Category Archive

In this final lecture, we establish a Ratner-type theorem for actions of the special linear group on homogeneous spaces. More precisely, we show:

Theorem 1. Let G be a Lie group, let be a discrete subgroup, and let be a subgroup isomorphic to . Let be an H-invariant probability measure on which is ergodic with respect to H (i.e. all H-invariant sets either have full measure or zero measure). Then is homogeneous in the sense that there exists a closed connected subgroup and a closed orbit such that 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 by conjugation, and when 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 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 , which is simpler in a different way (due to the semisimplicity of , and lack of compact factors).

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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.

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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 is replaced by . 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.

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In Lecture 11, we studied compact measure-preserving systems - those systems in which every function was almost periodic, which meant that their orbit was precompact in the 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 of . 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 over the complex numbers, but rather in the Hilbert module over the (commutative) von Neumann algebra . (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 into fibre measures for almost every and working fibre by fibre. We will discuss the connection between the two approaches below.]

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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 can be computed very explicitly (in fact, this average converges to the constant ). 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).

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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 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.

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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).

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

Exercise 1. Let be a probability space with separable. Then the Banach spaces are separable (i.e. have a countable dense subset) for every ; in particular, the Hilbert space is separable. Show that the claim can fail for . (We allow the spaces to be either real or complex valued, unless otherwise specified.)

Remark 1. In practice, the requirement that be separable is not particularly onerous. For instance, if one is studying the recurrence properties of a function on a non-separable measure-preserving system , one can restrict to the separable sub--algebra generated by the level sets for integer n and rational q, thus passing to a separable measure-preserving system 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.)

We are interested in the recurrence properties of sets or functions . The simplest such recurrence theorem is

Theorem 1. (Poincaré recurrence theorem) Let be a measure-preserving system, and let be a set of positive measure. Then . In particular, has positive measure (and is thus non-empty) for infinitely many n.

(Compare with Theorem 1 of Lecture 3.)

Proof. For any integer , observe that , and thus by Cauchy-Schwarz

(1)

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

(2)

On the other hand, . From this one easily obtains the asymptotic

(3)

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

. (4)

By shift-invariance we have , and the claim follows.

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.

Exercise 2. Prove the following generalisation of the Poincaré recurrence theorem: if is a measure-preserving system and is non-negative, then .

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

Exercise 4. Using the pigeonhole principle instead of the Cauchy-Schwarz inequality (and in particular, the statement that if , then the sets cannot all be disjoint), prove the weaker statement that for any set E of positive measure in a measure-preserving system, the set 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.)

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 in various spaces, and in particular in .

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In our final lecture on topological dynamics, we discuss a remarkable theorem of Furstenberg that classifies a major type of topological dynamical system - distal systems - in terms of highly structured (from an algebraic point of view) systems, namely towers of isometric extensions. This theorem is also a model for an important analogous result in ergodic theory, the Furstenberg-Zimmer structure theorem, which we will turn to in a few lectures. We will not be able to prove Furstenberg’s structure theorem for distal systems here in full, but we hope to illustrate some of the key points and ideas.

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In this lecture, we move away from recurrence, and instead focus on the structure of topological dynamical systems. One remarkable feature of this subject is that starting from fairly “soft” notions of structure, such as topological structure, one can extract much more “hard” or “rigid” notions of structure, such as geometric or algebraic structure. The key concept needed to capture this structure is that of an isometric system, or more generally an isometric extension, which we shall discuss in this lecture. As an application of this theory we characterise the distribution of polynomial sequences in torii (a baby case of a variant of Ratner’s theorem due to (Leon) Green, which we will cover later in this course).

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In this lecture, we use topological dynamics methods to prove some other Ramsey-type theorems, and more specifically the polynomial van der Waerden theorem, the hypergraph Ramsey theorem, Hindman’s theorem, and the Hales-Jewett theorem. In proving these statements, I have decided to focus on the ultrafilter-based proofs, rather than the combinatorial or topological proofs, though of course these styles of proof are also available for each of the above theorems.

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In the previous lecture, we established single recurrence properties for both open sets and for sequences inside a topological dynamical system . In this lecture, we generalise these results to multiple recurrence. More precisely, we shall show

Theorem 1. (Multiple recurrence in open covers) Let be a topological dynamical system, and let be an open cover of X. Then there exists such that for every , we have for infinitely many r.

Note that this theorem includes Theorem 1 from the previous lecture as the special case . This theorem is also equivalent to the following well-known combinatorial result:

Theorem 2. (van der Waerden’s theorem) Suppose the integers are finitely coloured. Then one of the colour classes contains arbitrarily long arithmetic progressions.

Exercise 1. Show that Theorem 1 and Theorem 2 are equivalent.

Exercise 2. Show that Theorem 2 fails if “arbitrarily long” is replaced by “infinitely long”. Deduce that a similar strengthening of Theorem 1 also fails.

Exercise 3. Use Theorem 2 to deduce a finitary version: given any positive integers m and k, there exists an integer N such that whenever is coloured into m colour classes, one of the colour classes contains an arithmetic progression of length k. (Hint: use a “compactness and contradiction” argument, as in my article on hard and soft analysis.)

We also have a stronger version of Theorem 1:

Theorem 3. (Multiple Birkhoff recurrence theorem) Let be a topological dynamical system. Then for any there exists a point and a sequence of integers such that as for all .

These results already have some application to equidistribution of explicit sequences. Here is a simple example (which is also a consequence of Weyl’s equidistribution theorem):

Corollary 1. Let be a real number. Then there exists a sequence of integers such that as .

Proof. Consider the skew shift system with . By Theorem 3, there exists and a sequence such that and both convege to . If we then use the easily verified identity

(1)

we obtain the claim.

Exercise 4. Use Theorem 1 or Theorem 2 in place of Theorem 3 to give an alternate derivation of Corollary 1.

As in the previous lecture, we will give both a traditional topological proof and an ultrafilter-based proof of Theorem 1 and Theorem 3; the reader is invited to see how the various proofs are ultimately equivalent to each other.

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We now begin the study of recurrence in topological dynamical systems - how often a non-empty open set U in X returns to intersect itself, or how often a point x in X returns to be close to itself. Not every set or point needs to return to itself; consider for instance what happens to the shift on the compactified integers . Nevertheless, we can always show that at least one set (from any open cover) returns to itself:

Theorem 1. (Simple recurrence in open covers) Let be a topological dynamical system, and let be an open cover of X. Then there exists an open set in this cover such that for infinitely many n.

Proof. By compactness of X, we can refine the open cover to a finite subcover. Now consider an orbit of some arbitrarily chosen point . By the infinite pigeonhole principle, one of the sets must contain an infinite number of the points counting multiplicity; in other words, the recurrence set is infinite. Letting be an arbitrary element of S, we thus conclude that contains for every , and the claim follows.

Exercise 1. Conversely, use Theorem 1 to deduce the infinite pigeonhole principle (i.e. that whenever is coloured into finitely many colours, one of the colour classes is infinite). Hint: look at the orbit closure of c inside , where A is the set of colours and is the colouring function.)

Now we turn from recurrence of sets to recurrence of individual points, which is a somewhat more difficult, and highlights the role of minimal dynamical systems (as introduced in the previous lecture) in the theory. We will approach the subject from two (largely equivalent) approaches, the first one being the more traditional “epsilon and delta” approach, and the second using the Stone-Čech compactification of the integers (i.e. ultrafilters).

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Before we begin or study of dynamical systems, topological dynamical systems, and measure-preserving systems (as defined in the previous lecture), it is convenient to give these three classes the structure of a category. One of the basic insights of category theory is that a mathematical objects in a given class (such as dynamical systems) are best studied not in isolation, but in relation to each other, via morphisms. Furthermore, many other basic concepts pertaining to these objects (e.g. subobjects, factors, direct sums, irreducibility, etc.) can be defined in terms of these morphisms. One advantage of taking this perspective here is that it provides a unified way of defining these concepts for the three different categories of dynamical systems, topological dynamical systems, and measure-preserving systems that we will study in this course, thus sparing us the need to give any of our definitions (except for our first one below) in triplicate.

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In this lecture, I define the basic notion of a dynamical system (as well as the more structured notions of a topological dynamical system and a measure-preserving system), and describe the main topics we will cover in this course.

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Next quarter, starting on Wednesday January 9, I will be teaching a graduate course entitled “Topics in Ergodic Theory“. As an experiment, I have decided to post my lecture notes on this blog as the course progresses, as it seems to be a good medium to encourage feedback and corrections. (On the other hand, I expect that my frequency of posting on non-ergodic theory topics is going to go down substantially during this quarter.) All of my class posts will be prefaced with the course number, 254A, and will be placed in their own special category.

The topics I plan to cover include

• Topological dynamics;
• Classical ergodic theorems;
• The Furstenberg-Zimmer structure theory of measure preserving systems;
• Multiple recurrence theorems, and the connections with Szemerédi-type theorems;
• Orbits in homogeneous spaces (and in particular, in nilmanifolds);
• (Special cases of) Ratner’s theorem, and applications to number theory (e.g. the Oppenheim conjecture).

If time allows I will cover some other topics in ergodic theory as well (I haven’t decided yet exactly which ones to discuss yet, and might be willing to entertain some suggestions in this regard.)

If this works out well then I plan to also do the same for my spring class, in which I will cover as much of Perelman’s proof of the Poincaré conjecture as I can manage. (Note though that this latter class will build upon a class on Ricci flow given by my colleague William Wylie in the winter quarter, which will thus be a de facto prerequisite for my spring course.)