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 2 There 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 1 In 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
Comments feed for this article
10 March, 2011 at 12:15 pm
Prashant V
Dear 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 Tao
As 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
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16 October, 2011 at 11:42 pm
Gil Kalai
Hi 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 Tao
Well, 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 Kalai
I 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
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