After re-reading the preliminary version of this survey text with Giovanni Forni, I noticed that it doesn’t make much sense to continue this series. Indeed, the results about the Lyapunov spectrum of square-tiled cyclic covers, or more generally cyclic covers, of the Riemann sphere that I was planning to describe were:
- either discussed in details in the survey text (and there is no point in exactly reproducing the parts of survey here)
- or already discussed in previous posts (e.g., here) in this blog.
In other words, my laziness to write the posts made that, after stopping the series for one year, there is nothing to add to the survey and/or the previous posts in this blog on this subject.
So, I will declare that this series came to its end (mainly because of the survey text’s fault! :) ), and I hope to discuss more recent developments in the Ergodic Theory on moduli spaces of Abelian differentials (e.g., Eskin-Kontsevich-Zorich formula and Eskin-Mirzakhani theorem) in future posts.
On the other hand, in order to avoid “wasting” this post, let’s take advantage of our previous post on the Kontsevich-Forni formula (preparatory to Pascal Hubert’s Bourbaki seminar talk delivered today) to discuss the positivity of the Lyapunov exponents of the Kontsevich-Zorich cocycle with respect to Masur-Veech measures and its applications to the weak mixing property of interval exchange transformations and translation flows.
1. Second Lyapunov exponent of KZ cocycle and weak mixing property for i.e.t.’s
1.1. Second Lyapunov exponent of KZ cocycle with respect to Masur-Veech measures
In this subsection we will give a sketch of proof of the following result:
Theorem 1 Let be a connected component of some stratum of and denote by the corresponding Masur-Veech measure. Then, .
This result is part of one of the main results of G. Forni’s 2002 paper showing that . However, we’ll not discuss the proof of the more general result because
- one already finds several of the ideas used to show during the sketch of proof of , and
- by sticking to the study of we avoid the (rather technical) discussion of characterizing Oseledets unstable subspaces of KZ cocycle via basic currents.
In any event, we start the sketch of proof of Theorem 1 by recalling (from Theorem 2 in this previous post) that the form , , is relevant in the study of first variation of the Hodge norm in view of the formula:
where , .
Also, recall that where and . Moreover, is KZ cocycle invariant and it contributes with the Lyapunov exponents. Hence, since KZ cocycle preserves the symplectic intersection form, we get that the Lyapunov exponents , come from the restriction of KZ cocycle to .
Denoting by where , , the complex-valued bilinear form on induced by (on ) via Hodge representation theorem, we obtain the following nice immediate consequence of this discussion:
Corollary 2 Let be an ergodic -invariant probability and suppose that for all . Then, .
Next, let us also recall from this previous post that, by taking a Hodge-orthonormal basis of , we have a matrix , , associated to , so that is (essentially) the curvature form of the holomorphic subbundle of the complex Hodge bundle equipped with the Gauss-Manin connection. The eigenvalues of have the form where is an eigenvalue of , i.e., induces a positive semi-definite form on . In particular, this geometrical interpretation hinted that should naturally enter into second variation formulas for the Hodge norm (of course, this should be compared with the fact that naturally enters into first variation formulas for the Hodge norm) and, a fortiori, the eigenvalues of should provide nice consequences to the study of Lyapunov exponents. In fact, as it was proposed by M. Kontsevich and proved by G. Forni, one can relate the eigenvalue of to Lyapunov exponents of KZ cocycle via the so-called Kontsevich-Forni formula:
Theorem 3 (M. Kontsevich, G. Forni) Let be a -invariant -ergodic probability on a connected component of some stratum of . Then, one has the following formula for the sum of non-negative Lyapunov exponents of KZ cocycle with respect to :
Remark 1 It is not hard to see that for all . In particular, since , one can rewrite the formula above as
Cf. Remark 11 of this previous post.
Corollary 4 Let be a -invariant -ergodic probability on a connected component of some stratum of . Suppose that . Then, for all .
Evidently, this corollary shows how one can prove Theorem 1: since Masur-Veech measures are fully supported, it suffices to check that for some . In other words, by Kontsevich-Forni formula (Theorem 3), we have that:
Corollary 5 If for some then .
By this corollary, the proof of Theorem 1 is reduced to the following theorem:
Theorem 6 In any connected component of a stratum of one can find some with .
Roughly speaking, the basic idea (somehow recurrent in Teichmüller dynamics) to show this result is to look for near the boundary of after passing to an appropriate compactification. More precisely, one shows that, by considering the so-called Deligne-Mumford compactification , there exists an open set near some boundary point such that for any “simply” because the “same” is true for .
A complete formalization of this idea is out of the scope of these notes as it would lead us to a serious discussion of Deligne-Mumford compactification, some variational formulas of J. Fay and A. Yamada, etc. Instead, we offer below a very rough sketch of proof of Theorem 6 based on some “intuitive” properties of Deligne-Mumford compactification (while providing adequate references for the omitted details).
The first step towards finding the boundary point is to start with the notion of Abelian differentials with periodic Lagrangian horizontal foliation:
Definition 7 Let be an Abelian differential on a Riemann surface . We say that the horizontal foliation is periodicwhenever all regular leaves of are closed, i.e., the translation surface can be completely decomposed into maximal cylinders corresponding to some closed regular geodesics in the horizontal direction.The homological dimension of with periodic horizontal foliation is the dimension of the (isotropic) subspace of generated by the waist curves of the horizontal maximal cylinders decomposing .
We say that has periodic Lagrangian horizontal foliation whenever its homological dimension is maximal (i.e., ).
In general, it is not hard to find Abelian differentials with periodic horizontal foliation: for instance, any square-tiled surface (or origami) verifies this property and the class of square-tiled surfaces is dense on (as they correspond to rational points of the strata).
Next, we claim that:
Lemma 8 contains Abelian differentials with periodic Lagrangian horizontal foliation.
Proof: Of course, the lemma follows once we can show that given with homological dimension , one can produce an Abelian differential with homological dimension . In this direction, given such an , we can select a closed curve disjoint from (i.e., zero algebraic intersection with) the waist curves of horizontal maximal cylinders of and in .
Then, let’s denote by the Poincaré dual of given by taking a small tubular neighborhoods of and taking a smooth function on such that
where (resp. ) is the connected component of (resp. ) to the right/left of with respect to its orientation of (see the figure below)
In this setting, since the waist curves of maximal cylinders of of generate a -dimensional isotropic subspace (as has homological dimension ) and is disjoint from ‘s, it is possible to check (see the proof of Lemma 4.4 of Forni’s 2002 paper) that the Abelian differential has homological dimension whenever is sufficiently small.
This completes the proof of the lemma.
Now, let’s fix with periodic Lagrangian horizontal foliation and let’s try to use to reach some nice boundary point on the Deligne-Mumford compactification of (whatever this means…). Intuitively, we note that horizontal maximal cylinders of and their waist curves looks like this
In particular, by applying Teichmüller flow and letting , we start to pinching off the waist curves . As it was observed by H. Masur (see also Section 4 of Forni’s 2002 paper and references therein), by an appropriate scalingprocess on , one can makes sense of a limiting object in the Deligne-Mumford compactification of looking like this:
Roughly speaking, this picture is intended to say that lives in a stable curve , i.e., a Riemann surface with nodes at the punctures obtained after pinching ‘s off, and it is a meromorphic quadratic differential with doublepoles (and strictly positive residues) at the punctures and the same zeroes of .
If has homological dimension , it is possible to check that lives in a sphere with paired punctures and has strictly positive residues on each of them. In this situation, certain variational formulas of J. Fay and A. Yamada allowing to show that as approaches , one has
whenever is a Hodge-orthonormal basis of the dual of the (-dimensional) subspace of generated by the waist curves ‘s of .
Remark 2 It is implicit in the formulas of J. Fay and A. Yamada the fundamental fact that can be interpreted as the “derivative of the period matrix”. See Section 4 of Forni’s 2002 paper for more comments.
In other words, up to orthogonal matrices, the matrix of the form approaches as . Hence, as , and, a fortiori, the rank of is as . Thus, this completes the sketch of proof of Theorem 6.
Remark 3 Actually, the fact that “approaches” can be used to show that
In a nutshell, the previous discussion around Theorem 6 can be resumed as follows: firstly, we searched (in ) some with periodic Lagrangian horizontal foliation; then, by using the Teichmüller flow orbit of and by letting , we spotted an open region of (near a certain “boundary” point ) where the form becomes an “almost” diagonal matrix with non-vanishing diagonal terms, so that the rank of is maximal. Here, we “insist” that the inspiration for spotting near the boundary of comes from the fact that is a sort of derivative of the so-called period matrix , and one knows since the works of J. Fay and A. Yamada that the period matrix (and therefore ) has nice asymptotic expansions near the boundary of . The following picture is a résumé of the discussion of this paragraph:
Remark 4 These arguments (concerning exclusively Masur-Veech measures) were extended by G. Forni in this article here to give the following far-reaching criterion for the non-uniform hyperbolicity of KZ cocycle with respect to a -invariant -ergodic probabilities (satisfying a certain local product structure property): if one can find in the support of with periodic Lagrangian horizontal foliation (i.e., there is some with homological dimension ), then .
1.2. Weak mixing property for i.e.t.’s and translation flows
The plan for this subsection is to vaguely sketch how the knowledge of the positivity of the second Lyapunov exponent of KZ cocycle with respect to Masur-Veech measures on connected components of strata was used by A. Avila and G. Forni to show weak mixing property for i.e.t.’s and translation flows.
For those interested in learning in a more serious way the ideas behind the work of A. Avila and G. Forni, I strongly recommend reading the extremely well-written survey of G. Forni on the occasion of the Brin prize 2011 won by Artur Avila.
Recall that a dynamical system preserving a probability is weak mixing whenever
for any measurable subsets . Equivalently, is weak mixing if given measurable subsets , there exists a subset of density one(i.e., ) such that
For the case of i.e.t.’s and translation flows, it is particularly interesting to consider the following spectral characterization of weak mixing.
An i.e.t. is weak mixing if for any there is no non-constant measurable function such that
for every .
Similarly, a (vertical) translation flow on a translation surface represented by the suspension of an i.e.t. , say with a (piecewise constant) roof function for (see the picture below)
is weak mixing if for any there is no non-constant measurable function such that
for every , .
This spectral characterization of weak mixing allowed W. Veech to setup a criterion of weak mixing for i.e.t.’s and translation flows: roughly speaking, in the case of translation flows , it says that if is not weak mixing, say the equation
has a non-constant measurable solution for some , then, by considering the times when the Teichmüller orbit of the translation surface comes back near itself, i.e., the times such that the is close , the KZ cocycle sends , where is thought as an element of the (relative) homology of , i.e., , near the integer lattice in (relative) homology:
Actually, this is a very crude approximation of Veech’s criterion: the formal statement depends on the relationship between Teichmüller flow/KZ cocycle and Rauzy-Veech-Zorich algorithm, and we will not try to recall it here.
Instead, we will close this post by saying that the idea to deduce weak mixing for “almost every” i.e.t.’s and translation flows is to carefully analyze the KZ cocycle in order to prove that “tends” to keep “typical” lines in homology sufficiently “far away” from the integral lattice when the second Lyapunov exponent (with respect to Masur-Veech measures ) is positive. In other words, one of the (many) key ideas in the original article of A. Avila and G. Forni is to show that Equation (1) can be contradicted for “almost every” i.e.t.’s and translation flows when , so that Veech’s criterion implies weak mixing property for “almost every” i.e.t.’s and translation flows.