Two common examples of ergodic measure-preserving systems, an irrational rotation of the circle with Lebesgue measure and the Bernoulli shift on with the product measure are typical members of the classes of compact and weak mixing systems, respectively. Due to the theory developed in ERT8, ERT9 and ERT10, we consider these two classes as well understood. Can one hope to describe general mps in terms of such building blocks? The surprising answer to this question is that if one allows for the relative notions of compact and weak mixing to play the part of building blocks, then this is indeed the case. This vague statement is made precise in the Furstenberg-Zimmer structure theorem, which we intend to prove soon.
Let be a homomorphism, with and . The present and next two posts are devoted to the definitions and study of the notions of compact and weakly mixing extensions, by adapting the usual notions of compact and weak mixing mps to a “relative” or “conditional” setting. The ideas are essentially the same as in the absolute cases, the main new trick being not to work in the Hilbert space over the complex numbers, but rather viewing the Hilbert space as a direct integral of a Hilbert bundle over the von Neumann algebra . Specifically, we consider the fiber Hilbert spaces , where is the disintegration associated to , form the Hilbert bundle over and identify with integrable sections of this Hilbert bundle. This will be properly explained in ERT16 (for weakly mixing extensions do not make use of such machinery).
As in the absolute case, there are two ways of defining relative notions of compactness and weak mixing. The first one, which I see as a qualitative one, is intrinsically more natural. As a matter of fact, once the two definitions are complementary, Furstenberg-Zimmer structure theorem basically follows from a transfinite induction. This definition relies on the analysis of generalized eigenfunctions.
On the other hand, to obtain Furstenberg multiple recurrence theorem, it is necessary a quantitative characterization of such objects. This is the other way of defining the concepts, in terms of ergodic averages and its expectations. See this post of Terry Tao for further details on this viewpoint.
In the present post, just like we did in the absolute case, we will define ergodicity and weak mixing from the quantitative viewpoint. This has two advantages: it is more concrete and demonstrates its usefulness for the purpose of multiple recurrence. ERT16 will describe the qualitative viewpoint. This will require the machinery of Hilbert bundles and generalized eigenfunctions. Then Furstenberg-Zimmer structure theorem will be established and, finally, for the purpose of multiple recurrence, we will show the two definitions are equivalent.
1. Ergodic extensions
Let and a factor of . We write for the corresponding homomorphism and, when is implicit, just .
Definition 1 is an ergodic extension of if the only -invariant sets of are images (modulo nulls sets) under of -invariant sets of .
If we denote the trivial one-point mps by , then is an extension of . Clearly, saying that is ergodic now becomes the assertion that is an ergodic extension.
Ergodic extensions behave well under composition and restriction. Specifically, if , then is ergodic if and only if and are both ergodic. This is straightforward, but for the purpose of completeness, I will give the proof.
- Assume is ergodic and let be -invariant (this means and satisfies ). By hypothesis, , where is -invariant. Observe that is invariant under and so is ergodic. Now, let be -invariant. Then is -invariant. By assumption, there is invariant under such that
- Assume and are ergodic and let be -invariant. Then for some -invariant . Then , where is -invariant and so .
Remember that induces a natural imbedding of into , . It is evident is ergodic if and only if every -invariant function of belongs to , that is, iff for any -invariant . By von Neummann’s theorem (see ERT1), the set of -invariant functions in is equal to , where
We thus have an alternative definition for ergodic extensions.
Observe that, as is continuous in the strong topology (see ERT14), it is enough to check the above equality in the weak topology.
Proof: Note that and so the above expression makes sense. Firstly,
and so the lemma reduces to saying that converges in the weak topology to zero. Let be a weak limit. Since converges to zero in the strong topology, and so, by ergodicity, . Finally, as each has null -expectation,
a passage of the weak limit (see the exercise below) gives that and so .
Exercise 1 Prove that is continuous in the weak topology.
Although equations (1) and (2) are equivalent, the natural place to work with then is downstairs, in . This is corroborated when we write the equations in the inhomogeneous case, which is the content of the next result. Actually, it characterizes ergodicity.
() Just apply the previous proposition to the functions and , noting that
() Let be -invariant. First assume that . Equation (3) gives that
and so . If is an arbitrary, the above argument implies that .
2. Weakly mixing extensions
Given , remember from ERT14 that the relative product mps is also an extension of , via any of the compositions or ( and are the natural projections).
Clearly, this extends Theorem 6 of ERT8: is a weakly mixing system iff is a weakly mixing extension of . Also, if is ergodic, then is ergodic, that is: every weakly mixing extension is ergodic. What we’ll do in the remaining of this post is to relativize the results of ERT8.
Proof: The idea is to apply Lemma 3 to the extension . We divide the proof in two parts.
Part 1. : as , we have . Observing that
Part 2. is arbitrary: apply Part 1 to the functions and .
We now prove the relative version of Theorem 6 of ERT8.
Proof: Let , , and . We’ll check (3). As generates a dense subset of , assume without loss of generality that . Due to the bilinearity in the functions , we can divide the proof into two parts.
Part 1. : we want to check equation (3). Observe that
Now observe that
Then we want to prove that
and this follows from the ergodicity assumption on and Proposition 3, because
Part 2. are arbitrary. Let and . Then
Corollary 8 If is weakly mixing, then is weakly mixing.
Proof: The assumption gives in particular that is ergodic. Three successive applications of Proposition 7 give that
is ergodic and so is weakly mixing.