A function with three fixed points

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.

Fixed point of a 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

Main article: 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 .

The notions of attracting fixed points, repelling fixed points, and periodic points are defined with respect to fixed-point iteration.

Fixed-point theorems

Main article: Fixed-point theorems

A fixed-point theorem is a result saying that at least one fixed point exists, under some general condition.[1]

For example, the Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, fixed-point iteration will always converge to a fixed point.

The Brouwer fixed-point theorem (1911) says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point.

The Lefschetz fixed-point theorem (and the Nielsen fixed-point theorem) from algebraic topology give a way to count fixed points.

Fixed point of a group action

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.

Topological fixed point property

Main article: Fixed-point property

A topological space is said to have the fixed point property (FPP) if for any continuous function

there exists such that .

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.[2]

Fixed points of partial orders

In domain theory, the notion and terminology of fixed points is generalized to a partial order. Let ≤ be a partial order over a set X and let f: XX be a function over X. Then a prefixed point (also spelled pre-fixed point, sometimes shortened to prefixpoint or pre-fixpoint)[citation needed] of f is any p such that f(p) ≤ p. Analogously, a postfixed point of f is any p such that pf(p).[3] The opposite usage occasionally appears.[4] Malkis justifies the definition presented here as follows: "since f is before the inequality sign in the term f(x) ≤ x, such x is called a prefix point."[5] A fixed point is a point that is both a prefixpoint and a postfixpoint. Prefixpoints and postfixpoints have applications in theoretical computer science.[6]

Least fixed point

Main article: Least fixed point

In order theory, the least fixed point of a function from a partially ordered set (poset) to itself is the fixed point which is less than each other fixed point, according to the order of the poset. A function need not have a least fixed point, but if it does then the least fixed point is unique.

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).[7]

Fixed-point combinator

Main article: Fixed point combinator

In combinatory logic for computer science, a fixed-point combinator is a higher-order function that returns a fixed point of its argument function, if one exists. Formally, if the function f has one or more fixed points, then

Fixed-point logics

Main article: Fixed-point logic

In mathematical logic, fixed-point logics are extensions of classical predicate logic that have been introduced to express recursion. Their development has been motivated by descriptive complexity theory and their relationship to database query languages, in particular to Datalog.

Applications

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In many fields, equilibria or stability are fundamental concepts that can be described in terms of fixed points. Some examples follow.

See also

Notes

  1. ^ Brown, R. F., ed. (1988). Fixed Point Theory and Its Applications. American Mathematical Society. ISBN 0-8218-5080-6.
  2. ^ Kinoshita, S. (1953). "On Some Contractible Continua without Fixed Point Property". Fund. Math. 40 (1): 96–98. doi:10.4064/fm-40-1-96-98. ISSN 0016-2736.
  3. ^ Smyth, Michael B.; Plotkin, Gordon D. (1982). "The Category-Theoretic Solution of Recursive Domain Equations" (PDF). Proceedings, 18th IEEE Symposium on Foundations of Computer Science. SIAM Journal of Computing (volume 11). pp. 761–783. doi:10.1137/0211062.
  4. ^ Patrick Cousot; Radhia Cousot (1979). "Constructive Versions of Tarski's Fixed Point Theorems" (PDF). Pacific Journal of Mathematics. 82 (1): 43–57. doi:10.2140/pjm.1979.82.43.
  5. ^ Malkis, Alexander (2015). "Multithreaded-Cartesian Abstract Interpretation of Multithreaded Recursive Programs Is Polynomial" (PDF). Reachability Problems. Lecture Notes in Computer Science. 9328: 114–127. doi:10.1007/978-3-319-24537-9_11. ISBN 978-3-319-24536-2. S2CID 17640585. Archived from the original (PDF) on 2022-08-10.
  6. ^ Yde Venema (2008) Lectures on the Modal μ-calculus Archived March 21, 2012, at the Wayback Machine
  7. ^ Yde Venema (2008) Lectures on the Modal μ-calculus Archived March 21, 2012, at the Wayback Machine
  8. ^ Coxeter, H. S. M. (1942). Non-Euclidean Geometry. University of Toronto Press. p. 36.
  9. ^ G. B. Halsted (1906) Synthetic Projective Geometry, page 27
  10. ^ Wilson, Kenneth G. (1971). "Renormalization Group and Critical Phenomena. I. Renormalization Group and the Kadanoff Scaling Picture". Physical Review B. 4 (9): 3174–3183. Bibcode:1971PhRvB...4.3174W. doi:10.1103/PhysRevB.4.3174.
  11. ^ Wilson, Kenneth G. (1971). "Renormalization Group and Critical Phenomena. II. Phase-Space Cell Analysis of Critical Behavior". Physical Review B. 4 (9): 3184–3205. Bibcode:1971PhRvB...4.3184W. doi:10.1103/PhysRevB.4.3184.
  12. ^ "P. Cousot & R. Cousot, Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints".