TITLE: Can we actually find any fixed points with Brouwer's theorem?
QUESTION [47 upvotes]: Background
At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is usually functional analytic and imposes strong conditions on the map $f:X \to X$ whereas the second class is usually algebraic topological and imposes strong conditions on the space $X$ itself.
A typical example of the first class of theorems is the fixed point theorem of Banach. While the spaces it applies to are fairly general (complete metric spaces), the function must have a Lipschitz constant strictly less than $1$. On the other hand, Brouwer's theorem falls into the second class. Any continuous map works, but the domain must be a compact and convex subset of Euclidean space (originally a disk). Of course, both these theorems have been vastly generalized from the versions that I am stating here.
Question
One fundamental advantage of the Banach theorem is that it actually provides a recipe for converging to the fixed point as part of the standard proof: just start at an initial point and iterate. The proofs of the Brouwer theorem that I have seen do no such thing. The best known proof (I think) is the one by contradiction: assuming the domain is a disk, if $f(x)$ and $x$ are always distinct then the ray from $f(x)$ through $x$ to the boundary of said disk provides a deformation-retraction from the disk to its boundary, aha!
Here is my question:

Is there any way to actually find a fixed point when using Brouwer's theorem?

A Possible Idea
One scheme that unfortunately fails is as follows. Consider the sequence of iterates $f^n(x)$ for $n \in \mathbb{N}$ and any initial $x$ in the domain. We have an infinite sequence in a compact set, and hence a convergent subsequence, so the limit point is a candidate. This won't work since a) we haven't used convexity at all, and b) one may just be converging to a periodic orbit of $f$.

REPLY [17 votes]: The paper "Exponential lower bounds for finding Brouwer fixed points"
Addendum by original poster: It was non-trivial to find a copy of this great paper of Hirsch, Papadimitriou and Vavasis. It does answer my general question quite clearly: finding Brouwer fixed points is exponentially hard in the worst case no matter what algorithm you use. Here is a link to this paper for all those who are interested and don't want to run into many, many pay-walls. I will take it down in a few days. -VN