Let be an abelian countable discrete group. A measure-preserving -system (or *-system for short*) is a probability space , equipped with a measure-preserving action of the group , thus

for all and , and

for all , with equal to the identity map. Classically, ergodic theory has focused on the cyclic case (in which the are iterates of a single map , with elements of being interpreted as a time parameter), but one can certainly consider actions of other groups also (including continuous or non-abelian groups).

A -system is said to be *strongly -mixing*, or strongly mixing for short, if one has

for all , where the convergence is with respect to the one-point compactification of (thus, for every , there exists a compact (hence finite) subset of such that for all ).

Similarly, we say that a -system is *strongly -mixing* if one has

for all , thus for every , there exists a finite subset of such that

whenever all lie outside .

It is obvious that a strongly -mixing system is necessarily strong -mixing. In the case of -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 -system necessarily strongly -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 -mixing*, which roughly speaking means that converges to for *most* . 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 , , and for a small but non-trivial number of pairs .

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 -systems, by a well-known counterexample of Ledrappier which can be described as follows. One can view a -system as being essentially equivalent to a stationary process of random variables in some range space indexed by , with being with the obvious shift map

In Ledrappier’s example, the take values in the finite field of two elements, and are selected at uniformly random subject to the “Pascal’s triangle” linear constraints

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 (known as Sierpinski’s triangle), one has

for all powers of two . This is enough to destroy strong -mixing, because it shows a strong correlation between , , and for arbitrarily large and randomly chosen . On the other hand, one can still show that and are asymptotically uncorrelated for large , giving strong -mixing. Unfortunately, there are significant obstructions to converting Ledrappier’s example from a -system to a -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 is replaced by the function field ring , which is a “dyadic” (or more precisely, “triadic”) model for the integers (cf. this earlier blog post of mine). In other words:

Theorem 2There exists a -system that is strongly -mixing but not strongly -mixing.

The idea is much the same as that of Ledrappier; one builds a stationary -process in which are chosen uniformly at random subject to the constraints

for all and all . Again, this system is manifestly not strongly -mixing, but can be shown to be strongly -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.

** — 1. The example — **

Let be any ball in , i.e. any set of the form for some and . One can then create a process adapted to this ball, by declaring to be uniformly distributed in the vector space of all tuples with coefficients in that obey (1) for all and . Because any translate of a line is still a line, we see that this process is stationary with respect to all shifts of degree at most . Also, if are nested balls, we see that the vector space projects surjectively via the restriction map to (since any tuple obeying (1) in can be extended periodically to one obeying (1) in ). As such, we see that the process is equivalent in distribution to the restriction of to . Applying the Kolmogorov extension theorem, we conclude that there exists an infinite process whose restriction to any ball has the distribution of . As each was stationary with respect to translations that preserved , we see that the full process is stationary with respect to the entire group .

Now let be a ball

which we divide into three equally sized sub-balls by the formula

By construction, we see that

where we use translation by to identify , , and together. As a consequence, we see that the projection map from to is surjective, and this implies that the random variables are independent. More generally, this argument implies that for any disjoint balls , the random variables and are independent.

Now we can prove strong -mixing. Given any measurable event and any , one can find a ball and a set depending only on such that and differ by at most in measure. On the other hand, for outside of , and are determined by the restrictions of to disjoint balls and are thus independent. In particular,

and thus

which gives strong -mixing.

On the other hand, we have almost surely, while each are uniformly distributed in and pairwise independent. In particular, if is the event that , we see that

and

showing that strong -mixing fails.

Remark 1In the Archimedean case , a constraint such as propagates itself to force complete linearity of , which is highly incompatible with strong mixing; in contrast, in the non-Archimedean case , such a constraint does not propagate very far. It is then tempting to relax this constraint, for instance by adopting an Ising-type model which penalises a configuration whenever quantities such as deviates from zero. However, to destroy strong -mixing, one needs infinitely many such penalisation terms, which roughly corresponds to an Ising model in an infinite-dimensional lattice. In such models, it seems difficult to find a way to set the “temperature” parameters in such a way that one has meaningful -correlations, without the system “freezing up” so much that -mixing fails. It is also tempting to try to truncate the constraints such as (1) to prevent their propagation, but it seems that any naive attempt to perform a truncation either breaks stationarity, or introduces enough periodicity into the system that -mixing breaks down. My tentative opinion on this problem is that a -counterexample is constructible, but one would have to use a very delicate and finely tuned construction to achieve it.

## 7 comments

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10 March, 2011 at 12:15 pm

Prashant VDear Terry,

Essentially the problem is whether a strongly mixing -system is strongly 3-mixing. If the answer to this is “yes”, then can we generalize and say that every $\textbf{Z}$-system is strongly $n$-mixing (e.g. using induction).

10 March, 2011 at 1:01 pm

Terence TaoAs far as I know there is no easy way to bootstrap the claim that strong 2-mixing implies strong 3-mixing, to the stronger claim that strong 2-mixing implies (say) strong 4-mixing. Indeed, to my knowledge, for any , the claim that strong n-mixing implies strong m-mixing for -systems remains open (and no implications between these claims are known, other than the trivial ones).

12 March, 2011 at 4:49 pm

A College Level Problem on Jensen’s Inequality | Gaurav Happy Tiwari[…] Rohlin’s problem on strongly mixing systems […]

16 October, 2011 at 11:42 pm

Gil KalaiHi Terry, perhaps Rohlin’s conjecture and the approach toward negative solution based on Ledrappier’s constructions and the further ideas in the post can become the starting point of a nice polymath project.

17 October, 2011 at 9:15 am

Terence TaoWell, if there is sufficient interest and ideas to get started, I could certainly consider this. But the above blog post basically represents the extent of my own progress on the problem; one would need new ideas in order to advance further. So at this stage I’m happy to just use this comment thread to discuss the problem, unless there is a significant new development.

17 October, 2011 at 2:37 pm

Gil KalaiI see, I thought remark 1 suggested some (perhaps vague) idea on how an example will look which involves some hypothetical probabilistic models similar to models coming from statistical physics with some fine properties. (But the remark was not detailed enough for me to understand what properties are needed.) I did not know how much further Remark 1 was persued. ( I am fairly ignorant about the poblem beside knowing a number of people who tried.) In any case, the problem looks of the right nature for a massive collective ‘head hitting against the wall’ activity, but I agree a strong start is needed.

27 January, 2013 at 5:06 am

Nivat’s conjecture after Cyr-Kra « Disquisitiones Mathematicae[…] 3-dots systems. This dynamical system has several interesting properties (see, e.g, this blog post of Terence Tao for the relationship between this example and Rokhlin’s mixing problem), but, for the sake of […]