Last Monday (January 7, 2013), Maxim Kontsevich gave an excellent talk entitled “Lyapunov exponents for variations of Hodge structures” at IHES. The talk was attend by a highly heterogenous audience (for instance, among the senior participants were Mark Pollicott, David Ruelle and Mikhail Gromov) and Maxim did a great job in explaining his ideas in a language that was comfortable to all of us.
Below I will transcript my notes of Maxim’s lecture. Of course, the eventual mistakes in what follows are my entire responsibility.
1. Lyapunov exponents
Let be a probability space and be an invertible measure-preserving transformation, that is, is a -action on .
Denote by a measurable -equivariant vector bundle, i.e., for each , the fiber of over is a -vector space of dimension depending measurably on , and we have linear isomorphisms depending measurably on .
In what follows, we will suppose that the fibers of can be equipped with a measurable family of norms such that the following -condition is fulfilled:
We will be interested in the dynamics of the linear isomorphisms . More precisely, denote by the linear isomorphisms obtained by successive compositions of , , , .
In this situation, let us consider the following quantity
measuring the average of the largest (logarithmic) growth of vectors in the fibers after iterations.
Note that is a sub-additive sequence (i.e., ). From this, one can deduce that the limit
exists. In the literature, is called the average top Lyapunov exponent.
Remark 1 We are talking about average Lyapunov exponents here because, for the time being, we are making no ergodicity assumption on .
It is worth to notice that the number depends on the choice of norms in a very mild way: if we replace by another norm satisfying
then is not changed.
Remark 2 More generally, the other Lyapunov exponents can be defined by considering the top Lyapunov exponents of the exterior powers of .
In this setting, the multiplicative ergodic theorem of V. Oseledets says that if is ergodic, then there are real numbers and an unique filtration of (defined at almost every ) such that
Remark 3 Actually, since is invertible, by “intersecting” the filtrations associated to and , we can decompose into a direct sum of subspace verifying the property above (for almost every ).
2. Lyapunov exponents for VHS of weight 1
2.1. General setting
Let us consider the following setup. Let be a complex projective curve, a finite set of points in , and
a representation of the fundamental group of .
We will assume that has unipotent monodromy around the “cusps”, i.e., the spectrum of the monodromy matrices , where is a small loop around , is contained in the unit circle .
This condition is natural: from the dynamical point of view, it corresponds to the -condition needed to get the existence of Lyapunov exponents (and from the algebro-geometrical point of view, it is relevant in the discussions of semisimplicity of monodromy representations).
Here, by “Lyapunov exponents”, we mean the following. Consider any metric in the conformal class determined by with area , and consider the Brownian motion on as the analog of the transformation . Now, using the representation , we can construct a vector bundle over , and, thus, we can talk about Lyapunov exponents.
The key observation here is that the Lyapunov spectrum is independent on the particular choice of the metric in the conformal class (and this is an incarnation of the conformal invariance of the Brownian motion).
In particular, one can compute these Lyapunov exponents in practice, by selecting the (hyperbolic) metric of curvature in the conformal class and we can replace the Brownian motion by the geodesic flow.
In order to appreciate the advantage of this replacement, let us assume that is a finite cover of the modular curve (this is the case if is the Teichmüller curve of a square-tiled surface). In this setting, one can compute the Lyapunov exponents with the aid of continued fractions as they code the geodesic flow on the modular curve.
Also, the replacement of the Brownian motion by the geodesic flow doesn’t change the Lyapunov exponents because typical trajectories of the Brownian motion do not deviate “too much” from geodesics on .
2.2. VHS of weight 1
Suppose now that (). The basic example here is a variation of Hodge structures (VHS) of weight 1, i.e., we consider a complex algebraic surface given by a fibration and, for each , we consider the first cohomology group with complex coefficients of the fiber over . By Hodge theory, we have that where , resp. , consists of holomorphic, resp. anti-holomorphic, -forms on the fiber over .
In this context, the Lyapunov exponents are symmetric with respect to the origin, i.e., and one has the following formula for the sum of the non-negative exponents:
Here, is the (orbifold) degree of the determinant line bundle associated to .
At this point, M. Gromov asked M. Kontsevich if he really meant . His answer was “yes”, and, actually, there is no typo here: even though Lyapunov exponents are real numbers tending to be irrational, their sum is rational by a sort of algebro-geometrical miracle. Then, the next question of M. Gromov was about the rationality of individual exponents, and M. Kontsevich’s answer was “in general nothing is known”.
A nice combinatorial application of this formula is the following one. Let and consider the (Nielsen) operations and . In this way, we see that the free group acts on .
Theorem 1 For every orbit of such that generate a transitive action on and the cycle structure of the commutator is (i.e., it is a single cycle of length ). Then,
Remark 4 As M. Kontsevich pointed out, maybe the number `9′ is `7′ and/or `31/27′ are something else, but this is not the main point of this theorem (see below). The reader interested in precise values should consult this article here.
The main fact behind this theorem is that this combinatorial application involves computations of Lyapunov exponents of square-tiled surfaces and, as far as he is aware of, there is currently no other way of getting this application without using the formula for Lyapunov exponents of VHS of weight 1.
Very roughly speaking, the proof of the formula goes as follows. The sum measures the growth of volume of real Lagrangian subspaces . Next, by computing with the so-called Hodge norm, one can check that, given a base point , the averages of the variations of the logarithm of the volume of after parallel transport to circles around don’t depend on , and, from this, one can deduce the formula. (This point was already previously discussed in more details in this previous post here).
3. Lyapunov exponents for VHS of higher weight
During the last summer (in August 2012), M. Kontsevich had some conversations with M. Möller, A. Eskin and A. Zorich, and the question of extending these formulas for other VHS was raised: indeed, from the point of view of Algebraic Geometry, there is no a priori reason to restrict oneself to VHS of weight 1.
After these conversations, M. Kontsevich started looking at several examples (and doing lots of numerical experiments), and, for the sake of this talk, he decided to focus on Calabi-Yau 3-folds (3CY for short).
More concretely, there are 14 examples of families of 3CY whose moduli space is . Recall that the 3rd cohomology group has dimension and the corresponding Hodge numbers are . In particular, from these 3CY we get monodromy representations
Among the 14 examples, M. Kontsevich found that:
- 7 cases are “good” in the sense that the formula holds and, as it turns out, they occur when the image is a thin subgroup of (in Peter Sarnak‘s nomenclature);
- 7 cases are “bad” in the sense that and, as it turns out, they occur when the image is “arithmetic”.
In order to understand what is going on here, let us consider the following general question: when is it possible to calculate the sums of exponents, or equivalently, the top Lyapunov exponent of a certain exterior power, using a formula like the one above?
We’ll try to answer this question in the following more or less general setting. Let be a local system of -vector spaces carrying polarized complex VHS whose lowest term of Hodge filtration of (dual of ) has rank , that is, contains a holomorphic line bundle .
Then, the claim is that, in “good circumstances”, .
More precisely, consider the projectivization of (whose fibers are ) and let denote some open subset invariant under the Gauss-Manin connection.
By the ergodic theorem, by looking at the unit circles inside the complex lines giving the top Lyapunov exponent, we get that, for almost every , there exists a circle forming a “fractal” set
supporting the unique invariant measure (for this last statement, I think that simplicity of the top exponent is implicitly used, but I’m not completely sure…) for the natural cocycle dynamics.
Now, let us consider the closure of this fractal set . In this language, the conjectural picture that M. Kontsevich has in mind is the following:
- We can find ;
- Otherwise, if is the only choice possible, then this happens because one has “logarithmic singularities” with positive density in , and they contribute non-trivially to a certain integral formula (thus explaining why in these cases).
At this point, M. Kontsevich gave two examples of good cases (where a open set can be done by hand).
Firstly, in the case of mirror quintic 3CY, the idea is that the open set given by the complement of the fixed direction of the unipotent matrices given by the monodromy around the cusps fits the desired requirements. Alternatively (and more explicitly in terms of periods), one can construct this open set by considering the multivalued function
on , and
In this notation, by putting,
then, the claim that (in the universal cover) allows to construct .
Secondly, let us consider the so-called hypergeometric local systems, that is, we take . Then, there exists an unique unipotent monodromy representation
- the spectrum of is , where is a small loop around ;
- the spectrum of is , where is a small loop around ;
- where is a small loop around and is a rank matrix.
Closing his lecture, Maxim Kontsevich pointed out that, in some sense, these examples work because of the nice features of the exterior power representation of the symplectic group . In particular, he told that one could construct even more interesting examples (not based on symplectic group) if one could answer the following (Lie) group-theoretical question:
Question. Find a real reductive group (not related to symplectic groups) and a representation such that:
- there is a action such that the subgroup is compact (here ),
- has a unique highest eigenvector, and
- there exists such that is not perpendicular to the highest vector in the previous item.
In principle, he believes that these kind of groups and representations might exist among certain examples Shimura varieties, but he ended his talk saying that he was not wishing to explain why he had this feeling.
4. A personal final remark on M. Kontsevich’s talk
Closing this post, let me mention a little (simple) remark concerning the Lyapunov exponents in the case of mirror quintic 3CY.
In fact, the first time I heard about this new work of M. Kontsevich was through my friend Hossein Movasati: indeed, Hossein was in the audience of a similar talk by Maxim in China (during Maxim’s visit to receive his Shaw Prize 2012).
After a few exchanges of emails with Hossein, I learned that, in the case of mirror quintic 3CY’s, the image is the monoid generated by the following two matrices:
Actually, the shape of these matrices was mentioned to me by A. Zorich and M. Kontsevich in August 2012, but I learned the precise value of their entries in this paper here of Hossein.
In principle, the positivity of the Lyapunov exponents in this case might have applications (if I understood correctly what Hossein told me), and so I decided to do the exercise of checking if this positivity property is true or not.
Firstly, let us notice that the formula allows to deduce the positivity of the sum but it leaves the possibility that . However, as we will see now, using the simplicity criterion developed by M. Möller, J.-C. Yoccoz and myself, it is possible to prove the simplicity of this Lyapunov spectrum, i.e., .
More precisely, let us recall that, by the works of Y. Guivarc’h and A. Raugi, I. Goldsheid and G. Margulis and, more recently, A. Avila and M. Viana, we know that, given a monoid of symplectic matrices, the random products of elements in have simple Lyapunov spectrum (i.e., non-zero Lyapunov exponents with multiplicity one) whenever the following conditions are satisfied:
- Pinching: there exists whose eigenvalues are all real with pairwise distinct modulus;
- Twisting: there exists a twisting matrix with respect to a pinching matrix , i.e., for all and all pairs of -invariant subspaces such that is isotropic and is coisotropic, one has .
In the context of symplectic matrices with integral coefficients, it was recently shown by M. Möller, J.-C. Yoccoz and myself (see this post here) that the pinching and twisting properties can be verified with the aid of the following “Galois-theoretical” criterion:
- , , ,
- , , , and
are positive integers that are not squares. Then, the matrices and are pinching and some product of powers of and is twisting with respect to , and, a fortiori, the monoid has the pinching and twisting properties.
In order to apply this discussion to our particular case of mirror quintic 3CY’s, we consider the matrices and . Their respective characteristic polynomials are
The corresponding discriminants are
It follows that one can apply Theorem~2 to conclude the simplicity of the Lyapunov spectrum of any monoid containing the matrices and .
Remark 5 We deduced that the Lyapunov exponents of random products of and are simple. In order to pass this information on random products to the geodesic flow, it suffices to use the main result in our paper with Alex Eskin (see this post here).