In mathematics, an iterated function is a function X → X (that is, a function from some set X to itself) which is obtained by composing another function f : X → X with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial object, the result of applying a given function is fed again in the function as input, and this process is repeated. For example on the image on the right:

L = ${\displaystyle {\mathit {F}}\,}$( K ),   M = ${\displaystyle {\mathit {F}}\,\circ {\mathit {F}}\,}$( K ) = ${\displaystyle {\mathit {F}}\;^{2}\,}$( K ),
with the circle‑shaped symbol of function composition.

Iterated functions are objects of study in computer science, fractals, dynamical systems, mathematics and renormalization group physics.