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 1 We say that is a compact system if
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 3 A 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 4 Given a mps and , are equivalent:
- is almost periodic.
- The restriction of to is a minimal homeomorphism of a compact metric space.
- For every , the set is syndetic.
Proof: (a) (b). Note that the referred restriction is a transitive isometry, by definition. The assertion then follows from the
Exercise 1 Let be a compact metric space and an isometry on . Then is transitive iff it is minimal.
(b) (c). It follows from
Exercise 2 Consider 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 5 Let be a mps and almost periodic. Then, for any and , the set
Proof: Suppose . Then for every . Hence,
The next theorem will be proved in ERT11.
Theorem 7 A 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 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.