Today we follow J.-C. Yoccoz’s 6th lecture and we begin our discussions around the simplicity of Lyapunov exponents of the Kontsevich-Zorich cocycle. By the end of this series of posts, we will focus on the case of square-tiled surfaces (along the lines of a work in progress by C.M., Martin Möller and J.-C. Yoccoz). However, since our future discussions will be based on the works of A. Avila and M. Viana, we dedicate today’s post to review (some aspects of) these works.
The main result of this article here of A. Avila and M. Viana (settling the Kontsevich-Zorich conjecture) is:
Theorem. The Lyapunov spectrum of the (Kontsevich-Zorich) cocycle is simple (i.e., all Lyapunov exponents have multiplicity ) with respect to the natural (Masur-Veech) measure on each connected component of each stratum of the moduli space of (unit area) Abelian differentials.
Roughly speaking, this theorem has two parts:
- a simplicity criterion for a certain class of cocycles;
- verification of the simplicity criterion in the particular case of the cocycle equipped with the Masur-Veech measure.
Of course, our goal here is not to make a complete discuss of the theorem of Avila and Viana, but instead only to revisit and adapt it in order to allow applications to the dynamics of with respect to other measures (than the Masur-Veech one).
In particular, for today we will content ourselves with the discussion of Avila-Viana simplicity criterion (and a straightforward adaptation of it to slightly more general contexts).
From the “historical” point of view, Avila-Viana simplicity criterion is inspired by previous results of H. Furstenberg, I. Goldsheid and G. Margulis, A. Raugi and Y. Guivarch, F. Ledrappier, C. Bonatti, X. Gomez-Mont and M. Viana, and C. Bonatti and M. Viana. Also, there are at least two versions of Avila-Viana criterion (serving for different purposes), one in this article here (published at Acta Math.) and the other in this article here (published at Portugaliae Math.). For the sake of today’s post, we will follow the version described in the paper published at Acta Math.
Let be a finite or countable alphabet. Define , and . Denote by and the natural (left) shift maps on and resp. Also, let and be the natural projections
We denote by the set of words of the alphabet . Given , let
Given a -invariant probability measure on , we denote by the unique -invariant probability measure with , and we define .
In the sequel, we will make the following bounded distortion assumption on :
Hypothesis (Bounded distortion). There exists a constant such that
for any .
This bounded distortion assumption says that, in some sense, is “not very far” from a Bernoulli measure. As an exercise, the reader can check that bounded distortion implies that is -ergodic.
In any case, such and are the bases dynamics. Now, our next assumptions concerns the class of cocycles we want to investigate over these bases.
Hypothesis (Locally constant integrable cocycles). We assume that our cocycle satisfies
- (the cocycle is locally constant) , where (“primitive case”) for , and ;
- (the cocycle is integrable) .
In the work of Avila and Viana, they treat the primitive case of together with the derived case of , even, as they apply their criterion to the case of a symplectic cocycle closely related to the Kontsevich-Zorich cocycle. In our context, given our previous discussions, it is interesting to consider the following derived cases “together”:
- , even (and the base field is );
- , (and the base field is );
- , (and the base field is ).
Remark. Actually, concerning the results stated below, it seems that we can also include , (and the base field is ) in the list of derived cases, but we will not do this as currently we do not see interesting “applications” of this case.
As a matter of fact, in the cases , up to changing the sign of the preserved Hermitian form, we may assume that .
Given , we write
By definition, it satisfies for any .
The ergodicity of (coming from the bounded distortion property) and the integrability of the cocycle allow us to apply the Oseledets theorem to deduce the existence of Lyapunov exponents
where is the dimension over . Here, we count Lyapunov exponents with multiplicities.
Remark. In the case or , one can think of Lyapunov exponents by seeing our spaces over , but instead of the list above, we must repeat or times each . So, when saying that we count the dimension over , we’re avoiding unnecessary repetitions of Lyapunov exponents, and keeping only “interesting” multiplicities.
In the derived cases, as we saw in previous posts, one has the following a priori constraints for the Lyapunov exponents:
- Symplectic (, even): ;
- Unitary (, , , ): and for .
Also, the unstable/stable Oseledets subspaces associated to positive/negative Lyapunov exponents are isotropic.
Definition. The Lyapunov spectrum of the cocycle is simple if
- for in the primitive case and the derived case ;
- for in the derived cases , .
In other words, we say that a cocycle is simple when its Lyapunov spectrum is as simple as possible given the constraints presented above.
Definition. We say that is admissible if
- without further restrictions in the primitive case or the derived case ;
- or in the derived cases , .
Given admissible, we denote by :
- the Grassmanian of -planes of in the primitive case ;
- the Grassmanian of -planes which are isotropic if , and coisotropic if in the derived case ;
- the Grassmanian of -planes over which are isotropic if , and coisotropic if in the derived case .
At this point, we are ready to make our main assumptions on our cocycle :
- Pinching. There exists such that the spectrum of is simple.
- Twisting (“strong form”). For all , admissibles, , , , there exists such that .
In this language, the simplicity criterion of Avila-Viana (compare with Theorem 7.1 in Avila-Viana’s article) can be stated as follows:
Theorem 1 (A. Avila and M. Viana). Under these assumptions, the Lyapunov spectrum of is simple.
Remark. In their article, A. Avila and M. Viana considered exclusively the derived case , but, as we’re going to sketch below, their arguments can be adapted (in a straightforward way) to give the statement above.
We begin by noticing that one can consider the cocycle over the invertible dynamics because the Lyapunov spectrum is not affected by this procedure.
The main result towards the proof of Theorem 1 above is:
Theorem 2 (A. Avila and M. Viana). For every admissible, there exists a map , verifying the properties:
- Invariance: the map satisfies ;
- for -almost every , and as ;
- for all , we have for a set of positive -measure.
- is the terminal word of of length ;
- is an ellipsoid of semi-major axis of lengths ;
- is the subspace generated by the largest semi-major axis.
This result corresponds to Theorem A.1 in Avila-Viana’s article. As the reader can check (from the [one-page] subsection A.6 [Proof of Theorem 7.1] of this article), it is not hard to deduce Theorem 1 from Theorem 2.
Just to say a few words on why is this so, we observe that Theorem 2 provides us with (verifying the properties above) for each . By using the same theorem with the time “reversed”, one gets (verifying analogs of the properties of ) for . By the third property above, one can show the “transversality property”: for almost every . Also, the second property above implies that is associated to the largest exponents and is associated to the smallest exponents, so that the transversality property permits to “separate” the th exponent from the th exponent (in the sense that ) for any admissible, and the simplicity of the Lyapunov spectrum of (in the sense we defined above) follows.
This “reduces” our considerations to the discussion of Theorem 2.
-Proof of Theorem 2-
Definition. A u-state (“u” standing for “unstable”) is a probability measure on such that , where is the natural projection, and there exists a constant with
for any Borelian , .
Roughly speaking, the previous condition says that u-states are almost product measures.
Example. Given any probability measure on , is a u-state with .
Proposition 1. There exists a u-state invariant under .
Proof. The argument is very classical and we will only sketch its main steps.
Even though the space may not be compact (in the case of an alphabet with countably many symbols), the space of probability measures on projecting to is compact in the weak-* topology. In particular, for each , it follows that the space of u-states with is a convex compact set.
Now we observe that:
Exercise. If be a u-state, then is also a u-state with .
This exercise is a direct (3 lines) computation (whose solution can be found in Lemma A.2 of Avila-Viana’s article).
Using this exercise, one can complete the proof of the proposition by the standard Krylov-Bogolyubov argument: by letting be an arbitrary u-state, and by putting , we have that, by convexity, compactness and the statement of the exercise, the Cesaro averages of accumulate on -invariant u-states (with ).
The following result is a simple application of the martingale convergence theorem (for a direct [half-page] proof of it, see Lemma A.4 of Avila-Viana’s article):
Proposition 2. Let be a probability measure on with . For , and Borelian, let
Then, for -almost every , converges to some . Here, we recall that is the terminal word of of length .
By Proposition 1, we can fix a -invariant u-state. Denote by , where is the natural projection. Of course, is a probability measure on . Given , we define the following sequence of probabilities on :
By the -invariance of , we have
Since is a u-state, we obtain
On the other hand, by definition,
so that we get
Of course, since the definition of u-state is “symmetric”, we can “reverse” the role of and in the argument above. The conclusion is
for any Borelian .
Corollary. The probability measure is equivalent to any accumulation point of the sequence .
As it is explained in the end of subsection A.4 of Avila-Viana’s article, the proof of Theorem 2 (i.e., Theorem A.1 in their article) is complete once we can prove the following fact:
Proposition 3. For -almost every , there exists a subsequence , , converging to a Dirac mass.
Indeed, this proposition says that accumulates a Dirac mass. Since, by the corollary above, is equivalent to any accumulation point of , it follows that itself is a Dirac mass, say . Then, it is possible to check (cf. end of subsection A.4 of Avila-Viana’s article) that has the desired properties in the statement of Theorem 2: for instance, the fact that has the desired invariance properties (first item of Theorem 2) follows from the -invariance of , while the other two remaining items need (of course) the pinching and twisting assumptions.
In any event, this “reduces” our task to the proof of Proposition 3. We start by noticing that the pinching and twisting assumptions on imply that
- a) there exists such that, for each admissible and for every pair of -invariant subspaces , one has ;
- b) there exists , and such that, for each admissible and for every , there exists with and the angle between and is . Here, is the subspace associated to largest exponents of .
The verification of a) and b) is not hard: by the pinching assumption, there are only finitely many -invariant subspaces (for admissible), so that a) follows from the twisting assumption; also, for each , we can use the twisting assumption to find , and a neighborhood of in such that and the angle between and is for every ; then, by compactness of (and Lebesgue covering lemma), b) follows.
Lemma 1 (cf. Lemma A.6 of Avila-Viana’s article). Let and a probability measure on . There exists and, for each , there exists such that, for , we have
where (here ) and is the ball of radius centered at .
This lemma is harder to state than to explain: geometrically, it says that, independently of the (long) word one picks, we can choose an appropriate “start” () so that the word obtained by the concatenation of and has the property that concentrates the most of the mass of any probability measure on on a tiny ball .
For the proof of this lemma, we will need the following fact: the family is equicontinuous with respect to on to the complement of a neighborhood of -planes non-transverse to , that is, on the set of -planes making an angle with the -dimensional subspace (for some fixed ). Below we exemplify this fact in the particular case of matrices:
Example. Any matrix can be decomposed as where and are orthogonal and is diagonal. Since the orthogonal matrices act on Grassmanians in an equicontinuous way, it suffices to understand the action of the diagonal part. In the case of a matrix , we write
with . Suppose we are interested in studying the action on -planes (as the case of lines is too simple). Then, the -direction is , and, in this case, a -plane makes angle with the direction if and only if it contain a pair of vectors
whose coordinates verify . Of course, we see that in this part of the Grassmanian of -planes, we can use and as coordinates in order to study the action of . Since
we get that the action of (on this part of the Grassmanian of -planes) in terms of the coordinates and is
Thus, we see that the action of this family on the complement of a neighborhood of is equicontinuous.
Proof of Lemma 1. Given an arbitrary probability , we apply . In this way, we get a probability whose mass concentrates nearby some point of invariant under . Now, we use (given by item a) above) in order to get probability measures concentrated nearby a point of of in general position. Next, we re-apply to get probability measures concentrated nearby the point .
Here we see a subtle but crucial gain of information: in the first application of , we get nearby some isotropic -plane invariant under , but after adjusting with the “twisting” , in the second application of , we get nearby a very specific isotropic -plane, namely, associated to the largest exponents of .
Continuing the argument, we take and, by using item b) above, we select such that the angle between
is . Then, by applying , we get probability measures concentrated nearby a point uniformly transverse to (i.e., making angle with) .
By the equicontinuity of the family on the complement of a -neighborhood of , we see that, by applying , one ends up with a probability measure concentrated nearby , where .
Coming back to the proof of Proposition 3, we recall that our task is to prove that for -almost every , there exists a subsequence converging to a Dirac mass (with the aid of Lemma 1 above). Here, we take a slightly different route from Avila-Viana’s article and, instead of following the arguments of their Lemma A.7, we will conclude with a density point plus bounded distortion argument (in the spirit of the proof of the ergodicity of Bernoulli shifts).
We wish to apply Lemma 1 above with (as, by definition, ).
In order to do so, we fix , and we claim that for -almost every , there are infinitely many integers such that with .
The proof of this claim is by contradiction. Suppose that, for some fixed , the set of points for which the claim is false has positive -measure. By definition, we can write , where is the set of such that has not the desired form (i.e., with ) for every . Since for all , and has positive -measure, we can find such that still has positive -measure.
Now, we take a density point . By definition, given , we can find such that
for all .
On the other hand, the bounded distortion property ensures the existence of a constant such that
for every and . Of course, may go to as , but since is fixed in this argument here (as we stressed out above), this is not a serious issue.
By taking and , we have
for all . In particular, by denoting by the complement of , one has
for every and . Combining this with the estimate coming from the bounded distortion property, we find that
for every and . In other words, we conclude that
for all and . Of course, this is a contradiction because, by the definition , one has for and .
So we proved the claim that, for each fixed , the set of such that there are infinitely many integers such that with has full -measure.
Since the intersection of a countable family of sets of full measure still has full measure, the claim implies the slightly stronger property that for -almost every , one has that, for every , there are infinitely many integers such that with .
As one can check from the statement of Lemma 1, this slightly stronger property is exactly what we need to be able to apply Lemma 1 to and . The conclusion is: for -almost every , we can extract a subsequence converging to a Dirac mass, so that the proof of Proposition 3 (and today’s post) is complete.
Next time, we will start the discussion of some strategies towards the verification of the pinching and twisting assumptions in concrete examples.