Editing Dyadic transformation (section)

==Relation to the Bernoulli process==
[[Image:Exampleergodicmap.svg|thumb|The map ''T'' : [0,&thinsp;1) → [0,&thinsp;1), <math>x \mapsto 2x \bmod 1</math> preserves the [[Lebesgue measure]].]]

The map can be obtained as a [[homomorphism]] on the [[Bernoulli process]]. Let <math>\Omega = \{H,T\}^{\mathbb{N}}</math> be the set of all semi-infinite strings of the letters <math>H</math> and <math>T</math>. These can be understood to be the flips of a coin, coming up heads or tails. Equivalently, one can write <math>\Omega = \{0,1\}^{\mathbb{N}}</math> the space of all (semi-)infinite strings of binary bits. The word "infinite" is qualified with "semi-", as one can also define a different space <math>\{0,1\}^{\mathbb{Z}}</math> consisting of all doubly-infinite (double-ended) strings; this will lead to the [[Baker's map]]. The qualification "semi-" is dropped below. 

This space has a natural [[shift space|shift operation]], given by
:<math>T(b_0, b_1, b_2, \dots) = (b_1, b_2, \dots)</math>
where <math>(b_0, b_1, \dots)</math> is an infinite string of binary digits.  Given such a string, write

:<math>x = \sum_{n=0}^\infty \frac{b_n}{2^{n+1}}.</math>
The resulting <math>x</math> is a [[real number]] in the [[unit interval]] <math>0 \le x \le 1.</math> The shift <math>T</math> induces a [[homomorphism]], also called <math>T</math>, on the unit interval. Since <math>T(b_0, b_1, b_2, \dots) = (b_1, b_2, \dots),</math> one can easily see that <math>T(x)=2x\bmod 1.</math>  For the doubly-infinite sequence of bits <math>\Omega = 2^{\mathbb{Z}},</math> the induced homomorphism is the [[Baker's map]].

The dyadic sequence is then just the sequence
:<math>(x, T(x), T^2(x), T^3(x), \dots)</math>
That is, <math>x_n = T^n(x).</math>

===The Cantor set===
Note that the sum
:<math>y=\sum_{n=0}^\infty \frac{b_n}{3^{n+1}}</math>
gives the [[Cantor function]], as conventionally defined. This is one reason why the set <math>\{H,T\}^\mathbb{N}</math> is sometimes called the [[Cantor set]].