As is well-known, the Brouwer fixed point theorem states that any continuous map from the unit disk in to itself has a fixed point. The standard proof uses the computation of the singular homology groups of spheres. The proof fails, and indeed this is no longer true, for more general compact spaces. However, the following result shows that there is a form of “approximate periodicity” that one can deduce using only elementary facts from general topology.
Consider a homeomorphism for a compact metric space, i.e. a discrete dynamical system. We will prove:
Theorem 1 (Birkhoff Recurrence Theorem) There exists and a sequence with as .
More can actually be said; I’ll return to this topic in the future. One doesn’t need to be a homeomorphism.
Before we prove this, we need an auxiliary notion. Say that a homeomorphism is minimal if for every , is dense in .
I claim that is minimal iff there is no proper closed with (such are called -invariant). This is straightforward. Indeed, if is not minimal, we can take . If there is such a -invariant , then for is not dense in .
Lemma 2 Let be a homeomorphism, compact. Then there is a -invariant such that is minimal.
Since is a bijection, the intersection of -invariant subsets is still -invariant. Partially order the collection of nonempty -invariant subsets by the relation if . Any chain has an upper bound then because is compact and by the previous remark, so the lemma follows at once from Zorn’s lemma.
We can now prove Birkhoff’s recurrence theorem. Choose to be -invariant and with minimal. Then any point of will do, I claim. This follows from the next lemma.
Lemma 3 Let be minimal, compact. Then for any , .
I claim that for any open, we have
Indeed, we have (for the complement is closed and -invariant), and by compactness one can write for some , . Apply to both sides now.
As a result, it follows that if is arbitrary, then for some , which proves the lemma. And thus the Birkhoff theorem.