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 2Let be a homeomorphism, compact. Then there is a -invariant such that is minimal. (more…)