Posted by: matheuscmss | May 10, 2013

## A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces

Martin Möller, Jean-Christophe Yoccoz and I have just upload to ArXiv our paper “A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces“. In some sense, the main mathematical content of this paper was already discussed in a certain amount of details in this blog (see these five posts here), and, thus, this short post will just give some quick “historical comments” on this paper.

This article started with a discussion we had in 2009 about some possible extensions of the celebrated work of A. Avila and M. Viana on the Kontsevich-Zorich conjecture, i.e., simplicity of Lyapunov spectrum of the Kontsevich-Zorich cocycle with respect to Masur-Veech measures, to other ergodic $SL(2,\mathbb{R})$-invariant probability measures on moduli spaces of Abelian differentials/translation surfaces.

Of course, we were aware of examples of ergodic $SL(2,\mathbb{R})$-invariant probability measures associated to certain square-tiled surfaces constructed by G. Forni and his coauthor showing that the simplicity of Lyapunov spectrum is simply false in general, that is, for square-tiled surfaces of genus $g\geq 3$.

On the other hand, some numerical experiments by A. Zorich (with Mathematica), V. Delecroix and myself (with SAGE) seemed to indicate that the examples found by G. Forni were rare in the sense that, after fixing a stratum (i.e., the orders of zeroes) of the Abelian differentials, almost all (say, all but finitely many) square-tiled surfaces generated ergodic $SL(2,\mathbb{R})$-invariant probability measures such that the Lyapunov spectrum of the Kontsevich-Zorich cocycle is simple.

For this reason, we decided to investigate sufficient conditions for the simplicity of Lyapunov spectrum of square-tiled surfaces (or more precisely, the ergodic $SL(2,\mathbb{R})$-invariant probability measures associated to them) with the hope of showing eventually that the simplicity property for almost all square-tiled surfaces in some stratum (of genus $g\geq 3$).

Our first attempt (still in 2009) was based on the computation of the Zariski closure of the group of matrices of the Kontsevich-Zorich cocycle (restricted to the symplectic complement of the so-called tautological Lyapunov subspaces). More precisely, this group of matrices is naturally identified to a subgroup of the symplectic group $Sp(2g-2,\mathbb{Z})$ and one could expect, after listing the possible Zariski closed subgroups of $Sp(2g-2,\mathbb{R})$ at least for low genus $g$ (say $g=3, 4$), to derive some geometric criterion (say particularly “easy” to apply to square-tiled surfaces) ensuring that a subgroup of $Sp(2g-2,\mathbb{Z})$ has full Zariski closure.

Unfortunately, after some partial success in the case $g=3$, we failed to produce a reasonable general criterion “easily applicable to square-tiled surfaces” out of the strategy of the previous paragraph (essentially because putting your hands in the Zariski closure of subgroups of $Sp(2g-2,\mathbb{Z})$ is tricky for large $g$).

After this first (partially successful, partially frustrated) attempt, we started looking in 2010 for simplicity criteria of Lyapunov spectra closer to Avila-Viana work. More precisely, we began to search for a systematic way of verifying the so-called pinching and twisting conditions of Avila-Viana. Very roughly speaking, the pinching condition asks for a matrix $A$ with simple real eigenvalues of distinct modulus and the twisting condition asks for a matrix $B$ putting the eigenspaces of $A$ in general position. Of course, the advantage of Avila-Viana simplicity criterion is that it ensures simplicity without requiring any computation with Zariski closures. Furthermore, the pinching condition is normally easy to check: indeed, if your cocycle has simple spectrum then some matrix is pinching and, “conversely”, if you have a hard time finding some pinching matrix then there is probably a good reason for this to happen and it is likely that your Lyapunov spectrum is not simple. So, this leaves us with the question of checking the twisting condition in a systematic way.

In general, the twisting condition is a subtle linear algebra problem: we want a single matrix $B$ putting all eigenspaces of a given pinching matrix $A$ at the same time (and this explains why A. Avila and M. Viana got this property in the context of the Kontsevich-Zorich conjecture through a subtle induction scheme).

Fortunately, in the context of square-tiled surfaces, our matrices have integer coefficients. Therefore, it is intuitive that, if the splitting field of a matrix $B$ is disjoint from the splitting field of the pinching matrix $A$ (in the sense that $\mathbb{Q}$ is the unique common subfield of these splitting fields), then some products of powers of $B$ and $A$ put all eigenspaces of $A$ in general position: indeed, the condition on splitting fields says that $B$ does not share invariant subspaces with $A$ and this kind of property is not very far from twisting.

As it turns out, it is not so simple to transform this intuition into a theorem, but we managed to do this along the following lines. First, we convert the twisting condition into a combinatorial statement about the completeness of certain graphs. Secondly, we show that the verification of these graphs are complete can be reduced to prove that some related graphs are mixing. Finally, we show the mixing property of these graphs by proving that our graphs have “lots of arrows” (using elementary Galois theory to “spread arrows around”) and by proving that the non-mixing cases of these graphs with lots of arrows imply the existence of common invariant subspaces of $A$ and $B$ or $B^2$. Here, this last step follows from a somewhat lengthy combinatorial study of these graphs in Section 4 (occupying 11 pages of our paper).

Anyhow, once we get our Galois-criterion for simplicity of the Lyapunov spectrum of square-tiled surfaces, we apply it for the stratum $\mathcal{H}(4)$, that is, the minimal stratum of genus $3$ Abelian differentials with a single zero of order $4$.

In fact, our particular choice of $\mathcal{H}(4)$ for our application is not by chance: in some sense, it is the smallest stratum where the simplicity of Lyapunov spectrum of square-tiled surfaces is not know, and, more importantly, there is a conjectural classification of ($SL(2,\mathbb{R})$-orbits of) square-tiled surfaces by V. Delecroix and S. Lelievre based on numerical experiments by them. Actually, this classification is very close in spirit to the classification of square-tiled surfaces in the minimal stratum $\mathcal{H}(2)$ in genus $2$ by P. Hubert and S. Lelievre, and C. McMullen.

In a nutshell, the conjecture of Delecroix-Lelievre predicts the existence of exactly $9$ distinct types of $SL(2,\mathbb{R})$-orbits of square-tiled surfaces in $\mathcal{H}(4)$. Furthermore, it is not hard to produce explicit  representatives of each of these orbits. In our paper, we compute (mostly by hand, but with some computer-assisted calculations with $41$ polynomials of degree $4$) the splitting fields of certain matrices of the Kontsevich-Zorich cocycle over each of these $9$ types of orbits of square-tiled surfaces. The outcome of this calculation is the following: one can apply the Galois-criterion for these $9$ types of orbits if some discriminants of these polynomials of degree $4$ are not squares and this last property holds for all but finitely many square-tiled surfaces in these orbits because of Siegel’s theorem (implying that a “reasonable” degree $4$ polynomial takes squares as its values only finitely many times).

In summary, from our Galois-criterion of simplicity and Siegel’s theorem, we deduce (unconditionally) that there are infinitely many square-tiled surfaces in $\mathcal{H}(4)$ with simple Lyapunov spectrum, and, conditionally to Delecroix-Lelievre conjecture, we get that all but finitely many square-tiled surfaces in $\mathcal{H}(4)$ have simple Lyapunov spectrum.

Closing this post, let us mention that there is little hope of improving our conditional statement above to “all square-tiled surfaces in $\mathcal{H}(4)$ have simple Lyapunov spectrum“. Indeed, even though we believe that this last statement is true, the fact that we used Siegel’s theorem (to derive our conditional statement) makes that the current best bounds on the size of the eventual “exceptional set” are doubly exponential (i.e., they are numbers of the form $\exp(\exp(600))$ where $600$ is a bound on the coefficients of the degree $4$ polynomials mentioned earlier) simply because the current quantitative versions of Siegel theorem (such as this one by Y. Bilu) provide only doubly exponential bounds (in some sense, this is not a surprise because the bounds are exponential on the “heights” and the “heights” are logarithmic on the coefficients of polynomials). In particular, a simple-minded strategy based on using some computer program to rule out exceptions for square-tiled surfaces with $n$ squares for $5\leq n\leq \exp(\exp(600))$ is out of reach.

## Responses

1. “(…) and, thus, this short post will just give some quick “historical comments” on this paper.”
This is it, this post turned out to be a great discussion on “how to do mathematics”, reminds me of this one: