Alex Eskin and I have just upload to Arxiv our note “A coding-free simplicity criterion for the Lyapunov exponents of Teichmüller curves”.
The basic idea behind this note started with an informal conversation with Alex (last August at I.H.E.S.). The central point of our discussion was the fact that the current proofs in the literature of simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle (KZ cocycle) for -invariant measures use at some point specific codings (countable Markov partitions) of the Teichmüller flow adapted to the support of the measure at hand.
For example, the celebrated article of A. Avila and M. Viana on the simplicity of KZ cocycle for Masur-Veech measures (formely known as Kontsevich-Zorich conjecture) depends at some stage on a combinatorial analysis of Rauzy-Veech algorithm (a coding of Teichmüller flow that is well-adapted for Masur-Veech measures), and, more recently, in a work (still in preparation) by Martin Möller, Jean-Christophe Yoccoz and myself (discussed in these five previous posts here) where a simplicity criterion of Lyapunov spectra of -invariant measures associated to square-tiled surfaces is developed with the aid of a coding coming from a finite extension of the continued fraction algorithm.
Evidently, the use of a coding makes the arguments technically easier, but it has a drawback that its construction is not always a piece of cake. In particular, it is natural to ask whether one can get rid of codings when studying the Lyapunov exponents of the Kontsevich-Zorich cocycle.
As it turns out, in the case of Teichmüller curves (i.e., closed -orbits in the moduli space of Abelian differentials of genus ), one can bypass the usage of codings because of the following argument (whose idea came from a conversation between Alex Eskin and Alex Furman some years ago) explained in Section 2 of our note.
By definition, the study of Lyapunov exponents of KZ cocycle is “essentially” the study of the growth of singular values of the matrices obtained as follows. One picks a sequence of long piece of “generic” trajectory, say for “typical”, one considers the elements of the mapping-class group associated to the closed loop obtained by closing with a very small segment between and (assuming that is generic enough so that its geodesic orbit is recurrent and thus we can find appropriate ‘s) and one looks at the matrix corresponding to the action on the first homology of .
In practice, the “problem” with this “strategy” of study of Lyapunov exponents of KZ cocycle is that it is not easy to use it to prove nice statements and/or compute about Lyapunov exponents because we don’t have any control of the sequence of elements ! In particular, a priori, we can’t pretend that the matrices are selected via some random walk process.
In part, this is why the works of Y. Guivarc’h and A. Raugi, and I. Goldsheid and G. Margulis can’t be directly applied to the study of Lyapunov exponents of KZ cocycle. Also, this explains in part why (technically speaking) codings are so useful (and why they may be hard to construct): in some sense, a nice coding allows us to pass from a problem with memory of studying a linear cocycle over trajectories of a deterministic dynamical system (e.g., the geodesic flow ) to a problem almost without memory of studying linear cocycles over well-behaved countable Markov shifts.
Remark 1 In the case of the article of A. Avila and M. Viana, the “almost no memory” feature is formally expressed by the bounded distortion condition in Appendix A of their paper.
Fortunately, in the case of Teichmüller curves, one can overcome the use of codings by noticing that they are very well-behaved supports of -invariant measures. Indeed, a Teichmüller curve is naturally isomorphic to a homogenous space where is a lattice (finite covolume subgroup) of (in the literature is called Veech group). In this language, the Kontsevich-Zorich cocycle is a linear cocycle of symplectic integral matrices over the Teichmüller geodesic flow on , i.e., the KZ cocycle correspond to a (monodromy) representation . A priori, one still has the problem that the sequence of matrices is “unknown”, but here a profound theorem of H. Furstenberg comes to our rescue. Informally, this theorem says that there is a choice of non-zero weights on the elements of leading to a probability measure such that one can really pretend that the matrices are obtained by a random walk with law . Formally, this translates into the fact that, for an appropriate choice of on , one has that the so-called Poisson boundary of is (see, e.g., this survey of A. Furman for a gentle introduction to the Poisson boundary).
Intuitively, in the case of a Teichmüller curve with finitely generated Veech group , say , one can think of Furstenberg’s theorem as follows. We can think of as the unit cotangent bundle of the hyperbolic surface , and we can think of as the hyperbolic disk tilled by copies of a fundamental domain of such that the crossing through sides of a fundamental domain correspond to applying one of the elements . (A nice concrete example is the modular curve, where , and ) In this situation, a typical geodesic in is a geodesic ray on from the origin landing at a -typical point in its boundary , and the matrices are obtained by multiplying a certain number of times some of the elements depending on how our geodesic ray crosses the several boundaries of our (-invariant) tilling of . Of course, by naively looking at this situation, one could be tempted to try to look at the random walk on obtained by assigning probability to each as a way of approximating the elements related to typical geodesic rays. A good way of testing this idea is to look at the distribution on the boundary of the points where our random walks “rays” land, and, after thinking a bit, it is not hard to convince oneself that there is no a priori reason that the probability on assigning weight to each has distribution on equal to the distribution of the geodesic rays (namely, the Lebesgue measure ). Nevertheless, the theorem of H. Furstenberg says that we can change into another (fully supported) probability on such that the distribution on the boundary (i.e., whose Poisson boundary) is precisely .
Anyhow, it is clear from our discussion so far that, in the case of Teichmüller curves, the homogeneity of the support of -invariant measures allow to apply Furstenberg’s theorem to convert a problem with memory (Lyapunov exponents of KZ cocycle) into a problem without memory (growth of singular values of random products of matrices in ). In particular, one can apply any simplicity theorem for random walks on groups of matrices (such as the results of Guivarc’h and Raugi, Goldsheid and Margulis, and Avila and Viana) to Teichmüller curves and this precisely the content of Theorem 1 in our note.
Remark 2 A natural question motivated by this discussion is the extension of Furstenberg’s theorem to the non-homogenous setting of general -invariant measures supported on high-dimensional affine orbifolds in moduli spaces of Abelian differentials. Interestingly enough, this question is open (to the best of my knowledge) despite the fact that several results from the theory of homogenous spaces after Furstenberg’s theorem now have counterparts in Teichmüller dynamics (see, e.g., the paper of A. Eskin and M. Mirzakhani on Ratner-like theorems).
Once we know that we don’t need codings to study Lyapunov exponents of KZ cocycle over Teichmüller curves, we illustrate this by considering the so-called class of Prym Teichmüller curves (previously discussed in this blog here). For our particular application in Section 3 of the note, we considered the Lyapunov spectrum of Prym Teichmüller curves in the minimal stratum of genus because:
- among the 4 non-negative Lyapunov exponents , we know:
- as it was pointed out in Remark 7 of this post here, some numerical experiments indicate that the Lyapunov spectra of these Teichmüller curves are simple;
- the commensurability classes of the Veech groups of these Teichmüller curves are essentially unknown when they are not arithmetic (i.e., associated to square-tiled surfaces), so that a coding-based study of Lyapunov exponents is probably hard to implement.
Here, we showed that, for an infinite family of non-arithmetic Prym Teichmüller curves in , the two unknown Lyapunov exponents are distinct, i.e., . The proof of this fact consists of:
- studying some concrete models (prototypes) of translation surfaces in these Prym Teichmüller curves (introduced by E. Lanneau and D. Nguyen; see also this post here),
- considering elements elements of the Veech group based on the geometry (cylinder decomposition) of the concrete models,
- computing (by brute force) the matrices , and in the image of the monodromy representation ,
- showing (again by brute force) that certain products of the matrices , and are complicated enough so that the Galois criterion (combined with Faltings theorem) discussed in these posts here applies to give the separation of the “unknown” Lyapunov exponents and of random products of matrices in the image of ,
- applying the coding-free simplicity criterion to conclude that the previous statement for random products has its counterpart for the KZ cocycle over the corresponding Teichmüller curves.
Closing this post, let us make the following two comments concerning our application to Prym Teichmüller curves in .
Remark 3 We believe that the statement is true for all Prym Teichmüller curves in . However, in our note we contented ourselves with proving this only for an infinity family of such Teichmüller curves because the main point of the note is not the application to Prym Teichmüller curves, but the coding-free simplicity criterion. In other words, the “Prym application” serves only to illustrate the point that the coding-free simplicity criterion can tackle some situations that are hard to treat by coding-based methods.
Remark 4 By carefully inspecting the previous paragraph, the reader will notice that we make no attempt of comparing and with the other two Lyapunov exponents. Indeed, there is a good reason for this: the Lyapunov exponents and belong to a certain “irreducible block” of the KZ cocycle while the exponents and belong to their own “irreducible blocks”; however, all current simplicity statements (for random products or in general) are only able to distinguish Lyapunov exponents within an irreducible block (but unfortunately not across irreducible blocks), and this is why the “best” one can hope to say with the current technology is .