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 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: 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. The formal definition of an iterated function on a set X follows. Let X be a set and f: X → X be a function. Defining f n as the n-th iterate of f (a notation introduced by Hans Heinrich Bürmann and John Frederick William Herschel), where n is a non-negative integer, by: and where idX is the identity function on X and f g denotes function composition. That is, (f g)(x) = f (g(x)), 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 use ∘ to denote the compositional meaning, writing f^∘n(x) for the n-th iterate of the function f(x), as in, for example, f^∘3(x) meaning f(f(f(x))). For the same purpose, f n was used by Benjamin Peirce whereas Alfred Pringsheim and Jules Molk suggested ^nf(x) instead. In general, the following identity holds for all non-negative integers m and n, This is structurally identical to the property of exponentiation that aman = am + n, i.e. the special case f(x) = ax. In general, for arbitrary general (negative, non-integer, etc.) indices m and n, this relation is called the translation functional equation, cf. Schröder's equation and Abel equation. On a logarithmic scale, this reduces to the nesting property of Chebyshev polynomials, Tm(Tn(x)) = Tm n(x), since Tn(x) = cos(n arccos(x)).