**1. The dichotomy between structure and randomness **

The main tool used in ERT1 and ERT2 was: given a mps , we decomposed the space into two pieces: one **structured**, formed by the fixed (or periodic, depending on the case) functions, and other **random**, formed by the functions for which the Cesàro averages converge to zero. This represents an example of the dichotomy that surrounds Ergodic Ramsey Theory: structure vs. randomness. This is actually briefly discussed in the end of ERT2 and in a broader way in this paper of Terence Tao.

This idea is also used in other branches of Mathematics, specially in combinatorics, harmonic analysis, number theory, etc. We can cite many situations:

- Theorems about the existence of ergodic averages.
- Szemerédi regularity lemma.
- Roth theorem on the existence of arithmetic progressions of length three in sets of positive density.
- Gowers norms.
- All proofs (up to my knowledge) of Szemerédi theorem.
- Green-Tao theorem on the existence of arithmetic progressions in the primes.
- The
*Julia set*of holomorphic functions is either connected or a Cantor set. - Mané-Bochi ’02: a -generic conservative diffeomorphim in surfaces either has all Lyapunov exponents zero almost everywhere or is Anosov.
- Avila-Moreira ’03: For almost every , the quadratic function is either regular (has a periodic attractor) or stochastic (has an invariant absolutely continuous probability with positive Lyapunov exponent).
- Avila-Forni ’07: almost every interval exchange transformation is either an irrational rotation or weak mixing.
- Avila ’10: for Schrodinger operators with a one-frequency and typical real analytic potential, the spectrum is either subcritical or supercritical.

Each situation has a notion of structure/randomness. The one we are interested is multiple recurrence of mps. Let us see this from the spectral theory point of view.

Given a mps , denote also by the *Koopman-von Neumann operator*, defined by

When necessary, we use the notation to denote this operator. Many of its spectral properties are related to ergodic properties of . We investigate the eigenvalues/eigenfunctions of . If the eigenfunctions form a basis of , then is determined. In fact, let be the multiset (repeated with multiplicity) of eigenvalues of and, for each , the eigenfunction associated to . If

then

In this case, we say that has *pure point spectrum* and is a *compact system*. This constitutes the structured notion we were looking for.

In contrast, when has no eigenvalues other than and it is simple, we say that has *continuous spectrum* and is a *weak mixing system*. It forms the random part.

As pure point/continuous spectrum are opposite notions, there is a hope that every mps can be decomposed into two parts: one compact and other weak mixing. This is not true at all. Instead, can be decomposed in several parts in such a way that every part is an extension of the previous one and it is a compact or weak mixing extension of the smaller one. In other words, the dynamics of is broken into many parts in which every braking is obtained from the previous one by adding one of the two dynamical prototypes we discussed above.

Formally speaking, given two mps and , we say that is an *extension* of if there is a surjective measurable map such that

We denote this by and is called a *factor* of .

Theorem 1 (Furstenberg structural theorem)Given a mps , there exists anordinaland a family of factors of , for every , such that:

is a single point.is acompact extensionof for everysuccessor ordinal.is theinverse limitof , for everylimit ordinal.is aweak mixing extensionof .

Above, inverse limit is in the sense that . This result will be discussed in Lebesgue-full detail in the last post. In order to understand it, we need to study four concepts:

- Weak mixing systems.
- Compact systems.
- Weak mixing extensions.
- Compact extensions.

These will be the topics of this and the next 2 or 3 lectures.

**2. Weak mixing systems **

The definition used below is different from the one we assumed above, but don’t worry: they will be shown to coincide. Actually, we will obtain various equivalent definitions of weak mixing.

Exercise 1Consider a bounded sequence of nonnegative real numbers. Prove:

If , thenConclude that strong mixing implies weak mixing.If , thenConclude that weak mixing implies ergodicity.

Exercise 2is weak mixing if and only if

for every .

Proposition 3is weak mixing if and only if

for every such that .

We leave the proof to the reader, which may be found in the book *Topics in ergodic theory* of W. Parry. The notion of weak mixing means that, in some sense, almost all the system behaves in a strong mixing way. This is what says the following lemma.

Lemma 4Consider a bounded sequence of nonnegative real numbers. Then

if and only if there exists a set of zero density such that

*Proof:* () Define, for each , the set

is an ascending chain of subsets of . They are the sets that may give problems in the convergence of to zero. Each of them has zero density, because

and then

In this way, take an increasing sequence of integers such that . Define

By definition,

It remains to prove that has zero density. Consider an integer , let us say, with . As ,

and then

which, by (2), implies that

Then, has zero density.

() Let such that , for every . Given , we want to prove that

for every large enough. By hypothesis, there is for which

and

Then, for ,

Taking such that

we get

which concludes the proof.

Taking and , Lemma 4 implies that is weak mixing if and only if, for every , there exists of zero density such that

By approximation, (4) is equivalent to the existence, for every , of a set of zero density such that

Another characterization comes from Proposition 3: is weak mixing if and only if

for every such that . In fact, by Lemma 4, converges a Cesàro to zero if and only if the same happens to .

Weak mixing, at first impression, seems an artificial notion, obtained by the relaxation of strong mixing. This is not the case: first because it also has the natural spectral characterization discussed in section 1 and second, as was already discussed above, weak mixing is an important part of every mps. In other contexts, it is abundant. For example, Avila and Forni proved that almost every interval exchange transformation is either an irrational rotation or weak mixing, in contrast to an older result of Katok which proved that these are never strong mixing.

**3. Product characterization of weak mixing **

Consider two mps and .

Definition 5The product mps of and is the quadruple

where is the -algebra generated by and is the probability measure on defined by

Theorem 6Given a mps , the following are equivalent.

is weak mixing.is weak mixing.is ergodic, for every ergodic mps .is ergodic.

*Proof:* (i) (ii). It is enough to check (4) for a generating algebra of . Let . By assumption, there exist of zero density such that

and

The set has zero density and satisfies

proving that is weak mixing.

(ii) (i). Given , there exists of zero density such that

that is,

(i) (iii). Follows from the exercise below.

Exercise 3Consider two bounded sequences and of real numbers. If and converge a Cesàro to and , respectively, then converges a Cesàro to .

(iii) (iv). If is the trivial mps with consisting of a single point, we conclude that is ergodic. Then, takin , it follows that is ergodic.

(iv) (i). First, note that is ergodic. Given ,

converges a Cesàro to

proving the assertion. Then

which converges to

**4. Spectral characterization of weak mixing **

We now characterize weak mixing in terms of spectral properties. At this point, it is interesting to introduce the

Theorem 7If is an unitary operator on the Hilbert space and , then there is a unique finite Borel measure on the circle such that

When has continuous spectrum, is a continuous measure (it has no atoms), for every such that .In this case, Fubini theorem guarantees that gives zero measure to the diagonal . This in turn implies the

Theorem 8is weak mixing if and only if has continuous spectrum.

*Proof:* () Suppose is an eigenfunction associated to . The function defined by is an eigenfunction of associated to . By Theorem 6, is constant and the same happens to .

() Let us check (6). Take such that . Using Theorem 7,

Decompose , where is the diagonal. For , the summand

converges to zero as uniformly in . Since assigns zero measure to , we’re done.

**5. Conditions for weak mixing **

In this section we resume all conditions obtained above for a a mps be weak mixing.

- For any ,
- For any ,
- For any such that ,
- For any such that ,
- For any , there exists of zero density such that
- For any , there exists of zero density such that
- is ergodic.
- is ergodic, for every ergodic.
- is weak-mixing.
- has continuous spectrum.

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

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