Iterated function

In mathematics, iterated functions are the objects of study in fractals and dynamical systems. An iterated function is a function which is composed with itself, repeatedly.

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

Let X be a set and

<math>f:X\rightarrow X<math>

be a mapping. Define the n'--th iterate

<math>f^n<math>

of the map by

<math>f^0=\textrm{id}_X<math>

where <math>\textrm{id}_X<math> is the identity function on X, and

<math>f^{n+1} = f \circ f^n<math>.

In the above, <math>f \circ g<math> denotes the function composition of functions; that is, <math>(f \circ g)(x)=f(g(x))<math>. The sequence fⁿ is called a Picard sequence, named after Charles Emile Picard. For a fixed x in X, the sequence of values fⁿ(x) is called the orbit of x.

If fⁿ(x) = f^n+m(x) for some integer m, the orbit is called a periodic orbit. The smallest such value of m for a given x is called the period of the orbit.

If m=1, that is, if f(x) = x for some x in X, then x is called a fixed point of the iterated sequence. The set of fixed points is often denoted as Fix(f). There exist a number of fixed-point theorems that guarantee the existence of fixed points in various situations, including the Banach fixed point theorem and the Brouwer fixed point theorem.

Examples

Famous iterated functions include the Mandelbrot set and Iterated function systems.

If f is the action of a group element on a set, then the iterated function corresponds to a free group.

Means of study

Iterated functions can be studied with the Artin-Mazur zeta function and with transfer operators.

References

• Vasile I. Istratescu, Fixed Point Theory, An Introduction, D.Reidel, Holland (1981). ISBN 90–277–1224–7