Brouwer fixed-point theorem


Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. Brouwer. It states that for any continuous function mapping a nonempty compact convex set to itself, there is a point such that. The simplest forms of Brouwer's theorem are for continuous functions from a closed interval in the real numbers to itself or from a closed disk to itself. A more general form than the latter is for continuous functions from a nonempty convex compact subset of Euclidean space to itself.
Among hundreds of fixed-point theorems, Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics. In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the Jordan curve theorem, the hairy ball theorem, the invariance of dimension and the Borsuk–Ulam theorem. This gives it a place among the fundamental theorems of topology. The theorem is also used for proving deep results about differential equations and is covered in most introductory courses on differential geometry. It appears in unlikely fields such as game theory. In economics, Brouwer's fixed-point theorem and its extension, the Kakutani fixed-point theorem, play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners Kenneth Arrow and Gérard Debreu.
The theorem was first studied in view of work on differential equations by the French mathematicians around Henri Poincaré and Charles Émile Picard. Proving results such as the Poincaré–Bendixson theorem requires the use of topological methods. This work at the end of the 19th century opened into several successive versions of the theorem. The case of differentiable mappings of the -dimensional closed ball was first proved in 1910 by Jacques Hadamard and the general case for continuous mappings by Brouwer in 1911.

Statement

The theorem has several formulations, depending on the context in which it is used and its degree of generalization. The simplest is sometimes given as follows:
This can be generalized to an arbitrary finite dimension:
A slightly more general version is as follows:
An even more general form is better known under a different name:

Importance of the pre-conditions

The theorem holds only for functions that are endomorphisms and for nonempty sets that are compact and convex. The following examples show why the pre-conditions are important.

The function ''f'' as an endomorphism

Consider the function
with domain . The range of the function is . Thus, f is not an endomorphism.

Boundedness

Consider the function
which is a continuous function from to itself. As it shifts every point to the right, it cannot have a fixed point. The space is convex and closed, but not bounded.

Closedness

Consider the function
which is a continuous function from the open interval to itself. Since the point is not part of the interval, there is no point in the domain such that. The set is convex and bounded, but not closed. On the other hand, the function does have a fixed point in the closed interval, namely. The closed interval is compact, the open interval is not.

Convexity

Convexity is not strictly necessary for Brouwer's fixed-point theorem. Because the properties involved are invariant under homeomorphisms, Brouwer's fixed-point theorem is equivalent to forms in which the domain is required to be a closed unit ball. For the same reason it holds for every set that is homeomorphic to a closed ball.
The following example shows that Brouwer's fixed-point theorem does not work for domains with holes. Consider the function, which is a continuous function from the unit circle to itself. Since -x≠x holds for any point of the unit circle, f has no fixed point. The analogous example works for the n-dimensional sphere. The unit circle is closed and bounded, but it has a hole . The function f have a fixed point for the unit disc, since it takes the origin to itself.
A formal generalization of Brouwer's fixed-point theorem for "hole-free" domains can be derived from the Lefschetz fixed-point theorem.

Illustrations

The theorem has several "real world" illustrations. Here are some examples.
  1. Take two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up the other one and place it, in any fashion, on top of the first so that the crumpled paper does not reach outside the flat one. There will then be at least one point of the crumpled sheet that lies directly above its corresponding point of the flat sheet. This is a consequence of the n = 2 case of Brouwer's theorem applied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet immediately beneath it.
  2. Take an ordinary map of a country, and suppose that that map is laid out on a table inside that country. There will always be a "You are Here" point on the map which represents that same point in the country.
  3. In three dimensions a consequence of the Brouwer fixed-point theorem is that, no matter how much you stir a cocktail in a glass, when the liquid has come to rest, some point in the liquid will end up in exactly the same place in the glass as before you took any action, assuming that the final position of each point is a continuous function of its original position, that the liquid after stirring is contained within the space originally taken up by it, and that the glass maintain a convex volume. Ordering a cocktail shaken, not stirred defeats the convexity condition. In that case, the theorem would not apply, and thus all points of the liquid disposition are potentially displaced from the original state.

    Intuitive approach

Explanations attributed to Brouwer

The theorem is supposed to have originated from Brouwer's observation of a cup of gourmet coffee.
If one stirs to dissolve a lump of sugar, it appears there is always a point without motion.
He drew the conclusion that at any moment, there is a point on the surface that is not moving.
The fixed point is not necessarily the point that seems to be motionless, since the centre of the turbulence moves a little bit.
The result is not intuitive, since the original fixed point may become mobile when another fixed point appears.
Brouwer is said to have added: "I can formulate this splendid result different, I take a horizontal sheet, and another identical one which I crumple, flatten and place on the other. Then a point of the crumpled sheet is in the same place as on the other sheet."
Brouwer "flattens" his sheet as with a flat iron, without removing the folds and wrinkles. Unlike the coffee cup example, the crumpled paper example also demonstrates that more than one fixed point may exist. This distinguishes Brouwer's result from other fixed-point theorems, such as Stefan Banach's, that guarantee uniqueness.

One-dimensional case

In one dimension, the result is intuitive and easy to prove. The continuous function f is defined on a closed interval and takes values in the same interval. Saying that this function has a fixed point amounts to saying that its graph intersects that of the function defined on the same interval which maps x to x.
Intuitively, any continuous line from the left edge of the square to the right edge must necessarily intersect the green diagonal. To prove this, consider the function g which maps x to f − x. It is ≥ 0 on a and ≤ 0 on b. By the intermediate value theorem, g has a zero in ; this zero is a fixed point.
Brouwer is said to have expressed this as follows: "Instead of examining a surface, we will prove the theorem about a piece of string. Let us begin with the string in an unfolded state, then refold it. Let us flatten the refolded string. Again a point of the string has not changed its position with respect to its original position on the unfolded string."

History

The Brouwer fixed point theorem was one of the early achievements of algebraic topology, and is the basis of more general fixed point theorems which are important in functional analysis. The case n = 3 first was proved by Piers Bohl in 1904. It was later proved by L. E. J. Brouwer in 1909. Jacques Hadamard proved the general case in 1910, and Brouwer found a different proof in the same year. Since these early proofs were all non-constructive indirect proofs, they ran contrary to Brouwer's intuitionist ideals. Although the existence of a fixed point is not constructive in the sense of constructivism in mathematics, methods to approximate fixed points guaranteed by Brouwer's theorem are now known.

Before discovery

At the end of the 19th century, the old problem of the stability of the solar system returned into the focus of the mathematical community.
Its solution required new methods. As noted by Henri Poincaré, who worked on the three-body problem, there is no hope to find an exact solution: "Nothing is more proper to give us an idea of the hardness of the three-body problem, and generally of all problems of Dynamics where there is no uniform integral and the Bohlin series diverge."
He also noted that the search for an approximate solution is no more efficient: "the more we seek to obtain precise approximations, the more the result will diverge towards an increasing imprecision".
He studied a question analogous to that of the surface movement in a cup of coffee. What can we say, in general, about the trajectories on a surface animated by a constant flow? Poincaré discovered that the answer can be found in what we now call the topological properties in the area containing the trajectory. If this area is compact, i.e. both closed and bounded, then the trajectory either becomes stationary, or it approaches a limit cycle. Poincaré went further; if the area is of the same kind as a disk, as is the case for the cup of coffee, there must necessarily be a fixed point. This fixed point is invariant under all functions which associate to each point of the original surface its position after a short time interval t. If the area is a circular band, or if it is not closed, then this is not necessarily the case.
To understand differential equations better, a new branch of mathematics was born. Poincaré called it analysis situs. The French Encyclopædia Universalis defines it as the branch which "treats the properties of an object that are invariant if it is deformed in any continuous way, without tearing". In 1886, Poincaré proved a result that is equivalent to Brouwer's fixed-point theorem, although the connection with the subject of this article was not yet apparent. A little later, he developed one of the fundamental tools for better understanding the analysis situs, now known as the fundamental group or sometimes the Poincaré group. This method can be used for a very compact proof of the theorem under discussion.
Poincaré's method was analogous to that of Émile Picard, a contemporary mathematician who generalized the Cauchy–Lipschitz theorem. Picard's approach is based on a result that would later be formalised by another fixed-point theorem, named after Banach. Instead of the topological properties of the domain, this theorem uses the fact that the function in question is a contraction.