Editing Bernoulli process (section)

==Dynamical systems==
The Bernoulli process can also be understood to be a [[dynamical system]], as an example of an [[ergodic system]] and specifically, a [[measure-preserving dynamical system]], in one of several different ways. One way is as a [[shift space]], and the other is as an [[Markov odometer|odometer]]. These are reviewed below.

===Bernoulli shift===
{{main article|Bernoulli scheme|Dyadic transformation}}
One way to create a dynamical system out of the Bernoulli process is as a [[shift space]]. There is a natural translation symmetry on the product space <math>\Omega=2^\mathbb{N}</math> given by the [[shift operator]]
:<math>T(X_0, X_1, X_2, \cdots) = (X_1, X_2, \cdots)</math>

The Bernoulli measure, defined above, is translation-invariant; that is, given any cylinder set <math>\sigma\in\mathcal{B}</math>, one has
:<math>P(T^{-1}(\sigma))=P(\sigma)</math>
and thus the [[Bernoulli measure]] is a [[Haar measure]]; it is an [[invariant measure]] on the product space.

Instead of the probability measure <math>P:\mathcal{B}\to\mathbb{R}</math>, consider instead some arbitrary function <math>f:\mathcal{B}\to\mathbb{R}</math>. The [[pushforward measure|pushforward]] 
:<math>f\circ T^{-1}</math>
defined by <math>\left(f\circ T^{-1}\right)(\sigma) = f(T^{-1}(\sigma))</math> is again some function <math>\mathcal{B}\to\mathbb{R}.</math> Thus, the map <math>T</math> induces another map <math>\mathcal{L}_T</math> on the space of all functions <math>\mathcal{B}\to\mathbb{R}.</math> That is, given some <math>f:\mathcal{B}\to\mathbb{R}</math>, one defines
:<math>\mathcal{L}_T f = f \circ T^{-1}</math>
The map <math>\mathcal{L}_T</math> is a [[linear operator]], as (obviously) one has <math>\mathcal{L}_T(f+g)= \mathcal{L}_T(f) + \mathcal{L}_T(g)</math> and <math>\mathcal{L}_T(af)= a\mathcal{L}_T(f)</math> for functions <math>f,g</math> and constant <math>a</math>. This linear operator is called the [[transfer operator]] or the ''Ruelle–Frobenius–Perron operator''.  This operator has a [[spectrum]], that is, a collection of [[eigenfunction]]s and corresponding eigenvalues. The largest eigenvalue is the [[Frobenius–Perron theorem|Frobenius–Perron eigenvalue]], and in this case, it is 1.  The associated eigenvector is the invariant measure: in this case, it is the Bernoulli measure. That is, <math>\mathcal{L}_T(P)= P.</math>

If one restricts <math>\mathcal{L}_T</math> to act on polynomials, then the eigenfunctions are (curiously) the [[Bernoulli polynomial]]s!<ref>Pierre Gaspard, "''r''-adic one-dimensional maps and the Euler summation formula", ''Journal of Physics A'', '''25''' (letter) L483-L485 (1992).</ref><ref>Dean J. Driebe, Fully Chaotic Maps and Broken Time Symmetry, (1999) Kluwer Academic Publishers, Dordrecht Netherlands {{ISBN|0-7923-5564-4}}</ref>  This coincidence of naming was presumably not known to Bernoulli.

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

The above can be made more precise. Given an infinite string of binary digits <math>b_0, b_1, \cdots</math> write
:<math>y=\sum_{n=0}^\infty \frac{b_n}{2^{n+1}}.</math>
The resulting <math>y</math> is a real number in the unit interval <math>0\le y\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, \cdots) = (b_1, b_2, \cdots),</math> one can see that <math>T(y)=2y\bmod 1.</math>  This map is called the [[dyadic transformation]]; for the doubly-infinite sequence of bits <math>\Omega=2^\mathbb{Z},</math> the induced homomorphism is the [[Baker's map]].

Consider now the space of functions in <math>y</math>. Given some <math>f(y)</math> one can find that
:<math>\left[\mathcal{L}_T f\right](y) = \frac{1}{2}f\left(\frac{y}{2}\right)+\frac{1}{2}f\left(\frac{y+1}{2}\right)</math>

Restricting the action of the operator <math>\mathcal{L}_T</math> to functions that are on polynomials, one finds that it has a [[discrete spectrum]] given by 
:<math>\mathcal{L}_T B_n= 2^{-n}B_n</math>
where the <math>B_n</math> are the [[Bernoulli polynomials]]. Indeed, the Bernoulli polynomials obey the identity
:<math>\frac{1}{2}B_n\left(\frac{y}{2}\right)+\frac{1}{2}B_n\left(\frac{y+1}{2}\right) = 2^{-n}B_n(y)</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]].

===Odometer===
{{main article|Markov odometer}}
Another way to create a dynamical system is to define an [[Markov odometer|odometer]]. Informally, this is exactly what it sounds like: just "add one" to the first position, and let the odometer "roll over" by using [[carry bit]]s as the odometer rolls over. This is nothing more than base-two addition on the set of infinite strings.  Since addition forms a [[group (mathematics)]], and the Bernoulli process was already given a topology, above, this provides a simple example of a [[topological group]].

In this case, the transformation <math>T</math> is given by
:<math>T\left(1,\dots,1,0,X_{k+1},X_{k+2},\dots\right) = \left(0,\dots,0,1,X_{k+1},X_{k+2},\dots \right).</math> 

It leaves the Bernoulli measure invariant only for the special case of <math>p=1/2</math> (the "fair coin"); otherwise not. Thus, <math>T</math> is a [[measure preserving dynamical system]] in this case, otherwise, it is merely a [[conservative system]].