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?

### Like this:

Like Loading...

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 be a non-empty closed set. Let . The map is continuous.

Consider the map . 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 if and only if or , and thus if and only if .

LikeLiked by 1 person