In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function 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. Definition The formal definition of an iterated function on a set X follows. Let be a set and be a function. Defining as the n-th iterate of (a notation introduced by Hans Heinrich Bürmann and John Frederick William Herschel), where n is a non-negative integer, by: and where is the identity function on and denotes function composition. That is, , always associative. Because the notation may refer to both iteration (composition) of the function or exponentiation of the function (the latter is commonly used in trigonometry), some mathematicians choose to use to denote the compositional meaning, writing for the -th iterate of the function , as in, for example, meaning . For the same purpose, was used by Benjamin Peirce whereas Alfred Pringsheim and Jules Molk suggested instead. Abelian property and iteration sequences In general, the following identity holds for all non-negative integers and , This is structurally identical to the property of exponentiation that , i.e. the special case . In general, for arbitrary general (negative, non-integer, etc.) indices and , 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, , since . The relation also holds, analogous to the property of exponentiation that . The sequence of functions is called a Picard sequence, named after Charles Émile Picard. For a given in , the sequence of values is called the orbit of . If for some integer , the orbit is called a periodic orbit. The smallest such value of for a given is called the period of the orbit. The point itself is called a periodic point. The cycle detection problem in computer science is the algorithmic problem of finding the first periodic point in an orbit,