Iterated function

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 number, the result of applying a given function is fed again in the function as input, and this process is repeated.

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

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

Let X be a set and f: X → X be a function.

Define f ⁿ as the n-th iterate of f, where n is a non-negative integer, by:

and

where id_X is the identity function on X and f○g denotes function composition. That is,

always associative.

Because the notation f ⁿ may refer to both iteration (composition) of the function f or exponentiation of the function f (the latter is commonly used in trigonometry), some mathematicians choose to write f °ⁿ for the n-th iterate of the function f.

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