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, a process called iteration.

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

Let X be a set and

$f:X\backslash rightarrow\; X$

be a function. Define the n 'th iterate

$f^n$

of f by

$f^0=\backslash textrm\{id\}\_X$

where $\backslash textrm\{id\}\_X$ is the identity function on X, and

$f^\{n+1\}\; =\; f\; \backslash circ\; f^n$.

In the above, $f\; \backslash circ\; g$ denotes function composition; that is, $(f\; \backslash circ\; g)(x)=f(g(x))$. The sequence fⁿ is called a Picard sequence, named after Charles Émile 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. The point x itself is called a periodic point.

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.

Upon iteration, one may find that there are sets that shrink and converge towards a single point. In such a case, the point that is converged to is known as an attractive fixed point. Conversely, iteration may give the appearance of points diverging away from a single point; this would be the case for an unstable fixed point.

When the points of the orbit converge to one or more limits, the set of accumulation points of the orbit is known as the limit set or the ω-limit set.

The ideas of attraction and repulsion generalize similarly; one may categorize iterates into stable sets and unstable sets, according to the behaviour of small neighborhoods under iteration.

The the idea of iteration can be generalized so that the iteration count n becomes a continuous parameter; in this case, such a system is called a flow.

If f and g are two iterated functions, and there exists a homeomorphism h such that $g=h^\{-1\}\backslash circ\; f\; \backslash circ\; h$, then f and g are said to be topologically conjugate. Clearly, topological conjugacy is preserved under iteration, as one has that $g^n=h^\{-1\}\backslash circ\; f^n\; \backslash circ\; h$, so that if one can solve one iterated function system, one has solutions for all topologically conjugate systems. For example, the tent map is topologically conjugate to the logistic map.

Examples

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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

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

See also

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• Rotation number
• Sarkovskii's theorem

References

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

Dynamical systems | Fractals | Sequences | Fixed points