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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.

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).

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.

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.

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).

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.

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.

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.)

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