Fixed points of the ball

Brouwer’s theorem shows that any map of the unit ball to itself has a fixed point. Evidently the collection of fixed points is a closed subset of the ball. What closed subsets are realized as the fixed point set of a continuous map of the ball into itself?

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Fixed points of the ball

One thought on “Fixed points of the ball

  1. Claim: every non-empty closed subset.
    By Brower’s theorem we know that such a set must be non empty and as already stated we know that the set must be closed.
    For the converse let F be a non-empty closed set. Let p \in F. The map f(x) = d(x,F) is continuous.
    Consider the map x \mapsto \frac{x-p}{f(x) + 1}+p = (1+f(x))^{-1} x + (1-(1+f(x))^{-1}) p. By the RHS it is clear that this is a well defined map, since the ball is convex. By the LHS it clear that this fixes a point x if and only if x = p or f(x) = 0, and thus if and only if x \in F.

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