Editing Bernoulli process (section)

===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.