So, now it’s time to connect the topological notions of dynamical systems with ergodic theory (which makes use of measures).  Our first example will use the notion of topological transitivity, which we now introduce.  The next example will return to the story about recurrent points, which I talked a bit about yesterday.

Say that a homeomorphism ${T: X \rightarrow X}$ of a compact metric space ${X}$ is topologically transitive if there exists ${x \in X}$ with ${T^{\mathbb{Z}}x}$ dense in ${X}$.  (For instance, a minimal homeomorphism is obviously topologically transitive.)  Let ${\{ U_n \}}$ be a countable basis for the topology of ${X}$. Then the set of all such ${x}$ (with ${T^{\mathbb{Z}}x}$ dense) is given by $\displaystyle \bigcap_n \bigcup_{i \in \mathbb{Z}} T^i U_n.$

In particular, if it is nonempty, then each ${\bigcup_{i \in \mathbb{Z}} T^i U_n}$ is dense—being ${T}$-invariant and containing ${U_n}$—and this set is a dense ${G_{\delta}}$ by Baire’s theorem.

Proposition 1 Let ${X}$ have a Borel probability measure ${\mu}$ positive on every nonempty open set, and let ${T: X \rightarrow X}$ be measure-preserving and ergodic. Then the set of ${x \in X}$ with ${\overline{T^{\mathbb{Z}}x}=X}$ is of measure 1, so ${T}$ is topologically transitive.

Indeed, each ${\bigcup_{i \in \mathbb{Z}} T^i U_n}$ must have measure zero or one by ergodicity, so measure 1 by hypothesis. Then take the countable intersection.

Poincaré recurrence

We now move to the abstract measure-theoretic framework, not topological.

Theorem 2 (Poincaré recurrence) Let ${T: X \rightarrow X}$ be a measure-preserving transformation on a probability space ${X}$. If ${E \subset X}$ is measurable, then there exists ${F \subset E}$ with ${\mu(E-F)=0}$ such that for each ${x \in F}$, there is a sequence ${n_i \rightarrow \infty}$ with ${T^{n_i} x \in E}$.

In other words, points of ${F}$ are ${T}$-frequently in ${E}$.

To see this, consider the sets $\displaystyle F_N := \bigcup_{i \geq N} T^{-i} E .$

Then ${F_0 \supset E}$ and ${\mu(F_0)=\mu(F_N)}$ since ${T^{-1} F_N = F_{N+1}}$. Set ${F' = \bigcap_N F_N}$ and ${F = E \cap F'}$. Then $\displaystyle E - F \subset E - F',$

and this latter set is the limit of the increasing sequence of sets ${E-F_N \subset F_0 - F_N}$, each of which has measure zero. This proves the theorem.

Topological application of Poincaré recurrence

Let ${X}$ be a compact metric space, ${T: X \rightarrow X}$ a continuous map. Say that ${x \in X}$ is ${T}$-recurrent if there exists a sequence ${n_i \rightarrow \infty}$ with ${T^{n_i} x \rightarrow x}$. Birkhoff’s theorem states that there always exists a ${T}$-recurrent point if ${T}$ is a homeomorphism.

Theorem 3 If ${T}$ preserves the probability measure ${\mu}$, then almost all points of ${X}$ are ${T}$-recurrent.

Let ${\{U_n\}}$ be a countable basis. Then ${x}$ is not ${T}$-recurrent if ${x \in U_n}$ for some ${n}$ and ${T^{\mathbb{N}} \cap U_n = \emptyset}$, as is easily seen (check the two cases ${x}$ a periodic point and ${x}$ non-periodic separately). So $\displaystyle \mathrm{non} \ \mathrm{recurrent} \ \mathrm{points} = \bigcup \left( U_n - \bigcup_{i \in \mathbb{N}} T^{-i} U_n \right).$

However, each set in this union has ${\mu}$-measure zero by Poincaré’s theorem, which proves this one.

We will show that any continuous ${T}$ has a probability measure ${\mu}$ with respect to which ${T}$ is measure-preserving, so

Corollary 4 If ${T}$ is a continuous transformation of the compact metric space ${X}$, then ${T}$-recurrent points exist.