Editing Dyadic transformation (section)

===Frobenius–Perron operator===
Denote the collection of all open sets on the Cantor set by <math>\mathcal{B}</math> and consider the set <math>\mathcal{F}</math> of all arbitrary functions <math>f:\mathcal{B}\to\mathbb{R}.</math> The shift <math>T</math> induces a [[pushforward measure|pushforward]] 
:<math>f\circ T^{-1}</math>
defined by <math>\left(f \circ T^{-1}\right)\!(x) = f(T^{-1}(x)).</math> This is again some function <math>\mathcal{B}\to\mathbb{R}.</math> In this way, 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>
This linear operator is called the [[transfer operator]] or the ''Ruelle–Frobenius–Perron operator''.  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]]. Again, <math>\mathcal{L}_T(\rho)= \rho</math> when <math>\rho(x)=1.</math>