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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Claim: every non-empty closed subset.
be a non-empty closed set. Let
. The map
is continuous.
. 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
.
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
Consider the map
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