We now turn attention to the structured situation. Consider a measure-preserving system . Remember that, by ERT8, this is the case when the Koopman-von Neumann operator has a basis of eigenvectors. Assume that is the multiset of eigenvalues and is the eigenvector associated to .

Definition 1We say that is acompact systemif

Let’s investigate an example: given , consider the rotation

If , , then

that is, each is an eigenvector. We know, by Fourier analysis, that generates the space , where is the Lebesgue measure on . This implies that is compact. In this setting, it is straightforward to prove multiple recurrence. Assume that is an interval and fix . We want to show the existence of such that

This is easy: given , take such that

for every . This implies that

whenever . If is such that ,

which implies that

Also, note that the set of satisfying is syndetic, so that the above argument actually proves that

The same argument holds for any with positive measure. In fact, by Lebesgue density theorem, given , there exists such that

which is the main inequality used in the above argument. We get our first result.

for any .

Note that the main tool is that, for various , the functions

are simultaneously close to each other. This is what we are looking for: structure (given by the algebraic structure of ) implying “almost periodicity”, as will be explained in the next section.

**1. Almost periodicity **

Consider a mps .

Definition 3A function is almost periodic if the set is pre-compact in .

In other words, if is a compact subset of , considered with the norm topology.

We will see below that the systems for which every element of is almost periodic are exactly the compact ones.

At first impression, this does not seen the expected definition we want, but the following proposition clarifies the apparent uncorrelation.

Proposition 4Given a mps and , are equivalent:

is almost periodic.The restriction of to is a minimal homeomorphism of a compact metric space.For every , the setis syndetic.

*Proof:* (a) (b). Note that the referred restriction is a transitive isometry, by definition. The assertion then follows from the

Exercise 1Let be a compact metric space and an isometry on . Then is transitive iff it is minimal.

(b) (c). It follows from

Exercise 2Consider a homeomorphism of the metric space . Then is minimal iff, for any open and , the set

is syndetic. If in addition is an isometry, the above condition is equivalent to

being syndetic for any and .

(c) (a). Given , the syndeticity of guarantees that the closure of is covered by finitely many balls . This proves the compacity of .

Using (c), we obtain

Proposition 5Let be a mps and almost periodic. Then, for any and , the set

is syndetic.

*Proof:* Suppose . Then for every . Hence,

Corollary 6(Multiple recurrence for almost periodic functions) Let be a non-negative almost periodic function such that . Then, for every ,

The next theorem will be proved in ERT11.

Theorem 7A mps is compact if and only if every is almost periodic.

The if part is easily obtained by means of simultaneous diophantine approximations. For the converse, we need an algebraic characterization of compact systems, to be established in the next lecture.

Corollary 6 says that compact systems are multiply recurrent in the sense of (1). In particular, considering , we get

Corollary 8(Multiple Poincaré Recurrence for compact mps) If is compact and , , then

for every .

The next post will characterize compact system. They are the **Kronecker systems**: rotations in abelian groups. This is why the example of studied above deserves attention.

**Previous posts:** ERT0, ERT1, ERT2, ERT3, ERT4, ERT5, ERT6, ERT7, ERT8, ERT9.

## Leave a Reply