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Let {G = (G,+)} be an abelian countable discrete group. A measure-preserving {G}-system {X = (X, {\mathcal X}, \mu, (T_g)_{g \in G})} (or {G}-system for short) is a probability space {(X, {\mathcal X}, \mu)}, equipped with a measure-preserving action {T_g: X \rightarrow X} of the group {G}, thus

\displaystyle  \mu( T_g(E) ) = \mu(E)

for all {E \in {\mathcal X}} and {g \in G}, and

\displaystyle  T_g T_h = T_{g+h}

for all {g, h \in G}, with {T_0} equal to the identity map. Classically, ergodic theory has focused on the cyclic case {G={\bf Z}} (in which the {T_g} are iterates of a single map {T = T_1}, with elements of {G} being interpreted as a time parameter), but one can certainly consider actions of other groups {G} also (including continuous or non-abelian groups).

A {G}-system is said to be strongly {2}-mixing, or strongly mixing for short, if one has

\displaystyle  \lim_{g \rightarrow \infty} \mu( A \cap T_g B ) = \mu(A) \mu(B)

for all {A, B \in {\mathcal X}}, where the convergence is with respect to the one-point compactification of {G} (thus, for every {\epsilon > 0}, there exists a compact (hence finite) subset {K} of {G} such that {|\mu(A \cap T_g B) - \mu(A)\mu(B)| \leq \epsilon} for all {g \not \in K}).

Similarly, we say that a {G}-system is strongly {3}-mixing if one has

\displaystyle  \lim_{g,h,h-g \rightarrow \infty} \mu( A \cap T_g B \cap T_h C ) = \mu(A) \mu(B) \mu(C)

for all {A,B,C \in {\mathcal X}}, thus for every {\epsilon > 0}, there exists a finite subset {K} of {G} such that

\displaystyle  |\mu( A \cap T_g B \cap T_h C ) - \mu(A) \mu(B) \mu(C)| \leq \epsilon

whenever {g, h, h-g} all lie outside {K}.

It is obvious that a strongly {3}-mixing system is necessarily strong {2}-mixing. In the case of {{\bf Z}}-systems, it has been an open problem for some time, due to Rohlin, whether the converse is true:

Problem 1 (Rohlin’s problem) Is every strongly mixing {{\bf Z}}-system necessarily strongly {3}-mixing?

This is a surprisingly difficult problem. In the positive direction, a routine application of the Cauchy-Schwarz inequality (via van der Corput’s inequality) shows that every strongly mixing system is weakly {3}-mixing, which roughly speaking means that {\mu(A \cap T_g B \cap T_h C)} converges to {\mu(A) \mu(B) \mu(C)} for most {g, h \in {\bf Z}}. Indeed, every weakly mixing system is in fact weakly mixing of all orders; see for instance this blog post of Carlos Matheus, or these lecture notes of myself. So the problem is to exclude the possibility of correlation between {A}, {T_g B}, and {T_h C} for a small but non-trivial number of pairs {(g,h)}.

It is also known that the answer to Rohlin’s problem is affirmative for rank one transformations (a result of Kalikow) and for shifts with purely singular continuous spectrum (a result of Host; note that strongly mixing systems cannot have any non-trivial point spectrum). Indeed, any counterexample to the problem, if it exists, is likely to be highly pathological.

In the other direction, Rohlin’s problem is known to have a negative answer for {{\bf Z}^2}-systems, by a well-known counterexample of Ledrappier which can be described as follows. One can view a {{\bf Z}^2}-system as being essentially equivalent to a stationary process {(x_{n,m})_{(n,m) \in {\bf Z}^2}} of random variables {x_{n,m}} in some range space {\Omega} indexed by {{\bf Z}^2}, with {X} being {\Omega^{{\bf Z}^2}} with the obvious shift map

\displaystyle  T_{(g,h)} (x_{n,m})_{(n,m) \in {\bf Z}^2} := (x_{n-g,m-h})_{(n,m) \in {\bf Z}^2}.

In Ledrappier’s example, the {x_{n,m}} take values in the finite field {{\bf F}_2} of two elements, and are selected at uniformly random subject to the “Pascal’s triangle” linear constraints

\displaystyle  x_{n,m} = x_{n-1,m} + x_{n,m-1}.

A routine application of the Kolmogorov extension theorem allows one to build such a process. The point is that due to the properties of Pascal’s triangle modulo {2} (known as Sierpinski’s triangle), one has

\displaystyle  x_{n,m} = x_{n-2^k,m} + x_{n,m-2^k}

for all powers of two {2^k}. This is enough to destroy strong {3}-mixing, because it shows a strong correlation between {x}, {T_{(2^k,0)} x}, and {T_{(0,2^k)} x} for arbitrarily large {k} and randomly chosen {x \in X}. On the other hand, one can still show that {x} and {T_g x} are asymptotically uncorrelated for large {g}, giving strong {2}-mixing. Unfortunately, there are significant obstructions to converting Ledrappier’s example from a {{\bf Z}^2}-system to a {{\bf Z}}-system, as pointed out by de la Rue.

In this post, I would like to record a “finite field” variant of Ledrappier’s construction, in which {{\bf Z}^2} is replaced by the function field ring {{\bf F}_3[t]}, which is a “dyadic” (or more precisely, “triadic”) model for the integers (cf. this earlier blog post of mine). In other words:

Theorem 2 There exists a {{\bf F}_3[t]}-system that is strongly {2}-mixing but not strongly {3}-mixing.

The idea is much the same as that of Ledrappier; one builds a stationary {{\bf F}_3[t]}-process {(x_n)_{n \in {\bf F}_3[t]}} in which {x_n \in {\bf F}_3} are chosen uniformly at random subject to the constraints

\displaystyle  x_n + x_{n + t^k} + x_{n + 2t^k} = 0 \ \ \ \ \ (1)

for all {n \in {\bf F}_3[t]} and all {k \geq 0}. Again, this system is manifestly not strongly {3}-mixing, but can be shown to be strongly {2}-mixing; I give details below the fold.

As I discussed in this previous post, in many cases the dyadic model serves as a good guide for the non-dyadic model. However, in this case there is a curious rigidity phenomenon that seems to prevent Ledrappier-type examples from being transferable to the one-dimensional non-dyadic setting; once one restores the Archimedean nature of the underlying group, the constraints (1) not only reinforce each other strongly, but also force so much linearity on the system that one loses the strong mixing property.

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In the previous lecture, we studied the recurrence properties of compact systems, which are systems in which all measurable functions exhibit almost periodicity – they almost return completely to themselves after repeated shifting. Now, we consider the opposite extreme of mixing systems – those in which all measurable functions (of mean zero) exhibit mixing – they become orthogonal to themselves after repeated shifting. (Actually, there are two different types of mixing, strong mixing and weak mixing, depending on whether the orthogonality occurs individually or on the average; it is the latter concept which is of more importance to the task of establishing the Furstenberg recurrence theorem.)

We shall see that for weakly mixing systems, averages such as \frac{1}{N} \sum_{n=0}^{N-1} T^n f \ldots T^{(k-1)n} f can be computed very explicitly (in fact, this average converges to the constant (\int_X f\ d\mu)^{k-1}). More generally, we shall see that weakly mixing components of a system tend to average themselves out and thus become irrelevant when studying many types of ergodic averages. Our main tool here will be the humble Cauchy-Schwarz inequality, and in particular a certain consequence of it, known as the van der Corput lemma.

As one application of this theory, we will be able to establish Roth’s theorem (the k=3 case of Szemerédi’s theorem).

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