So…It’s finally my term to lecture on the ergodic theory seminar! (background, our goal is to go through the ergodic proof of Szemerédi’s theorem as in Furstenberg’s book). My part is the beginning of the discussion on weak mixing and prove the multiple recurrence theorem in the weak mixing case, the weak mixing assumption shall later be removed (hence the theorem is in fact true for any ergodic system) and hence prove Szemerédi’s theorem via the correspondence principal discussed in the last lecture.
Given two measure preserving systems and , we denote the product system by where is the smallest -algebra on including all products of measurable sets.
Definition: A m.p.s. is weakly mixing if is ergodic.
Note that weak mixing ergodic
as for non-ergodic systems we may take any intermediate measured invariant set the whole space to produce an intermediate measured invariant set of the product system.
For any , let .
Ergodic for all with positive measure,
Weakly mixing for all with positive measure, .
Hence is weakly mixing for all with positive measure, . We’ll see later that this is in fact but let’s say for now.
As a toy model for the later results, let’s look at the proof of following weak version of ergodic theorem:
Theorem 1: Let be ergodic m.p.s., then
Proof: Let is unitary on .
Any weak limit point of the above set is -invariant, hence ergodicity implies they must all be constant functions.
then we have
Therefore the set has only one limit point under the weak topology.
Since the closed unit ball in Hilbert space is weakly compact, hence converges weakly to the constant valued function .
Next, we apply the above theorem on the product system and prove the following:
Theorem 2: For weakly mixing,
Proof: For , let where
By theorem 1, we have
Set hence by theorem 1, we have
By , we have
This establishes theorem 2.
We now prove that the following definition of weak mixing is equivalent to the original definition.
Theorem 3: weakly mixing iff for all ergodic, is ergodic.
proof:“” is obvious as if has the property that its product with any ergodic system is ergodic, then is ergodic since we can take its product with the one point system.
This implies that the product of the system with itself is ergodic, which is the definition of being weakly mixing.
“” Suppose weakly mixing.
is ergodic iff all invariant functions are constant a.e.
For any , let , let ; hence .
By theorem 1, since is ergodic on ,
On the other hand, let
By theorem 2, since is weak mixing hence
by direct computation i.e. subtract the left from the right and obtain a perfect square.
Approaches to as by .
Combining the two parts we get
Since the linear combination of functions of the form is dense in (in particular the set includes all characteristic functions of product sets and hence all simple functions with basic sets being product sets)
We have shown that for a dense subset of the sequence of functions converge weakly to the constant function. (Since it suffice to check convergence a dense set of functional in the dual space)
Hence for any , the average weakly converges to the constant function .
For any -invariant function, the average is constant, hence this implies all invariant functions are constant a.e. Hence we obtain ergodicity of the product system.
Establishes the theorem.