Editing Transfer operator (section)

==Definition==
The iterated function to be studied is a map <math>f\colon X\rightarrow X</math> for an arbitrary set <math>X</math>. 

The transfer operator is defined as an operator <math>\mathcal{L}</math> acting on the space of functions <math>\{\Phi\colon X\rightarrow \mathbb{C}\}</math> as

:<math>(\mathcal{L}\Phi)(x) = \sum_{y\,\in\, f^{-1}(x)} g(y) \Phi(y)</math>

where <math>g\colon X\rightarrow\mathbb{C}</math> is an auxiliary valuation function. When <math>f</math> has a [[Jacobian matrix and determinant|Jacobian]] determinant <math>|J|</math>, then <math>g</math> is usually taken to be <math>g=1/|J|</math>.

The above definition of the transfer operator can be shown to be the point-set limit of the measure-theoretic [[Pushforward measure|pushforward]] of ''g'': in essence, the transfer operator is the [[direct image functor]] in the category of [[measurable space]]s.  The left-adjoint of the Perron&ndash;Frobenius operator is the [[Koopman operator]] or [[composition operator]]. The general setting is provided by the [[Borel functional calculus]].

As a general rule, the transfer operator can usually be interpreted as a (left-)[[shift operator]] acting on a [[shift space]]. The most commonly studied shifts are the [[subshifts of finite type]]. The adjoint to the transfer operator can likewise usually be interpreted as a right-shift. Particularly well studied right-shifts include the [[Jacobi operator]] and the [[Hessenberg matrix]], both of which generate systems of [[orthogonal polynomials]] via a right-shift.