Editing Cantor set (section)

=== Self-similarity ===
The Cantor set is the prototype of a [[fractal]]. It is [[self-similar]], because it is equal to two copies of itself, if each copy is shrunk by a factor of 3 and translated. More precisely, the Cantor set is equal to the union of two functions, the left and right self-similarity transformations of itself, <math>T_L(x)=x/3</math> and <math>T_R(x)=(2+x)/3</math>, which leave the Cantor set invariant up to [[homeomorphism]]: <math>T_L(\mathcal{C})\cong T_R(\mathcal{C})\cong \mathcal{C}=T_L(\mathcal{C})\cup T_R(\mathcal{C}).</math>

Repeated [[iterated function|iteration]] of <math>T_L</math> and <math>T_R</math> can be visualized as an infinite [[binary tree]]. That is, at each node of the tree, one may consider the subtree to the left or to the right. Taking the set <math>\{T_L, T_R\}</math> together with [[function composition]] forms a [[monoid]], the [[dyadic monoid]].

The [[automorphism]]s of the binary tree are its hyperbolic rotations, and are given by the [[modular group]]. Thus, the Cantor set is a [[homogeneous space]] in the sense that for any two points <math>x</math> and <math>y</math> in the Cantor set <math>\mathcal{C}</math>, there exists a homeomorphism <math>h:\mathcal{C}\to \mathcal{C}</math> with <math>h(x)=y</math>. An explicit construction of <math>h</math> can be described more easily if we see the Cantor set [[#Topological and analytical properties|as a product space]] of  countably many copies of the discrete space <math>\{0,1\}</math>. Then the map <math>h:\{0,1\}^\N\to\{0,1\}^\N </math> defined by <math>h_n(u):=u_n+x_n+y_n \mod 2</math> is an [[involution (mathematics)|involutive]] homeomorphism exchanging  <math>x</math> and <math>y</math>.