As is well-known, the Brouwer fixed point theorem states that any continuous map from the unit disk in ${\mathbb{R}^n}$ 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 ${T: X \rightarrow X}$ for ${X}$ a compact metric space, i.e. a discrete dynamical system. We will prove:

Theorem 1 (Birkhoff Recurrence Theorem) There exists ${x \in X}$ and a sequence ${n_i \rightarrow \infty}$ with ${T^{n_i} x \rightarrow x}$ as ${i \rightarrow \infty}$.

More can actually be said; I’ll return to this topic in the future. One doesn’t need ${T}$ to be a homeomorphism.

Before we prove this, we need an auxiliary notion. Say that a homeomorphism ${T: X \rightarrow X}$ is minimal if for every ${x \in X}$, ${T^{\mathbb{Z}} x }$ is dense in ${X}$.

I claim that ${T}$ is minimal iff there is no proper closed ${E \subset X}$ with ${TE = E}$ (such ${E}$ are called ${T}$-invariant). This is straightforward. Indeed, if ${T}$ is not minimal, we can take ${E = \overline{ T^{\mathbb{Z}} x}}$. If there is such a ${T}$-invariant ${E}$, then ${T^{\mathbb{Z}} e}$ for ${e \in E}$ is not dense in ${X}$.

Lemma 2 Let ${T: X \rightarrow X}$ be a homeomorphism, ${X}$ compact. Then there is a ${T}$-invariant ${E \subset X}$ such that ${T|_E: E \rightarrow E}$ is minimal. (more…)