We will now apply the machinery already developed to a few concrete problems.

Proposition 1 Let ${G}$ be a compact abelian group and ${T}$ the rotation by ${a \in G}$. Then ${T}$ is uniquely ergodic (with the Haar measure invariant) if ${a^{\mathbb{Z}}}$ is dense in ${G}$.

The proof is straightforward. Suppose ${\mu}$ is invariant with respect to rotations by ${a}$. Then for ${f \in C(G)}$, we have

$\displaystyle \int f(a^m x ) d \mu = \int f(x) d \mu, \quad \forall m \in \mathbb{Z}$

and hence

$\displaystyle \int f(bx ) d \mu = \int f(x) d \mu, \quad \forall m \in \mathbb{Z},$

for any ${b \in G}$, which means that ${\mu}$ must be Haar measure (which is unique).

Corollary 2 An irrational rotation of the unit circle ${S^1}$ is uniquely ergodic.

Application: Equidistribution

Theorem 3 Let ${\xi \in \mathbb{R}}$ be irrational and let ${f: \mathbb{R} \rightarrow \mathbb{C}}$ be continuous and ${2 \pi }$-periodic. Then$\displaystyle \boxed{ \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{i=0}^{N-1} f( n \xi) = \int_0^1 f(x) dx .}$ (more…)

So, let’s fix a compact metric space ${X}$ and a transformation ${T: X \rightarrow X}$ which is continuous. We defined the space ${M(X,T)}$ of probability Borel measures which are ${T}$-invariant, showed it was nonempty, and proved that the extreme points correspond to ergodic measures (i.e. measures with respect to which ${T}$ is ergodic). We are interested in knowing what ${M(X,T)}$ looks like, based solely on the topological properties of ${T}$. Here are some techniques we can use:

1) If ${T}$ has no fixed points, then ${\mu \in M(X,T)}$ cannot have any atoms (i.e. ${\mu(\{x\})=0, x \in X}$). Otherwise ${\{x, Tx , T^2x, \dots \}}$ would have infinite measure.

2) The set of recurrent points in ${X}$ (i.e. ${x \in X}$ such that there exists a sequence ${n_i \rightarrow \infty}$ with ${T^{n_i}x \rightarrow x}$) has ${\mu}$-measure one. We proved this earlier.

3) The set of non-wandering points has measure one. We define this notion now. Say that ${x \in X}$ is wandering if there is a neighborhood ${U}$ of ${X}$ such that ${T^{-n}(U) \cap U = \emptyset, \forall n \in \mathbb{N}}$. In other words, the family of sets ${T^{i}(U), i \in \mathbb{Z}_{\geq 0}}$ is disjoint. If not, say that ${x}$ is non-wandering. Any recurrent point, for instance, is non-wandering, which implies that the set of non-wandering points has measure one.

Here is an example. (more…)

Up until now, we have concentrated on a transformation ${T}$ of a fixed measure space. We now take a different approach: ${T}$ is fixed, and we look for appropriate measures (on a fixed ${\sigma}$-algebra).  First, we will show that this space is nonempty.  Then we will characterize ergodicity in terms of extreme points.

This is the first theorem we seek to prove:

Theorem 1 Let ${T: X \rightarrow X}$ be a continuous transformation of the compact metric space ${X}$. Then there exists a probability Borel measure ${\mu}$ on ${X}$ with respect to which ${T}$ is measure-preserving.

Consider the Banach space ${C(X)}$ of continuous ${f: X \rightarrow \mathbb{C}}$ and the dual ${C(X)^*}$, which, by the Riesz representation theorem, is identified with the space of (complex) Borel measures on ${X}$. The positive measures of total mass one form a compact convex subset ${P}$ of ${C(X)^*}$ in the weak* topology by Alaoglu’s theorem. Now, ${T}$ induces a transformation of ${C(X)}$: ${f \rightarrow f \circ T}$. The adjoint transformation of ${C(X)^*}$ is given by ${\mu \rightarrow T^{-1}(\mu}$, where for a measure ${\mu}$, ${T^{-1}(\mu)(E) := \mu(T^{-1}E)}$. We want to show that ${T^*}$ has a fixed point on ${P}$; then we will have proved the theorem.

There are fancier methods in functional analysis one could use, but to finish the proof we will appeal to the simple

Lemma 2 Let ${C}$ be a compact convex subset of a locally convex space ${X}$, and let ${T: C \rightarrow C}$ be the restriction of a continuous linear map on ${X}$. Then ${T}$ has a fixed point in ${C}$. (more…)

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 ${E}$. (more…)

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…)

Ergodicity

Let ${(X, \mu)}$ be a probability space and ${T: X \rightarrow X}$ a measure-preserving transformation. In many cases, it turns out that the averages of a function ${f}$ given by

$\displaystyle \frac{1}{N} \sum_{i=0}^{N-1} f \circ T^i$

actually converge a.e. to a constant.

This is the case if ${T}$ is ergodic, which we define as follows: ${T}$ is ergodic if for all ${E \subset X}$ with ${T^{-1}E = E}$, ${m(E)=1}$ or ${0}$. This is a form of irreducibility; the system ${X,T}$ has no smaller subsystem (disregarding measure zero sets). It is easy to see that this is equivalent to the statement: ${f}$ measurable (one could assume measurable and bounded if one prefers) and ${T}$-invariant implies ${f}$ constant a.e. (One first shows that if ${T}$ is ergodic, then ${\mu(T^{-1}E \Delta E )}$ implies ${\mu(E)=0,1}$, by constructing something close to ${E}$ that is ${T}$-invariant.)

In this case, therefore, the ergodic theorem takes the following form. Let ${f: X \rightarrow \mathbb{C}}$ be integrable. Then almost everywhere,

$\displaystyle \boxed{ \frac{1}{N} \sum_{i=0}^{N-1} f ( T^i (x)) \rightarrow \int_X f d\mu .}$

This is a very useful fact, and it has many applications. (more…)

Let ${X}$ be a measure space with measure ${\mu}$; let ${T: X \rightarrow X}$ be a measure-preserving transformation. Last time we looked at how the averages

$\displaystyle A_N := \frac{1}{N} \sum_{i=0}^{N-1} f \circ T^i$

behave in ${L^2}$. But, now we want pointwise convergence.

The pointwise ergodic theorem

We consider the pointwise ergodic theorem of Garrett George Birkhoff:

Theorem 1 (Birkhoff) Let ${f \in L^1(\mu)}$. Then the averages ${A_N}$ converge almost everywhere to a function ${f^* \in L^1(\mu)}$ with ${f^* \circ = f^*}$ a.e. (more…)

So, what’s ergodic theory all about?

The idea is that we are given a system together with some operation ${T}$ on it. For instance, ${T}$ could be a homeomorphism of a topological space, i.e. a discrete dynamical system. We are interested in studying the iterates of this process. In many case, averaging over the iterates of this process yields in the limit something that is actually invariant under this process.

For instance, suppose ${T}$ is a measure-preserving transformation of a measure space ${(X, \mu)}$ (which means if ${E}$ is measurable, then so is ${T^{-1}(E)}$ and ${\mu(T^{-1}(E)) = \mu(E)}$). How might one arise? Well, suppose ${M}$ is a compact symplectic manifold, ${X}$ a Hamiltonian vector field, and ${dV}$ the volume form. Then the flows ${\phi_t: M \rightarrow M}$ of ${X}$ leave invariant the volume form ${dV}$, so any such diffeomorphism ${\phi_t}$ is a measure-preserving transformation of the measure induced by the volume form. Anyway, back to the general story. Then the action of ${T}$ on a function ${f}$ is given by

$\displaystyle Tf(x) := f(T(x)).$

The Birkhoff ergodic theorem states that the averages

$\displaystyle \frac{1}{N} \sum_{i=0}^{N-1} T^i f$

converge a.e. to a function invariant under ${T}$, provided ${f \in L^1(\mu)}$. In many interesting cases, the invariant limit will actually be constant a.e. For instance, this is guaranteed if the transformation ${T}$ is ergodic, i.e. has no nontrivial invariant subsets. There are many spectacular applications to number theory of this result, e.g. the existence of Khintchine’s constant. Cf. also this post of Harrison Brown. (more…)