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In this final lecture, we establish a Ratner-type theorem for actions of the special linear group $SL_2({\Bbb R})$ on homogeneous spaces. More precisely, we show:

Theorem 1. Let G be a Lie group, let $\Gamma < G$ be a discrete subgroup, and let $H \leq G$ be a subgroup isomorphic to $SL_2({\Bbb R})$. Let $\mu$ be an H-invariant probability measure on $G/\Gamma$ which is ergodic with respect to H (i.e. all H-invariant sets either have full measure or zero measure). Then $\mu$ is homogeneous in the sense that there exists a closed connected subgroup $H \leq L \leq G$ and a closed orbit $Lx \subset G/\Gamma$ such that $\mu$ is L-invariant and supported on Lx.

This result is a special case of a more general theorem of Ratner, which addresses the case when H is generated by elements which act unipotently on the Lie algebra ${\mathfrak g}$ by conjugation, and when $G/\Gamma$ has finite volume. To prove this theorem we shall follow an argument of Einsiedler, which uses many of the same ingredients used in Ratner’s arguments but in a simplified setting (in particular, taking advantage of the fact that H is semisimple with no non-trivial compact factors). These arguments have since been extended and made quantitative by Einsiedler, Margulis, and Venkatesh.
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The last two lectures of this course will be on Ratner’s theorems on equidistribution of orbits on homogeneous spaces. Due to lack of time, I will not be able to cover all the material here that I had originally planned; in particular, for an introduction to this family of results, and its connections with number theory, I will have to refer readers to my previous blog post on these theorems. In this course, I will discuss two special cases of Ratner-type theorems. In this lecture, I will talk about Ratner-type theorems for discrete actions (of the integers ${\Bbb Z})$ on nilmanifolds; this case is much simpler than the general case, because there is a simple criterion in the nilmanifold case to test whether any given orbit is equidistributed or not. Ben Green and I had need recently to develop quantitative versions of such theorems for a number-theoretic application. In the next and final lecture of this course, I will discuss Ratner-type theorems for actions of $SL_2({\Bbb R})$, which is simpler in a different way (due to the semisimplicity of $SL_2({\Bbb R})$, and lack of compact factors).

In this lecture – the final one on general measure-preserving dynamics – we put together the results from the past few lectures to establish the Furstenberg-Zimmer structure theorem for measure-preserving systems, and then use this to finish the proof of the Furstenberg recurrence theorem.

Having studied compact extensions in the previous lecture, we now consider the opposite type of extension, namely that of a weakly mixing extension. Just as compact extensions are “relative” versions of compact systems, weakly mixing extensions are “relative” versions of weakly mixing systems, in which the underlying algebra of scalars ${\Bbb C}$ is replaced by $L^\infty(Y)$. As in the case of unconditionally weakly mixing systems, we will be able to use the van der Corput lemma to neglect “conditionally weakly mixing” functions, thus allowing us to lift the uniform multiple recurrence property (UMR) from a system to any weakly mixing extension of that system.

To finish the proof of the Furstenberg recurrence theorem requires two more steps. One is a relative version of the dichotomy between mixing and compactness: if a system is not weakly mixing relative to some factor, then that factor has a non-trivial compact extension. This will be accomplished using the theory of conditional Hilbert-Schmidt operators in this lecture. Finally, we need the (easy) result that the UMR property is preserved under limits of chains; this will be accomplished in the next lecture.

In Lecture 11, we studied compact measure-preserving systems – those systems $(X, {\mathcal X}, \mu, T)$ in which every function $f \in L^2(X, {\mathcal X}, \mu)$ was almost periodic, which meant that their orbit $\{ T^n f: n \in {\Bbb Z}\}$ was precompact in the $L^2(X, {\mathcal X}, \mu)$ topology. Among other things, we were able to easily establish the Furstenberg recurrence theorem (Theorem 1 from Lecture 11) for such systems.

In this lecture, we generalise these results to a “relative” or “conditional” setting, in which we study systems which are compact relative to some factor $(Y, {\mathcal Y}, \nu, S)$ of $(X, {\mathcal X}, \mu, T)$. Such systems are to compact systems as isometric extensions are to isometric systems in topological dynamics. The main result we establish here is that the Furstenberg recurrence theorem holds for such compact extensions whenever the theorem holds for the base. The proof is essentially the same as in the compact case; the main new trick is to not to work in the Hilbert spaces $L^2(X,{\mathcal X},\mu)$ over the complex numbers, but rather in the Hilbert module $L^2(X,{\mathcal X},\mu|Y, {\mathcal Y}, \nu)$ over the (commutative) von Neumann algebra $L^\infty(Y,{\mathcal Y},\nu)$. (Modules are to rings as vector spaces are to fields.) Because of the compact nature of the extension, it turns out that results from topological dynamics (and in particular, van der Waerden’s theorem) can be exploited to good effect in this argument.

[Note: this operator-algebraic approach is not the only way to understand these extensions; one can also proceed by disintegrating $\mu$ into fibre measures $\mu_y$ for almost every $y \in Y$ and working fibre by fibre. We will discuss the connection between the two approaches below.]

In the previous lecture, we studied the recurrence properties of compact systems, which are systems in which all measurable functions exhibit almost periodicity – they almost return completely to themselves after repeated shifting. Now, we consider the opposite extreme of mixing systems – those in which all measurable functions (of mean zero) exhibit mixing – they become orthogonal to themselves after repeated shifting. (Actually, there are two different types of mixing, strong mixing and weak mixing, depending on whether the orthogonality occurs individually or on the average; it is the latter concept which is of more importance to the task of establishing the Furstenberg recurrence theorem.)

We shall see that for weakly mixing systems, averages such as $\frac{1}{N} \sum_{n=0}^{N-1} T^n f \ldots T^{(k-1)n} f$ can be computed very explicitly (in fact, this average converges to the constant $(\int_X f\ d\mu)^{k-1}$). More generally, we shall see that weakly mixing components of a system tend to average themselves out and thus become irrelevant when studying many types of ergodic averages. Our main tool here will be the humble Cauchy-Schwarz inequality, and in particular a certain consequence of it, known as the van der Corput lemma.

As one application of this theory, we will be able to establish Roth’s theorem (the k=3 case of Szemerédi’s theorem).

The primary objective of this lecture and the next few will be to give a proof of the Furstenberg recurrence theorem (Theorem 2 from the previous lecture). Along the way we will develop a structural theory for measure-preserving systems.

The basic strategy of Furstenberg’s proof is to first prove the recurrence theorems for very simple systems – either those with “almost periodic” (or compact) dynamics or with “weakly mixing” dynamics. These cases are quite easy, but don’t manage to cover all the cases. To go further, we need to consider various combinations of these systems. For instance, by viewing a general system as an extension of the maximal compact factor, we will be able to prove Roth’s theorem (which is equivalent to the k=3 form of the Furstenberg recurrence theorem). To handle the general case, we need to consider compact extensions of compact factors, compact extensions of compact extensions of compact factors, etc., as well as weakly mixing extensions of all the previously mentioned factors.

In this lecture, we will consider those measure-preserving systems $(X, {\mathcal X}, \mu, T)$ which are compact or almost periodic. These systems are analogous to the equicontinuous or isometric systems in topological dynamics discussed in Lecture 6, and as with those systems, we will be able to characterise such systems (or more precisely, the ergodic ones) algebraically as Kronecker systems, though this is not strictly necessary for the proof of the recurrence theorem.

In this lecture, we describe the simple but fundamental Furstenberg correspondence principle which connects the “soft analysis” subject of ergodic theory (in particular, recurrence theorems) with the “hard analysis” subject of combinatorial number theory (or more generally with results of “density Ramsey theory” type). Rather than try to set up the most general and abstract version of this principle, we shall instead study the canonical example of this principle in action, namely the equating of the Furstenberg multiple recurrence theorem with Szemerédi’s theorem on arithmetic progressions.
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We continue our study of basic ergodic theorems, establishing the maximal and pointwise ergodic theorems of Birkhoff. Using these theorems, we can then give several equivalent notions of the fundamental concept of ergodicity, which (roughly speaking) plays the role in measure-preserving dynamics that minimality plays in topological dynamics. A general measure-preserving system is not necessarily ergodic, but we shall introduce the ergodic decomposition, which allows one to express any non-ergodic measure as an average of ergodic measures (generalising the decomposition of a permutation into disjoint cycles).

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 $L^2$.