Fixed-point property

A mathematical object ''X'' has the fixed-point property if every suitably well-behaved mapping from ''X'' to itself has a fixed point. The term is most commonly used to describe topological spaces on which every continuous mapping has a fixed point. But another use is in order theory, where a partially ordered set ''P'' is said to have the fixed point property if every increasing function on ''P'' has a fixed point.

Definition

Let ''A'' be an object in the concrete category C. Then ''A'' has the ''fixed-point property'' if every morphism (i.e., every function) $f: A \to A$ has a fixed point.

The most common usage is when C = Top is the category of topological spaces. Then a topological space ''X'' has the fixed-point property if every continuous map $f: X \to X$ has a fixed point.

Examples

Singletons

In the category of sets, the objects with the fixed-point property are precisely the singletons.

The closed interval

The closed interval ,1has the fixed point property: Let ''f'': ,1→ ,1be a continuous mapping. If ''f''(0) = 0 or ''f''(1) = 1, then our mapping has a fixed point at 0 or 1. If not, then ''f''(0) > 0 and ''f''(1) − 1 < 0. Thus the function ''g''(''x'') = ''f''(''x'') − x is a continuous real valued function which is positive at ''x'' = 0 and negative at ''x'' = 1. By the intermediate value theorem, there is some point ''x''₀ with ''g''(''x''₀) = 0, which is to say that ''f''(''x''₀) − ''x''₀ = 0, and so ''x''₀ is a fixed point.

The open interval does ''not'' have the fixed-point property. The mapping ''f''(''x'') = ''x''² has no fixed point on the interval (0,1).

The closed disc

The closed interval is a special case of the closed disc, which in any finite dimension has the fixed-point property by the Brouwer fixed-point theorem.

Topology

A retract ''A'' of a space ''X'' with the fixed-point property also has the fixed-point property. This is because if $r: X \to A$ is a retraction and $f: A \to A$ is any continuous function, then the composition $i \circ f \circ r: X \to X$ (where $i: A \to X$ is inclusion) has a fixed point. That is, there is $x \in A$ such that $f \circ r(x) = x$. Since $x \in A$ we have that $r(x) = x$ and therefore $f(x) = x.$

A topological space has the fixed-point property if and only if its identity map is universal.

A product of spaces with the fixed-point property in general fails to have the fixed-point property even if one of the spaces is the closed real interval.

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. More generally, according to the Schauder-Tychonoff fixed point theorem every compact and convex subset of a locally convex topological vector 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 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.Kinoshita, S. On Some Contractible Continua without Fixed Point Property. ''Fund. Math.'' 40 (1953), 96–98

References

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*{{cite book , first = Bernd , last = Schröder , title = Ordered Sets , publisher = Birkhäuser Boston , year = 2002

Fixed points (mathematics)