The iterated function to be studied is a map for an arbitrary set .

The formal definition of an iterated function on a set X follows.

Two important functionals can be defined in terms of iterated functions.

In numerical analysis, fixed-point iteration is a method of computing fixed points of iterated functions.

Fixed-point iteration, a general method to compute the fixed point of an iterated function.

Repeated composition of a function with itself is called iterated function.

Complex dynamics - the study of dynamical systems defined by iterated functions on complex number spaces.

This iterated function forms a dynamical system.

A similar notation is used for iterated functions.

The following provides a definition for the case of a system that is either an iterated function or has discrete-time dynamics.