# Fixed point (mathematics)

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

${\displaystyle f(x)=x^{2}-3x+4,}$
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 ${\displaystyle f}$ with the same domain and codomain, a point ${\displaystyle x_{0}}$ in the domain of ${\displaystyle f}$, the fixed-point iteration is

${\displaystyle x_{n+1}=f(x_{n}),\,n=0,1,2,\dots }$

which gives rise to the sequence ${\displaystyle x_{0},x_{1},x_{2},\dots }$ of iterated function applications ${\displaystyle x_{0},f(x_{0}),f(f(x_{0})),\dots }$ which is hoped to converge to a point ${\displaystyle x}$. If ${\displaystyle f}$ is continuous, then one can prove that the obtained ${\displaystyle x}$ is a fixed point of ${\displaystyle f}$.

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.

## Fixed point of a group action

In algebra, for a group G acting on a set X with a group action ${\displaystyle \cdot }$, x in X is said to be a fixed point of g if ${\displaystyle g\cdot x=x}$.

The fixed-point subgroup ${\displaystyle G^{f}}$ of an automorphism f of a group G is the subgroup of G:

${\displaystyle G^{f}=\{g\in G\mid f(g)=g\}.}$

Similarly the fixed-point subring ${\displaystyle R^{f}}$ of an automorphism f of a ring R is the subring of the fixed points of f, that is,

${\displaystyle R^{f}=\{r\in R\mid f(r)=r\}.}$

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

A topological space ${\displaystyle X}$ is said to have the fixed point property (FPP) if for any continuous function

${\displaystyle f\colon X\to X}$

there exists ${\displaystyle x\in X}$ such that ${\displaystyle f(x)=x}$.

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.[1]

## 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).[2] The opposite usage occasionally appears.[3] 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."[4] A fixed point is a point that is both a prefixpoint and a postfixpoint. Prefixpoints and postfixpoints have applications in theoretical computer science.[5]

### 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).[6]

## Fixed-point combinator

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

${\displaystyle {\textsf {fix}}\ f=f\ ({\textsf {fix}}\ f)\ ,}$

## Fixed-point logics

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.

## Fixed-point theorems

A fixed-point theorem is a result saying that at least one fixed point exists, under some general condition.[7]

## Applications

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

## Notes

1. ^ 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.
2. ^ 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.
3. ^ 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.
4. ^ 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.
5. ^ Yde Venema (2008) Lectures on the Modal μ-calculus Archived March 21, 2012, at the Wayback Machine
6. ^ Yde Venema (2008) Lectures on the Modal μ-calculus Archived March 21, 2012, at the Wayback Machine
7. ^ Brown, R. F., ed. (1988). Fixed Point Theory and Its Applications. American Mathematical Society. ISBN 0-8218-5080-6.
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. ^