Fixed point (mathematics)

In mathematics, a fixed point (sometimes shortened to fixpoint, also known as an invariant point) of a function is an element of the function's domain that is mapped to itself by the function. That is to say, "c" is a fixed point of the function "f" if "f"("c") = "c". This means "f"("f"(..."f"("c")...)) = "f" "n"("c") = "c", an important terminating consideration when recursively computing "f". A set of fixed points is sometimes called a "fixed set".
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.
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 projective geometry, a fixed point of a projectivity has been called a double point.
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.
Attracting fixed points.
An "attracting fixed point" of a function "f" is a fixed point "x"0 of "f" such that for any value of "x" in the domain that is close enough to "x"0, the iterated function sequence
converges to "x"0. An expression of prerequisites and proof of the existence of such a solution is given by the Banach fixed-point theorem.
The natural cosine function ("natural" means in radians, not degrees or other units) has exactly one fixed point, which is attracting. In this case, "close enough" is not a stringent criterion at all—to demonstrate this, start with "any" real number and repeatedly press the "cos" key on a calculator (checking first that the calculator is in "radians" mode). It eventually converges to about 0.739085133, which is a fixed point. That is where the graph of the cosine function intersects the line formula_3.
Not all fixed points are attracting. For example, "x" = 0 is a fixed point of the function "f"("x") = 2"x", but iteration of this function for any value other than zero rapidly diverges. However, if the function "f" is continuously differentiable in an open neighbourhood of a fixed point "x"0, and formula_4, attraction is guaranteed.
Attracting fixed points are a special case of a wider mathematical concept of attractors.
An attracting fixed point is said to be a "stable fixed point" if it is also Lyapunov stable.
A fixed point is said to be a "neutrally stable fixed point" if it is Lyapunov stable but not attracting. The center of a linear homogeneous differential equation of the second order is an example of a neutrally stable fixed point.
Multiple attracting points can be collected in an "attracting fixed set".
Applications.
In many fields, equilibria or stability are fundamental concepts that can be described in terms of fixed points. Some examples follow.
Topological fixed point property.
A topological space formula_5 is said to have the "fixed point property" (FPP) if for any continuous function
there exists formula_7 such that formula_8.
The FPP is a topological invariant, i.e. is preserved by any homeomorphism. The FPP is also preserved by any retraction.
According to the Brouwer fixed-point theorem, every compact and convex subset of a Euclidean space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP. In 1932 Borsuk asked whether compactness together with contractibility could be a necessary and sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.
Generalization to partial orders: prefixpoint and postfixpoint.
The notion and terminology is generalized to a partial order. Let ≤ be a partial order over a set "X" and let "f": "X" → "X" be a function over "X". Then a prefixpoint (also spelled pre-fixpoint) of "f" is any "p" such that "f"("p") ≤ "p". Analogously a postfixpoint (or post-fixpoint) of "f" is any "p" such that "p" ≤ "f"("p"). One way to express the Knaster–Tarski theorem is to say that a monotone function on a complete lattice has a least fixpoint that coincides with its least prefixpoint (and similarly its greatest fixpoint coincides with its greatest postfixpoint). Prefixpoints and postfixpoints have applications in theoretical computer science.