

<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

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>X\subset Y</math>，

<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 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]].