During my conversations with my friend Artur Avila around the 2x, 3x mod 1 problem and its applications, Artur explained me a cute argument (using the Fourier transform of measures) to show a weak version of Rudolph’s theorem (which is a partial answer to this problem). Of course, I believe it is a good idea to share it here, but, in order to keep this post more self-contained, I’ll not proceed directly to the argument. In fact, I’ll divide the exposition into 3 sections: the first one is a general introduction to Furstenberg problem, Rudolph’s theorem and the weak version of Rudolph’s theorem, the second section contains the proof of this weak version of Rudolph result and the third section briefly discusses Einsiedler, Katok and Lindenstrauss’ application of a (highly non-trivial) analog of Rudolph’s theorem to the so-called Littlewood conjecture (about simultaneous Diophantine approximations).
Disclaimer. While Artur’s explanation was clear and correct, any possible mistakes and errors appearing below are my responsability (of course).
-Furstenberg’s problem: measure rigidity of expanding rank-two semi-group actions-
Consider the following (uniformly expanding) dynamical system on the circle :
where is an integer. Despite its simple definition, is rich from the dynamical point of view: possesses plenty of ergodic invariant measures, e.g., the Lebesgue measure and the Dirac measures supported on its periodic points (in fact, due to the expanding features of , one can use the so-called thermodynamical formalism [explained in Bowen’s book] to construct a whole family of ergodic invariant measure “interpolating” the Lebesgue measure and the Dirac measures); since, by Birkhoff’s theorem, the ergodic invariant measures captures the statistical features of most orbits in its support, we see that the existence of several distinct ergodic measures for indicates the presence of several distinct statistical behaviours of the orbits of (which justifies the adjective “rich”). Of course, the dynamics of an individual uniformly expanding (and, more generally, hyperbolic) dynamical systems is well-understood nowadays (after the works of Anosov, Smale, Sinai, Ruelle and Bowen among several other authors) and we do not plan to discuss further this issue today. Instead, let us investigate the dynamics of the joint action of two expanding maps, say and . In other words, denoting by and , we are replacing the action
of the rank-one semi-group (i.e., the individual action of ) by the action
of the rank-two semi-group (i.e., the joint action of and ).
Remark. In general, a similar action by a rank-two semi-group can be obtained replacing by any pair of multiplicatively independent non-negative integers.
The basic topological result about the dynamics of is:
Theorem (H. Furstenberg). The sole infinite closed -invariant subset of the circle is the circle itself.
This theorem should be constrasted with the rank-one situation: while has only one closed invariant set, it is not hard to see that has a lot of invariant Cantor sets. In fact, since is semi-conjugated to a full Bernoulli shift with symbols (via a codification of the orbits using the Markov partition ), any subshift gives you an invariant Cantor set. More concretely, you can take a finite collection of disjoint closed intervals whose union doesn’t coincide with the whole circle and define the set of points of the circle whose -orbit never enters . Of course, is -invariant and, from the expanding features of , it follows that is a Cantor set (exercise).
In resume, Furstenberg’s result says that is more rigid than (dynamically speaking), which is quite unexpected because is the combination of the dynamics of two strongly chaotic endomorphisms, namely, and . Taking this theorem as a motivation, Furstenberg posed the following problem of the measure (i.e., ergodic-theoretical) rigidity of :
Furstenberg’s 2x,3x (mod 1) problem. Is it true that the Lebesgue measure is the sole non-atomic ergodic -invariant measure? Equivalently, is it true that the Lebesgue measure is the unique non-atomic ergodic measure which is both and -invariant?
In other words, this problem asks whether the intersection of two enormous sets of measures (the set -invariant probabilities and the set of -invariant probabilities) is relatively small (namely, the Lebesgue measure and the Dirac measures supported on periodic orbits).
While Furstenberg’s problem is still open (to the best of my knowledge), some important partial results are available nowadays. In particular, let me quote the following theorem of D. Rudolph:
Theorem (D. Rudolph). Let be a and invariant ergodic measure such that
Then, is the Lebesgue measure. Equivalently, if a invariant measure is not the Lebesgue measure, then the semigroup
is a group for almost every .
In this statement, denotes the metric (Kolmogorov-Sinai) entropy of the -invariant measure .
Remark. The motivation of the metric entropy condition in Rudolph’s theorem comes from the fact that the metric entropy of Dirac measures supported on periodic orbits is zero. Of course, Rudolph theorem is just a partial answer to Furstenberg question because non-atomic invariant measures can have zero entropy.
While we do not pretend to give a complete proof of Rudolph’s theorem (which is not very hard but involves some amount of abstract ergodic theory), we do plan to show in the next section the following fact:
Theorem 1 (weak version of Rudolph’s theorem). Let be a and invariant measure. Assume that and is ergodic. Then, is the Lebesgue measure.
-Weak Rudolph theorem: proof of theorem 1-
Take a probability measure verifying the assumptions of theorem 1 and let be the translation of on the circle .
Note that and are commuting maps. In particular, since is invariant, it follows that is invariant. Moreover, the assumption of ergodicity of with respect to implies that and are -ergodic.
On the other hand, the positive entropy assumption implies that the restriction of to the support of is not invertible. Equivalently, we can always find some pairs of generic points with the same image under . However, such a pair of points is always permuted by the translation . In particular, we have that the probability measures and aren’t mutually singular.
Putting all these facts together, we have two invariant ergodic measures ( and ) such that and aren’t mutually singular. It follows that , i.e., is -invariant.
At this stage, one has enough information to conclude that is the Lebesgue measure. Indeed, we claim that all of the non-zero Fourier modes of vanish, that is,
(1) for any .
In fact, let us start with the odd Fourier modes (i.e., where ). Using the -invariance of , we obtain
Since , we get
In particular, it follows that whenever is odd. Finally, we can use the -invariance of (with a similar argument) to obtain
for every ,
that is, the even Fourier modes can be deduced from the odd ones (more precisely, writing a non-zero even number as where is odd, we have ). Consequently, our claim (1) about the vanishing of the Fourier modes of is proved. As we know (1) forces to be the Lebesgue measure, so that the proof of theorem 1 is complete.
-Measure rigidity of higher-rank hyperbolic actions: Einsiedler-Katok-Lindenstrauss theorem-
In this final section, we briefly outline a recent application of the measure rigidity of higher-rank hyperbolic actions to a partial solution of a number-theoretical problem.
A well-known fruitful interaction in Mathematics occurs between Dynamical Systems and Number Theory. Among the several applications of dynamical ideas to number-theoretical problems, one finds Margulis’ solution to Oppenheim conjecture (see e.g. this blog post of Terence Tao). Here, the basic idea is to convert the study of the values of indefinite quadratic forms in variables () into the study of the dynamics of an action of a higher-rank group in the space of unimodular lattices in . Another (more recent) application of these ideas was performed by M. Einsiedler, A. Katok and E. Lindenstrauss in the study of the so-called Littlewood conjecture:
Littlewood conjecture. For every , it holds
Here denotes the fractional part of .
Again, the basic idea is to convert this problem into the study of the dynamics of the action of the group of positive diagonal matrices in .
However, although the basic strategy of Margulis and Einsiedler-Katok-Lindenstrauss is the same, there is a subtle and important difference: while in Margulis’ context the acting group contains only unipotent (parabolic) elements, in Einsiedler-Katok-Lindenstrauss’s setting the acting group contains only hyperbolic elements.
In particular, at the present moment, we have the following picture: one can completely classify the invariant measures of the -action (by Ratner’s theorems), but one can say something interesting about the invariant measures of the -action only when they have positive entropy. In other terms, it is “morally” more easy to show measure rigidity statements (in the spirit of Furstenberg’s problem) for the -action than -action because the individual elements of the group are already parabolic, i.e., rigid. Here, the positive entropy condition for the measure rigidity statement (for the action) is certainly motivated by Rudolph’s result.
In any case, the moral philosophy here is the following: the number-theoretical problems quoted above (Oppenheim and Littlewood conjectures) can be reduced to the complete classification of the invariant measures of adequate higher-rank group actions. Since such a complete classification is possible for the -action, Margulis completely solved Oppenheim conjecture; but because our current technology only allows us to classify invariant measures of positive entropy, Einsiedler-Katok-Lindenstrauss partially (almost) solved Littlewood conjecture. More precisely, they proved the following result:
Theorem (Einsiedler, A. Katok and E. Lindenstrauss). The set of (possible) exceptions to Littlewood conjecture has Hausdorff dimension 0.
Closing this post, let me say that this last result deserves further explanation and I plan to discuss it a little bit more in a future post (probably after the series of posts around Trieste’s conference). By the way, the curious reader may consult (besides the original article) these expository notes of Akshay Venkatesh for a nice introduction. Ciao!