In Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. One reason is that... mathematics, iterated functions are the objects of study in A fractal is a geometric object which can be divided into parts, each of which is similar to the original object. Fractals are said to possess infinite detail, and are generally self-similar and independent of scale. In many cases a fractal can be generated by a repeating pattern, typically... fractals and In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. Types of dynamical systems A dynamical system is called discrete if time is measured in... dynamical systems. An iterated function is a function which is In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. The functions f: X → Y and g: Y →... composed with itself, repeatedly.

The formal definition of an iterated function on a This article is about sets in mathematics. For other meanings, see Set (disambiguation). Sets are one of the most important and fundamental concepts in modern mathematics. Basic set theory, having only been invented at the end of the 19th century, is now a ubiquitous part of mathematics education, being introduced... set X follows:

Let X be a set and

be a In mathematics and related technical fields, the term map or mapping is often a synonym for function. Along these lines, a partial map is a partial function, and a total map is a total function. In many specific branches of mathematics, the term is used for a function with a... mapping. Define the n'--th iterate

fⁿ

of the map by

f⁰ = id_X

where id_X is the An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. Formally, if M is a set, we define the identity function idM on M to be that function with domain and codomain M which satisfies... identity function on X, and

.

In the above, denotes the In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. The functions f: X → Y and g: Y →... function composition of functions; that is, . The sequence fⁿ is called a Picard sequence, named after Charles Emile Picard (July 24, 1856 - December 11, 1941) was a French mathematician. See: Picard theorem Picard variety Picard functor Picard-Lefschetz formula Picard-Lindelöf theorem Categories: People stubs | 1856 births | 1941 deaths | French mathematicians ... Charles Emile Picard. For a fixed x in X, the This is a page about mathematics. For other usages of sequence, see: sequence (non-mathematical). In mathematics, a sequence is a list of objects (or events) which have been arranged in a linear fashion; such that each member comes either before, or after, every other member, and the order of... sequence of values fⁿ(x) is called the In the study of dynamical systems, an orbit is the sequence generated by iterating a map. An orbit is called closed if this sequence is finite. In simple terms, this means that the orbit will repeat itself. Such an orbit may be periodic, meaning that the entire sequence repeats. Othewise... 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 See also fixed-point arithmetic. In mathematics, a fixed point of a function is a point that is mapped to itself by the function. For example, if f is defined on the real numbers by f(x) = x2 − 3x + 4, then 2 is a fixed point of f, because... fixed point of the iterated sequence. The set of fixed points is often denoted as Fix(f). There exist a number of In mathematics, a fixed-point theorem is a result saying that a function will have at least one fixed point, under some conditions on that can be stated in general term. Results of this kind are amongst the most generally useful in mathematics. The Banach fixed point theorem gives a... fixed-point theorems that guarantee the existence of fixed points in various situations, including the The Banach fixed point theorem is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces, and provides a constructive method to find those fixed points. The theorem is named after Stefan Banach (1892-1945... Banach fixed point theorem and the In mathematics, the Brouwer fixed point theorem states that every continuous function from the closed unit ball D n to itself has a fixed point. In this theorem, n is any positive integer, and the closed unit ball is the set of all points in Euclidean n-space Rn... Brouwer fixed point theorem.

Examples

Famous iterated functions include the A rendering of the Mandelbrot set: black points represent the stable points under the iterative map In mathematics, the Mandelbrot set is a fractal that is defined as the set of points c in the complex plane for which the iteratively defined sequence: does not tend to infinity. The sequence... Mandelbrot set and Menger sponge, created by using IFS. Iterated function systems or IFS, are a kind of fractal that was conceived in its present form by John Hutchinson in 1981 and popularized by Michael Barnsleys book Fractals Everywhere. IFS fractals as they are normally called can be of any number of... Iterated function systems.

If f is the In mathematics, groups are often used to describe symmetries of objects. This is formalized by the notion of a group action: every element of the group acts like a bijective map (or symmetry) on some set. In this case, the group is also called a transformation group of the set... action of a group element on a set, then the iterated function corresponds to a The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many... free group.

Means of study

Iterated functions can be studied with the In mathematics, the Artin-Mazur zeta function is a tool for studying the iterated functions that occur in dynamical systems and fractals. It is defined as the formal power series , where is the set of fixed points of the n_th iterate of an iterated function f, and is the cardinality... Artin-Mazur zeta function and with In mathematics, the transfer operator encodes information about an iterated map and is frequently used to study the behavior of dynamical systems, statistical mechanics, quantum chaos and fractals. The iterated function to be studied is a map for an arbitrary set . The transfer operator is defined as an operator acting... transfer operators.

References

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