Editing Self-similarity (section)

==Definition==
A [[Compact space|compact]] [[topological space]] ''X'' is self-similar if there exists a [[finite set]] ''S'' indexing a set of non-[[surjective]] [[homeomorphism]]s <math>\{ f_s : s\in S \}</math> for which

:<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

:<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.

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.

A more general notion than self-similarity is [[self-affinity]].