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Concept# Fixed point (mathematics)

Summary

hatnote|1=Fixed points in mathematics are not to be confused with other uses of "fixed point", or stationary points where math|1=f(x) = 0.
In mathematics, a fixed point (sometimes shortened to fixpoint), also known as an invariant point, is a value that does not change under a given transformation. Specifically for functions, a fixed point is an element that is mapped to itself by the function.
Formally, c is a fixed point of a function f if c belongs to both the domain and the codomain of f, and f(c) = c.
For example, if f is defined on the real numbers by
then 2 is a fixed point of f, because f(2) = 2.
Not all functions have fixed points: for example, f(x) = x + 1, has no fixed points, since x is never equal to x + 1 for any real number. In graphical terms, a fixed point x means the point (x, f(x)) is on the line y = x, or in other words the graph of f has a point in common with that line.
Fixed-point iteration
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. Specifically, given a function with the same domain and codomain, a point in the domain of , the fixed-point iteration is
which gives rise to the sequence of iterated function applications which is hoped to converge to a point . If is continuous, then one can prove that the obtained is a fixed point of .
Points that come back to the same value after a finite number of iterations of the function are called periodic points. A fixed point is a periodic point with period equal to one.
In algebra, for a group G acting on a set X with a group action , x in X is said to be a fixed point of g if .
The fixed-point subgroup of an automorphism f of a group G is the subgroup of G:
Similarly the fixed-point subring of an automorphism f of a ring R is the subring of the fixed points of f, that is,
In Galois theory, the set of the fixed points of a set of field automorphisms is a field called the fixed field of the set of automorphisms.
Fixed-point property
A topological space is said to have the fixed point property (FPP) if for any continuous function
there exists such that .

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