Editing Iterated function (section)

==Invariant measure==
If one considers the evolution of a density distribution, rather than that of individual point dynamics, then the limiting behavior is given by the [[invariant measure]]. It can be visualized as the behavior of a point-cloud or dust-cloud under repeated iteration. The invariant measure is an eigenstate of the Ruelle-Frobenius-Perron operator or [[transfer operator]], corresponding to an eigenvalue of 1. Smaller eigenvalues correspond to unstable, decaying states.

In general, because repeated iteration corresponds to a shift, the transfer operator, and its adjoint, the [[Koopman operator]] can both be interpreted as [[shift operator]]s action on a [[shift space]]. The theory of [[subshifts of finite type]] provides general insight into many iterated functions, especially those leading to chaos.