Brouwer's Fixed Point Theorem | Research & Encyclopedia Articles

Brouwer's Fixed Point Theorem

The ball, Bⁿ, is the subset of Rⁿ (this is n-dimensional real space) equal to the set of all real n-tuples (x₁,..,x_n) such that the square root of (x₁² + ... + x_n²) is less than or equal to one. Brouwer's fixed point theorem states that if f is any continuous function from Bⁿ to Bⁿ then there is some point x in Bⁿ with the property that f(x) = x. x is called a fixed point of f. For example, B² is just the unit disk in the plane. The function f((x,y)) = (x * cos (a) - y * sin (a), x* sin (a) + y* cos (a)) for some angle a, is the map which rotates the disk by the angle a. It has one fixed point at (0,0). The geometric shape of the ball is not really important; the theorem is true for any polygon or polyhedron or indeed any topological space homeomorphic to Bⁿ.

Here is another example. Cut out a square piece of paper. Draw a grid on it and number each little square. Make a copy of it. Fold or crumple the copy without tearing and place it on top of the original. The theorem states that some little square on the original is directly below a little square on the copy that has the number as the square on the original. Of course, the two squares might not match up perfectly. The interesting thing is that it does not matter how small you make the little squares or how much the copy is crumpled.

Here is a proof sketch of Brouwer's theorem. Suppose that the theorem is false. Then there is a function, f say, from Bⁿ to Bⁿ with the property that for no point x of Bⁿ is it true that f(x) = x. Then for each x in Bⁿ draw the ray from f(x) to x. Let g(x) be the intersection point of this ray with S^n-1, which is the set of points (x₁,...,x_n) such that the square root of (x₁² + ... + x_n²) = 1. S^n-1 is called the n-1-dimensional sphere and it is the boundary of Bⁿ. Notice if x is in S^n-1 then g(x) = x. Hence g is a continuous map from Bⁿ to S^n-1 such that for every point x in S^n-1, g(x) = x. The fact is, such a function g cannot exist. The reason is that S^n-1 is 'contractible' in Bⁿ, i.e. it can be continuously deformed to a point. However, S^n-1 is not contractible inside itself. For example, the boundary circle of the unit disk can be shrunk continuously to the origin. However, a circle cannot be continuously deformed to a point in such a way that the deformation takes place inside the circle itself. To be precise, a contraction is a continuous map C from S^n-1 x [0,1] to Bⁿ such that C(x,0) = x for all x in S^n-1 and C(x,1) = C(y,1) for all x and y in S^n-1. If g exists, then the function H from S^n-1 x [0,1] to S^n-1 defined by H(x, t) = g(C(x, t)) is a contraction of S^n-1 in S^n-1. The fact that no such contraction of S^n-1 exists within S^n-1, can be proved by using a higher dimensional analogue of the fundamental group. Thus, H cannot exist. So g cannot exist either. This implies that the assumption that f exists is false. Hence Brouwer's theorem is not false. Therefore, it is true.

Brouwer rejected the validity of his proof for two reasons: first, it is nonconstructive because it does not show how to find any fixed points and second, it relies on the law of the excluded middle. This law states that a proposition is either true or false. The proof only shows that Brouwer's theorem is not false. According to the law of the excluded middle, the theorem must therefore be true. Brouwer, and the school of intuitionism, rejected the validity of that law and hence the validity of the above proof. No one has ever found a proof of Brouwer's theorem that avoids using the law of the excluded middle and many believe that so such proof exists.

Brouwer's theorem has inspired many other fixed point-theorems. These theorems are used to establish the existence of solutions to differential equations. Consider the differential equation dy/dx = F(x,y) for all x greater than or equal to zero and less than or equal to one with initial conditions y = 0 at x = 0. The solution y(x) satisfies y(x) = the integral from 0 to x of F(x, y(x)) with respect to x. We define an operation on continuous real functions of [0,1] called H by H(f) = the integral from 0 to x of F(x,f(x)) with respect to x. The solution to the differential equation is a fixed point of this operator, i.e. a function f such that H(f) = f. It is a fact that the space of continuous real functions on [0,1] satisfies the hypothesis of a fixed-point theorem similar to Brouwer's fixed point theorem so that the existence of a solution is guaranteed. Differential equations such as these are common in the calculus of variations and hydrodynamics.