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We now begin our study of measure-preserving systems $(X, {\mathcal X}, \mu, T)$, i.e. a probability space $(X, {\mathcal X}, \mu)$ together with a probability space isomorphism $T: (X, {\mathcal X}, \mu) \to (X, {\mathcal X}, \mu)$ (thus $T: X \to X$ is invertible, with T and $T^{-1}$ both being measurable, and $\mu(T^n E) = \mu(E)$ for all $E \in {\mathcal X}$ and all n). For various technical reasons it is convenient to restrict to the case when the $\sigma$-algebra ${\mathcal X}$ is separable, i.e. countably generated. One reason for this is as follows:

Exercise 1. Let $(X, {\mathcal X}, \mu)$ be a probability space with ${\mathcal X}$ separable. Then the Banach spaces $L^p(X, {\mathcal X}, \mu)$ are separable (i.e. have a countable dense subset) for every $1 \leq p < \infty$; in particular, the Hilbert space $L^2(X, {\mathcal X}, \mu)$ is separable. Show that the claim can fail for $p = \infty$. (We allow the $L^p$ spaces to be either real or complex valued, unless otherwise specified.) $\diamond$

Remark 1. In practice, the requirement that ${\mathcal X}$ be separable is not particularly onerous. For instance, if one is studying the recurrence properties of a function $f: X \to {\Bbb R}$ on a non-separable measure-preserving system $(X, {\mathcal X}, \mu, T)$, one can restrict ${\mathcal X}$ to the separable sub-$\sigma$-algebra ${\mathcal X}'$ generated by the level sets $\{ x \in X: T^n f(x) > q \}$ for integer n and rational q, thus passing to a separable measure-preserving system $(X, {\mathcal X}', \mu, T)$ on which f is still measurable. Thus we see that in many cases of interest, we can immediately reduce to the separable case. (In particular, for many of the theorems in this course, the hypothesis of separability can be dropped, though we won’t bother to specify for which ones this is the case.) $\diamond$

We are interested in the recurrence properties of sets $E \in {\mathcal X}$ or functions $f \in L^p(X, {\mathcal X}, \mu)$. The simplest such recurrence theorem is

Theorem 1. (Poincaré recurrence theorem) Let $(X,{\mathcal X},\mu,T)$ be a measure-preserving system, and let $E \in {\mathcal X}$ be a set of positive measure. Then $\limsup_{n \to +\infty} \mu( E \cap T^n E ) \geq \mu(E)^2$. In particular, $E \cap T^n E$ has positive measure (and is thus non-empty) for infinitely many n.

(Compare with Theorem 1 of Lecture 3.)

Proof. For any integer $N > 1$, observe that $\int_X \sum_{n=1}^N 1_{T^n E}\ d\mu = N \mu(E)$, and thus by Cauchy-Schwarz

$\int_X (\sum_{n=1}^N 1_{T^n E})^2\ d\mu \geq N^2 \mu(E)^2.$ (1)

The left-hand side of (1) can be rearranged as

$\sum_{n=1}^N \sum_{m=1}^N \mu( T^n E \cap T^m E ).$ (2)

On the other hand, $\mu( T^n E \cap T^m E) = \mu( E \cap T^{m-n} E )$. From this one easily obtains the asymptotic

$(2)\leq (\limsup_{n \to \infty} \mu( E \cap T^n E ) + o(1)) N^2,$ (3)

where o(1) denotes an expression which goes to zero as N goes to infinity. Combining (1), (2), (3) and taking limits as $N \to +\infty$ we obtain

$\limsup_{n \to \infty} \mu( E \cap T^n E ) \geq \mu(E)^2$ (4)

as desired. $\Box$

Remark 2. In classical physics, the evolution of a physical system in a compact phase space is given by a (continuous-time) measure-preserving system (this is Hamilton’s equations of motion combined with Liouville’s theorem). The Poincaré recurrence theorem then has the following unintuitive consequence: every collection E of states of positive measure, no matter how small, must eventually return to overlap itself given sufficient time. For instance, if one were to burn a piece of paper in a closed system, then there exist arbitrarily small perturbations of the initial conditions such that, if one waits long enough, the piece of paper will eventually reassemble (modulo arbitrarily small error)! This seems to contradict the second law of thermodynamics, but the reason for the discrepancy is because the time required for the recurrence theorem to take effect is inversely proportional to the measure of the set E, which in physical situations is exponentially small in the number of degrees of freedom (which is already typically quite large, e.g. of the order of the Avogadro constant). This gives more than enough opportunity for Maxwell’s demon to come into play to reverse the increase of entropy. (This can be viewed as a manifestation of the curse of dimensionality.) The more sophisticated recurrence theorems we will see later have much poorer quantitative bounds still, so much so that they basically have no direct significance for any physical dynamical system with many relevant degrees of freedom. $\diamond$

Exercise 2. Prove the following generalisation of the Poincaré recurrence theorem: if $(X, {\mathcal X}, \mu, T)$ is a measure-preserving system and $f \in L^1(X, {\mathcal X},\mu)$ is non-negative, then $\limsup_{n \to +\infty} \int_X f T^n f \geq (\int_X f\ d\mu)^2$. $\diamond$

Exercise 3. Give examples to show that the quantity $\mu(E)^2$ in the conclusion of Theorem 1 cannot be replaced by any smaller quantity in general, regardless of the actual value of $\mu(E)$. (Hint: use a Bernoulli system example.) $\diamond$

Exercise 4. Using the pigeonhole principle instead of the Cauchy-Schwarz inequality (and in particular, the statement that if $\mu(E_1) + \ldots + \mu(E_n) > 1$, then the sets $E_1,\ldots,E_n$ cannot all be disjoint), prove the weaker statement that for any set E of positive measure in a measure-preserving system, the set $E \cap T^n E$ is non-empty for infinitely many n. (This exercise illustrates the general point that the Cauchy-Schwarz inequality can be viewed as a quantitative strengthening of the pigeonhole principle.) $\diamond$

For this lecture and the next we shall study several variants of the Poincaré recurrence theorem. We begin by looking at the mean ergodic theorem, which studies the limiting behaviour of the ergodic averages $\frac{1}{N} \sum_{n=1}^N T^n f$ in various $L^p$ spaces, and in particular in $L^2$.