Posted by: matheuscmss | December 22, 2011

Neutral Oseledets bundles of the Kontsevich-Zorich cocycle

These days I gave two talks (the first one on Tuesday, December 13, entitled Some examples of cocycles with wild central Oseledets bundle during the conference Recent advances in modern dynamics held at the University of Warwick, and the second one on Wednesday, December 21, entitled Neutral Oseledets bundles of the Kontsevich-Zorich cocycle during the Christmas workshop of Karlsruhe University) around some results (from joint works with G. Forni and A. Zorich, and A. Avila and J.-C. Yoccoz) on the neutral Oseledets bundles of the Kontsevich-Zorich cocycle partly announced in this previous post here. Below the fold, the reader will find an expanded version of my lecture notes.

Acknowledgments. I would like to thank the organizers of the two conferences above (in particular, Corinna Ulcigrai and Gabriela Schmithüsen) for the invitation to deliver the talks at the origin of these notes.

-Some cyclic covers of the Riemann sphere {\overline{\mathbb{C}}}-

We consider the family of curves (in the moduli space) described by the following algebraic equation

\displaystyle C_6 = C_6(x_1,\dots,x_6)=\{y^6=(x-x_1)(x-x_2)(x-x_3)(x-x_4)(x-x_5)(x-x_6)\}

where {x_1,\dots, x_6\in\overline{\mathbb{C}}} are mutually distinct points of the Riemann sphere {\overline{\mathbb{C}}}. The natural projection {p:C_6\rightarrow\overline{\mathbb{C}}} , {p(x,y)=x}, is a (ramified) covering and the reader can use the Riemann-Hurwitz formula to check that {C_6(x_1,\dots,x_6)} is a family of genus 10 curves.

The group of deck transformations of {p} is naturally isomorphic to the cyclic group {\mathbb{Z}/6\mathbb{Z}} and it is generated by {T(x,y)=(x,\zeta_6y)} where {\zeta_6= \exp(2\pi i/6)}. In other words, {C_6} is a family of cyclic covers of the Riemann sphere. In particular, we can form the real, resp. complex Hodge bundle {H^1(C_6,\mathbb{R})}, resp. {H^1(C_6,\mathbb{C})} over the family {C_6} whose fiber over a point {C_6(x_1,\dots,x_6)} is its first cohomology group, and we can decompose these bundles into a direct sum of eigenspaces of the action {T^*} of {T} in the first cohomology group of {C_6}. For instance, in the case of the complex Hodge bundle, we observe that {(T^*)^6=\textrm{Id}} (because {T^6=\textrm{Id}}), so that the eigenvalues of {T^*} belong to the set {\{1,\zeta_6,\zeta_6^2,\zeta_6^3=-1,\zeta_6^4,\zeta_6^5\}} of {6}th roots of unity. Moreover, since the action of {T^*} preserves the natural Hodge filtration {H^1(C_6,\mathbb{C}) = H^{1,0}(C_6)\oplus H^{0,1}(C_6)} (into holomorphic and anti-holomorphic 1-forms), we can write

\displaystyle H^1(C_6,\mathbb{C})=\bigoplus\limits_{j=0}^5H^1_{\zeta_6^j}(C_6,\mathbb{C}) = \bigoplus\limits_{j=0}^5\left(H^{1,0}_{\zeta_6^j}(C_6)\oplus H^{0,1}_{\zeta_6^j}(C_6)\right)

where {H^{1,0}_{\zeta_6^j}(C_6)} and {H^{0,1}_{\zeta_6^j}(C_6)} are the eigenspaces associated to the eigenvalue {\zeta_6^j} of the action of {T^*} on {H^{1,0}(C_6)} and {H^{0,1}(C_6)}, and {H^1_{\zeta_6^j}(C_6,\mathbb{C})} is the eigenspace associated to the eigenvalue {\zeta_6^j} of the action of {T^*} on the complex Hodge bundle {H^1(C_6,\mathbb{C})}.

We observe that the eigenspace associated to the eigenvalue {1=\zeta_6^0} is trivial: indeed, any non-trivial cohomology class in this eigenspace would project under {p} into a non-trivial cohomology class of the Riemann sphere {\overline{\mathbb{C}}} (whose first cohomology group is trivial).

Moreover, the reader can verify that the following list is a basis of holomorphic 1-forms of {C_6}:

\displaystyle \begin{array}{cccc} dx/y^2, & & & \\ dx/y^3, & xdx/y^3, & & \\ dx/y^4, & xdx/y^4, & x^2dx/y^4, & \\ dx/y^5, & x^2dx/y^5, & x^3dx/y^5, & x^4dx/y^5. \end{array}

In particular, we see that {\textrm{dim}_{\mathbb{C}}H^{1,0}_{\zeta_6^j}(C_6)=j-1} for every {j=1,\dots,5}. Furthermore, since {\zeta_6^{6-j}} is complex conjugate to {\zeta_6^j}, we conclude that {H^{0,1}_{\zeta_6^j}(C_6)=\overline{H^{1,0}_{\zeta_6^{6-j}}(C_6)}}, so that the dimensions of all pieces of the above decomposition of the complex Hodge bundle {H^1(C_6,\mathbb{C})} are determined.

This kind of discussion (of certain cyclic covers of {\overline{\mathbb{C}}}) was recently performed by C. McMullen (in this article here) in connection with a natural monodromy representation of braid groups in this context. For instance, let us fix an initial configuration of points {x_1(0),\dots,x_6(0)} and let us consider a continuous closed path {[0,1]\ni t\mapsto (x_1(t),\dots,x_6(t))} of configurations of six points such that for each {t\in[0,1]}, {x_1(t),\dots,x_6(t)} are mutually distinct and {\{x_1(1),\dots,x_6(1)\}=\{x_1(0),\dots,x_6(0)\}} (side remark: this last condition means that we find at time {t=1} the same configuration we had at time {t=0} as a set; of course, we could require that the configuration is the same as an ordered set; evidently, this alternative has its own interest as it leads to the pure braid group, but we will not consider it here). Informally, such a closed path corresponds to (continuously) move around an initial configuration of six points {x_1(0),\dots x_6(0)\in\overline{\mathbb{C}}} in a certain way and, after a while, we come back to the initial configuration. By definition, the braid group {B_6} of configurations of six points is the group of isotopy classes of closed paths as above. Of course, any element of the braid group {B_6} define a way of deforming complex structures in the moduli space via the map {[0,1]\ni t\mapsto C_6(x_1(t),\dots,x_6(t))}, and, again by definition, the variation of Hodge structures along such paths induces (with the aid of the so-called Gauss-Manin connection) a monodromy representation

\displaystyle \rho_6: B_6\rightarrow\textrm{Aut}(H^1(C_6(x_1(0),\dots,x_6(0)),\mathbb{C}))

of the braid group {B_6}.

It is not hard to check that the action of {T^*} commutes with the monodromy representation {\rho_6}, so that the eigenspaces {H^1_{\zeta_6^j}(C_6,\mathbb{C})} of {T^*} can be used to diagonalize by blocks {\rho_6}. In order to analyze the restrictions of {\rho_6} on each of these eigenspaces, it is convenient to introduce the “intersection” form

\displaystyle (\alpha,\beta):=\frac{i}{2}\int \alpha\wedge\overline{\beta}

for {\alpha,\beta\in H^1(C_6,\mathbb{C})}. In the literature, this Hermitian form is known as Hodge form. It is a Hermitian form with signature {(g,g)} as its restriction to {H^{1,0}(C_6)} is positive-definite, while its restriction to {H^{0,1}(C_6)} is negative-definite. As a matter of fact, the Hodge form is preserved by Gauss-Manin connection, so that {\rho_6} acts on {H^1(C_6(x_1(0),\dots,x_6(0)),\mathbb{C})} by automorphisms preserving the Hodge form {(.,.)}.

Now, we consider the restriction of {\rho_6} to an eigenspace {H^1_{\zeta_6^j}(C_6,\mathbb{C})}. Since {H^1_{\zeta_6^j}(C_6,\mathbb{C}) = H^{1,0}_{\zeta_6^j}(C_6)\oplus H^{0,1}_{\zeta_6^j}(C_6)}, {\textrm{dim}_{\mathbb{C}}H^{1,0}_{\zeta_6^j}(C_6)=j-1} and {H^{0,1}_{\zeta_6^j}(C_6)=\overline{H^{1,0}_{\zeta_6^{6-j}}(C_6)}}, we see that {\rho_6} acts on {H^1_{\zeta_6^j}(C_6,\mathbb{C})} by automorphisms preserving the restriction of the Hodge form {(.,.)} to it, that is, an Hermitian form of signature {(\textrm{dim}_{\mathbb{C}}H^{1,0}_{\zeta_6^j}(C_6),\textrm{dim}_{\mathbb{C}}H^{0,1}_{\zeta_6^j}(C_6)) = (j-1,5-j)}. In other words, we have that

\displaystyle \rho_6^{(j)}:=\rho_6|_{H^1_{\zeta_6^j}(C_6,\mathbb{C})}: B_6\rightarrow U(j-1,5-j)

where {U(p,q)} is the group of matrices preserving a Hermitian form of signature {(p,q)}.

Of course, the representations {\rho_6^{(j)}} and {\rho_6^{(6-j)}} are complex conjugated, so we only need to understand them when {j=3, 4} and {5}.

For {j=5}, we have that {\rho_6^{(5)}} acts by {U(4,0)} matrices, that is, {\rho_6^{(5)}} acts by isometries (with respect to the positive definite Hermitian form of signature {(4,0)} obtained by the restriction of the Hodge form to {H^1_{\zeta_6^5}(C_6,\mathbb{C})}). Moreover, the facts that {H^1_{\zeta_6^5}(C_6,\mathbb{C}) = H^{1,0}_{\zeta_6^5}(C_6)} and {H^1_{\zeta_6^5}(C_6,\mathbb{C})} is “purely topological” (in the sense that it is defined as the eigenspace of the cohomological action of the automorphism {T}) imply that {H^{1,0}_{\zeta_6^5}(C_6)} is a fixed part (rigid factor) of the Jacobian of {C_6}. The definition of a fixed part is a complex torus {A} (of positive dimension) such that there exists an isogeny {Jac(C)\rightarrow J(C)\times A}, where {C} is a family of curves and {A} is independent of {C}. In the case of {C_6}, we can informally say that {H^{1,0}_{\zeta_6^5}(C_6)} generates a rigid factor {H^{1,0}_{\zeta_6^5}(C_6)/(H^1_{\zeta_6^5}(C_6,\mathbb{C})\cap H^1(C_6,\mathbb{Z}\oplus i\mathbb{Z}))} on the Jacobian of {C_6} because the fact that a 1-form inside {H^{1,0}_{\zeta_6^5}(C_6)} is holomorphic is independent on the Riemann surface structure {C_6(x_1,\dots,x_6)} (as {H^{1,0}_{\zeta_6^5}(C_6)=H^1_{\zeta_6^5}(C_6,\mathbb{C})} and {H^1_{\zeta_6^5}(C_6,\mathbb{C})} is independent of Riemann surface structures since it is “purely topological”).

For the other values of {j} (i.e., {j=3} and {4}), we do not have actions by isometries (nor rigid factors), but C. McMullen computed the image of monodromy representations (in particular {\rho_6^{(4)}(B_6)}) by relating them to Artin systems, complex reflections (and sometimes to Burau representations). As a consequence of his computations, he was able to show that the action of {B_6} on {H^1_{\zeta_6^j}(C_6,\mathbb{C})} is irreducible for every {j\neq 3}.

Partly motivated by this, G. Forni, A. Zorich and I (see Appendix B of this preprint here) decided to “insert some dynamics” into C. McMullen’s discussion by attaching an Abelian differential to each Riemann surface {C_6(x_1,\dots,x_6)} in such a way that the resulting object in the moduli space of Abelian differentials is invariant under the natural action of {SL(2,\mathbb{R})} on such moduli spaces (see this post here for a brief account on the natural {SL(2,\mathbb{R})}-action on the moduli space of Abelian differentials).

More precisely, we attach to {C_6(x_1,\dots,x_6)} the Abelian differential {\omega=(x-x_1)dx/y^3}. By checking at the ramification points {x_1,\dots,x_6} (notice that there is no ramification at {\infty}), one see that {\omega} has a zero of order {8} over {x_1} and zeroes of order {2} over the five points {x_2,\dots,x_6}. We encode this information by saying that {\omega\in\mathcal{H}(8,2^5)} (see this post here for an “explanation” of this notation). Observe that, by Riemann-Hurwitz formula, {2g(C_6)-2= 8 + (5\times 2)}, i.e., {g(C_6)=10} (a fact that we already knew). In this way, we have that {(C_6,\omega)} defines a locus {\mathcal{Z}} of {\mathcal{H}(8,2^5)} (the moduli space of genus 10 Abelian differentials with a zero of order {8} and five double zeroes).

We claim that {\mathcal{Z}} is invariant under the natural {SL(2,\mathbb{R})}-action on {\mathcal{H}(8,2^5)}. This can be proved by the following argument. By definition, {\omega} is {T^*}-anti-invariant, so that {\omega^2} is {T^*}-invariant, and hence {\omega^2} can be projected into Riemann sphere {\overline{\mathbb{C}}}. By direct verification, one sees that

\displaystyle p^*(\omega^2)=(x-x_1)(dx)^2/(x-x_2)\dots(x-x_6),

that is, {\omega^2} projects (under {p}) into a quadratic differential with a single zero of order {1} (simple zero) and five poles of order {-1} (simple poles), i.e., {p^*(\omega^2)\in\mathcal{Q}(1,-1^5)}. Now, by definitions, the natural {SL(2,\mathbb{R})}-action on {\mathcal{Z}} commutes with the covering map {p} (as {SL(2,\mathbb{R})} acts by post-composition with appropriate charts while the covering has to do with pre-compositionwith charts). By combining this with the fact that {\mathcal{Q}(1,-1^5)} is {SL(2,\mathbb{R})}-invariant (since this action don’t change the order of zeroes), we conclude that {\mathcal{Z}} is also {SL(2,\mathbb{R})}-invariant.

Remark 1 Just to “count dimensions”: {\mathcal{Z}} has dimension {4} because we need {3} parameters to determine the complex structure of {C_6(x_1,\dots,x_6)} (as we can always normalize {3} of them to be {0,1,\infty}), and {1} parameter to determine the Abelian differential, and {\mathcal{Q}(1,-1^5)} has dimension {4} as well (by using period coordinates, see this post here).

Observe that we have an intermediate covering {h:C_6\rightarrow C_2}, {h(x,y)=(x,y^3)}, where

\displaystyle C_2=C_2(x_1,\dots,x_6)=\{z^2=(x-x_1)\dots(x-x_6)\}.

One has {h^*(\omega)=(x-x_1)dx/z^2} is an Abelian differential with a double zero over {x_1} and no zeroes otherwise, i.e., {h^*(\omega)\in\mathcal{H}(2)}. By the same argument above, {(C_2,h^*(\omega))} defines a {SL(2,\mathbb{R})}-invariant locus of dimension {4} of {\mathcal{H}(2)}. Since {\mathcal{H}(2)} has dimension {4} (again by using period coordinates, see this post here), we get that the locus {(C_2,h^*(\omega))} is exactly {\mathcal{H}(2)}. In other words, {\mathcal{Z}} is a copy of (i.e., it is isomorphicto) {\mathcal{H}(2)} inside the moduli space of Abelian differentials of genus 10.

Just to get a “feeling” of what the {SL(2,\mathbb{R})} action on {\mathcal{Z}} looks like, we notice the following facts. It is not hard to check that the flat structure associated to {(C_2, h^*(\omega))} is described by the following octagon (whose opposite parallel sides are identified):

Flat structure associated to a {(C_2,h^*(\omega))\in\mathcal{H}(2)}.

Here, the vertices of this octagon are all identified to a single point corresponding to {x_1}. Moreover, since {x_2,\dots,x_6} are Weierstrass points of {C_2}, one can organize the picture in such a way that the four points {x_2,\dots, x_5} are located exactly at the middle points of the sides, and {x_6} is located at the “symmetry center” of the octagon. See the picture above for an indication of the relative positions of {x_1} (marked by a black dot) and {x_2,\dots, x_6} (marked by crosses). In this way, we obtain a concrete description of {\mathcal{H}(2)} where the {SL(2,\mathbb{R})} action is reasonably easy to understand: given {A\in SL(2,\mathbb{R})} and denoting by {\omega\in\mathcal{H}(2)} the Abelian differential associated to the planar figure above, we define {A\omega} to be the Abelian differential associated to the object obtained by letting the matrix {A} act on the planar figure above.

Moreover, since the locus {\mathcal{Z}} is defined by Abelian differentials {(C_6,\omega)} given by certain triple (ramified) covers of the Abelian differentials {(C_2,h^*(\omega))\in\mathcal{H}(2)}, one can check that the flat structure associated to {(C_6,\omega)} are described by the following picture:

Flat structure associated to a {(C_6,\omega)\in\mathcal{Z}}.

Here, we glue the half-sides determined by the vertices (black dots) and the crosses of these five pentagons in a cyclic way, so that every time we positively cross the side of a pentagon indexed by {j}, we move to the corresponding side on the pentagon indexed {j+1} (mod {5}). For instance, in the figure above we illustrated the effect of going around the singularity point over {x_6}.

In this language, the Teichmüller geodesic flow is the action of the diagonal subgroup {g_t=\textrm{diag}(e^t,e^{-t})} of {SL(2,\mathbb{R})}. From now on, we will restrict our attention to the unit area Abelian differentials inside our loci {\mathcal{Q}(1,-1^5)}, {\mathcal{H}(2)} and {\mathcal{Z}}. It is not hard to check that the {SL(2,\mathbb{R})}-action preserves the total area of Abelian differentials, and moreover, by the results of H. Masur and W. Veech, the action of the Teichmüller flow is ergodic in the subset of unit area elements of {\mathcal{Q}(1,-1^5)}, {\mathcal{H}(2)} (and, a fortiori, {\mathcal{Z}}) with respect to a natural (“Lebesgue-like”) {SL(2,\mathbb{R})}-invariant probability {\mu_{MV}} (sometimes called Masur-Veech measure). Please see this post here for more information on this.

Remark 2 In fact, the works of K. Calta and C. McMullen allow to classify all {SL(2,\mathbb{R})}-invariant probabilities supported on {\mathcal{H}(2)} (somehow in the spirit of Ratner’s theorem). In principle, most of the discussion below could be extended to all of these probability measures, but, for sake of simplicity, we will stick to the Masur-Veech measure in the sequel.

In view of the ergodicity of the Teichmüller flow, we can play the following game: starting with a “typical” {\omega\in\mathcal{Z}} (i.e., for {\mu_{MV}}-a.e. {\omega\in\mathcal{Z}}), we can run the Teichmüller flow {g_t} for a very long time {t} until we come back very close to {\omega}; by completing the trajectory segment {[\omega,g_t(\omega)]:=\{g_s(\omega):s\in[0,t]\}} with a small path connecting {\omega} to {g_t(\omega)}, we get a closed path {\gamma_t(\omega)}; then, we can look at the monodromy matrix on the (real and/or complex) Hodge bundle associated to all (homotopy classes of) paths {\gamma_t(\omega)} obtained in this way. In the literature, this “monodromy representation over the Teichmüller flow” is known as the Kontsevich-Zorich (KZ) cocycle {G_t^{KZ}}. For an introduction to the Kontsevich-Zorich cocycle the reader may consult this post here. Actually, we can think of the Kontsevich-Zorich cocycle as a part of the monodromy representation of the {SL(2,\mathbb{R})}-action on the Hodge bundle (through parallel transport with respect to Gauss-Manin connection).

The main goal of today’s post is the study of the Lyapunov spectrum (that is, the collection of Lyapunov exponents) of the Kontsevich-Zorich cocycle on the Hodge bundle over {\mathcal{Z}} as a prototypical example of the conjectural general behavior of KZ cocycle on the Hodge bundle of the support of any {SL(2,\mathbb{R})}-invariant probability supported in any stratum of moduli space of Abelian differentials. Before proceeding, let me just do some propaganda on why one should care about Lyapunov exponents of KZ cocycle. A first reason (coming from Dynamics) is the fact that these exponents can be shown to govern deviations of Birkhoff sums (ergodic averages) of interval exchange transformations, translation flows and billiards on rational polygons essentially because the Teichmüller flow and KZ cocycle act as “renormalization dynamics” for these (zero entropy) systems. A second reason (coming from Statistical Mechanics) is the fact that these exponents were recently shown to govern the rates of diffusion on Ehrenfest “wind-tree” model of Lorenz gases. A third reason (coming from Algebraic Geometry) is the fact that the sum of Lyapunov exponents can be related to “orbifold degrees” of the determinant bundle of the Hodge bundle (by some formulas derived by M. Kontsevich, G. Forni, I. Bouw and M. Möller, and, more recently, A. Eskin, M. Kontsevich and A. Zorich). The reader is encouraged to consult A. Zorich’s survey, these posts here, and these articles here as some references for these three motivations for the study of Lyapunov exponents of KZ cocycle.

Remark 3 In fact, during the discussion below, we will consider exclusively the non-negative Lyapunov exponents of KZ cocycle: indeed, it is known that the Lyapunov spectrum of KZ cocycle is symmetric with respect to the origin (i.e., whenever {\lambda} is a Lyapunov exponent then {-\lambda} is also a Lyapunov exponent) due to a certain “symplecticity” (see this post here for more details).

We begin with the remark that {SL(2,\mathbb{R})}, and hence KZ cocycle, acts by monodromy (with respect to Gauss-Manin connection) on the Hodge bundle over {\mathcal{Z}}. Therefore, the decomposition of {H^1(C_6,\mathbb{C})} in terms of eigenspaces of {T^*} and the Hodge form are preserved by them. In particular, the restriction of {G_t^{KZ}} to {H^1_{\zeta_6^j}(C_6,\mathbb{C})} acts by matrices inside the group {U(j-1,5-j)}.

As a consequence, for {j=1} and {5}, {G_t^{KZ}} acts on {H^1_{\zeta_6^j}(C_6,\mathbb{C})} by isometries, and hence the Lyapunov exponents are all zero (as they measure exponential rates of growth). A nice information coming out of this is the continuity (and actually real-analyticity) of this part of the neutral Oseledets bundle {E^0_{\mu_{MV}}} (associated to the vectors with zero Lyapunov exponents): indeed, as we just saw, for {j=1} or {5}, one has {E^0_{\mu_{MV}}\cap H^1_{\zeta_6^j}(C_6,\mathbb{C}) = H^1_{\zeta_6^j}(C_6,\mathbb{C})}, and, in general, {H^1_{\zeta_6^j}(C_6,\mathbb{C})} depends continuously (actually analytically) on the base point {C_6(x_1,\dots,x_6)}. The reader should notice that this continuity of the neutral Oseledets bundle is somehow a “precious” information (since, in general, Oseledets theorem ensures only its measurability).

Remark 4 For some known examples (such as square-tiled cyclic covers, a family of cyclic covers of {\overline{\mathbb{C}}} branched at {4} points giving rise to interesting square-tiled surfaces, see these articles here for more information on them), it is possible to show continuity (and analyticity) of the neutral Osedelets bundle along the following lines. The variation of the Hodge form along cohomology classes is driven by the second fundamental form (Kodaira-Spencer map) {B} of the Gauss-Manin connection on the Hodge bundle in the sense that we can write down variational formulas of the form

\displaystyle \frac{d}{dt}(G_t^{KZ}(\alpha),G_t^{KZ}(\beta)) = B(\alpha,\beta)

Thus, if the annihilator {Ann(B)} of {B} is Teichmüller flow invariant, it is a natural candidate for the neutral Oseledets bundle {E^0_{\mu}}. Furthermore, it is not hard to check that {Ann(B)} depends continuously (actually analytically) on the base point. Hence, the continuity (and analyticity) of {E^0_{\mu}} follows whenever the equality {E^0_{\mu}= Ann(B)} is true. Actually, in the case of square-tiled cyclic covers, G. Forni, A. Zorich and I were capable (in Appendix A of this paper here) of verifying that the Teichmüller flow invariance of {Ann(B)} and the equality {Ann(B)=E^0_{\mu}}. Of course, this may lead one to conjecture that this maybe always true, but, as we are going to see below, this is not quite the case. :)

On the other hand, for {j=3}, we have that {H^1_{\zeta_6^3}(C_6,\mathbb{C})} is isomorphic to {H^1(C_2,\mathbb{C})} (as a {SL(2,\mathbb{R})}-module) through the natural injective map {h^*:H^1(C_2,\mathbb{C})\rightarrow H^1(C_6,\mathbb{C})}. In particular, {G_t^{KZ}|_{H^1_{\zeta_6^3}(C_6,\mathbb{C})}} has the same exponents as the Kontsevich-Zorich cocycle on the Hodge bundle over {\mathcal{H}(2)}. By the results of M. Bainbridge, in this particular (genus 2) situation, the (non-negative) Lyapunov exponents are {1} and {1/3}.

At this stage, it remains “only” to understand the (non-negative) Lyapunov exponents of the restriction of {G_t^{KZ}} to {H^1_{\zeta_6^j}(C_6,\mathbb{C})} for {j=2} and {4}. Actually, since {G_t^{KZ}|_{H^1_{\zeta_6^2}(C_6,\mathbb{C})}} is conjugated to {G_t^{KZ}|_{H^1_{\zeta_6^4}(C_6,\mathbb{C})}}, it suffices to discuss one of these cases, say {j=4}. Here, we have the information that {G_t^{KZ}|_{H^1_{\zeta_6^4}(C_6,\mathbb{C})}} has monodromy {U(3,1)}. We claim that this is sufficient to conclude that {G_t^{KZ}|_{H^1_{\zeta_6^4}(C_6,\mathbb{C})}} has {2} vanishing exponents at least. Indeed, this is a direct consequence of the following more general proposition about {U(p,q)} cocycles:

Proposition 1 Suppose that {G_t} is a {U(p,q)} cocycle, i.e., a linear cocycle on a (measurable) complex vector bundle {V} over an ergodic flow {g_t} (with respect to a probability {\mu}) on the base {M} of {V} preserving a (measurable) family of Hermitian forms {(.,.)} of signature {(p,q)}. Assume that {G_t} is a {\log}-integrable cocycle (with respect to some measurable family of norms on the fibers of {V}), so that the conditions of Oseledets theorem are met. Then, {G_t} has {|q-p|} vanishing exponents at least.

Remark 5 I’m sure that this proposition was known to experts in random products of matrices (such as A. Raugi and Y. Guivarch, and I. Goldscheid and G. Margulis) because its proof is completely elementary (as we’re going to see). However, I was unable to locate a precise reference where this appears for the first time.

The proof of this proposition has two ingredients. The first one is the fact that the stable {E^-_{\mu}} and unstable {E^+_{\mu}} Oseledets spaces (associated to negative and positive Lyapunov exponents of {G_t} resp.) are “exiled” to the light-cone (null cone) {\Sigma} of the Hermitian form {(.,.)}:

Lemma 2 One has {E^-_{\mu}, E^+_{\mu}\subset \Sigma:=\{v\in V: (v,v)=0\}}.

Proof: Take {v\in E^{\mp}_{\mu}}. On one hand, {(G_t(v),G_t(v))=(v,v)} for every {t\in\mathbb{R}} because the Hermitian form {(.,.)} is preserved by {G_t} (by hypothesis). On the other hand, the fact that {v\in E^{\mp}_{\mu}} implies that the norm of {G_t(v)} decays (exponentially fast) to zero as {t\rightarrow\pm\infty}.

At this point, one is tempted to say that the exponential decay of {G_t(v)} implies {(G_t(v),G_t(v))\rightarrow0} as {t\rightarrow\pm\infty}, so that one would have {(v,v)=0}, that is, {E^{\mp}_{\mu}\subset\Sigma}. But we should be a little bit careful (as we’re dealing with measurable families of Hermitian forms and norms). The formal argument goes as follows. Since our vector bundle {V} is finite-dimensional, we can “compare” the (measurable) family of Hermitian form with the (measurable) family of norms in the sense that a Cauchy-Schwarz inequality is true up to a multiplicative factor maybe depending (measurably) on the base point {x\in M}, i.e., {|(v,w)_x|\leq C_x\|v\|_x\cdot\|w\|_x}. By Luzin’s theorem, these comparisons can be made uniform on large compact sets (of almost full {\mu}-measure), so that, by the ergodicity of the flow {g_t}, we may assume {(G_t(v),G_t(v))} is uniformly controlled by the norms of {G_t(v)} for a sequence of times going to infinity. In any event, the conclusion is that {v\in E^{\mp}_{\mu}} implies {(v,v)=0}, and this completes the proof of the lemma. \Box

The second ingredient is the following simple linear algebra lemma:

Lemma 3 Let {V\subset\mathbb{C}^{p+q}} be a complex vector subspace contained in the light-cone of a Hermitian form {(.,.)} of signature {(p,q)}. Then, {\textrm{dim}_{\mathbb{C}}V\leq \min\{p,q\}}.

Proof: We can always choose our coordinates so that {(z,w)=z_1\overline{w_1}+\dots+z_p\overline{w_p}-z_{p+1}\overline{w_{p+1}}-\dots-z_{p+q}\overline{w_{p+q}}}. Without loss of generality, suppose that {p\leq q}. Reasoning by contradiction, let’s assume that {\textrm{dim}_{\mathbb{C}}V>p}. Then, one could find {p+1} linearly independent vectors {v^{(1)},\dots, v^{(p+1)}\in V}. Using these vectors, we can define {p+1} vectors {w^{(j)}=(v^{(j)}_1,\dots,v^{(j)}_p)\in\mathbb{C}^p} ({j=1,\dots, p+1}) by temporarily “forgetting” the last {q} coordinates of {v^{(j)}}. Now, we consider a non-trivial linear combination of {w^{(j)}} equal to {0}:

\displaystyle \sum\limits_{j=1}^{p+1}a_j w^{(j)}=0

where {(a_1,\dots,a_{p+1})\neq (0,\dots,0)}. It follows that the vector {v:=\sum\limits_{j=1}^{p+1}a_jv^{(j)}\in V} has its first {p} coordinates equal to zero, i.e., {v=(\underbrace{0,\dots,0}_p,v_{p+1},\dots,v_{p+q})} On the other hand, since {v\in V\subset \Sigma}, one has {0=(v,v)=-|v_{p+1}|^2-\dots-|v_{p+q}|^2}, so that {v=0}. In other words, we found a vanishing non-trivial linear combination {v=\sum\limits_{j=1}^{p+1}a_jv^{(j)}} of the {p+1} linearly independent vectors {v^{(j)}}, a contradiction. \Box

Putting these two ingredients (lemmas) together, we find that {\textrm{dim}_{\mathbb{C}}E^-_{\mu}+\textrm{dim}_{\mathbb{C}}E^+_{\mu}\leq 2\min\{p,q\}}. Since the total dimension of the fibers of {V} is {p+q}, we find that the neutral Oseledets bundle {E^0_{\mu}} has dimension {\textrm{dim}_{\mathbb{C}}E^0_{\mu}\geq p+q-2\min\{p,q\}=|q-p|}, so that the proof of Proposition 1 is complete.

Going back to our concrete example, we have that the restriction of {G_t^{KZ}} to {H^1_{\zeta_6^4}(C_6,\mathbb{C})} has 2 vanishing exponents (due to the monodromy {U(3,1)}).

Actually, it is possible to show that the remaining two Lyapunov exponents are non-zero, and, by using a formula for the sum of Lyapunov exponents due to A. Eskin, M. Kontsevich and A. Zorich, and by computing the so-called Siegel-Veech constant of {\mathcal{Z}}, one can determine their explicit value: {4/9} and {-4/9}. The details of this computation will appear in a forthcoming article by G. Forni, A. Zorich and myself.

In any event, given the discussion in Remark 4 above, one can ask whether {E^0_{\mu_{MV}}(\zeta_6^4):= E^0_{\mu_{MV}}\cap H^1_{\zeta_6^4}(C_6,\mathbb{C})} coincides with the annihilator of the second fundamental form {B} of Gauss-Manin connection restricted to {H^1_{\zeta_6^4}(C_6,\mathbb{C})}, and/or whether {E^0_{\mu_{MV}}(\zeta_6^4)} is continuous. In the forthcoming article by G. Forni, A. Zorich and myself (alluded to in the previous paragraph), we show that {E^0_{\mu_{MV}}(\zeta_6^4)} doesn’t coincide with the annihilator of {B}, but this still leaves open the possibility that {E^0_{\mu_{MV}}(\zeta_6^4)} is continuous.

Remark 6 During an exposition at Rennes (in January 2011), Y. Guivarch asked whether {G_t^{KZ}} still acts isometrically on {E^0_{\mu_{MV}}(\zeta_6^4)} (now that one can’t use variational formulas involving {B} to deduce this property) or whether one has genuine subexponential growth in this subbundle. As it turns out, {G_t^{KZ}} acts isometrically on {E^0_{\mu_{MV}}(\zeta_6^4)} by the following argument: one has that {E^0_{\mu_{MV}}(\zeta_6^4)} is outside the light-cone {\Sigma} because the stable and unstable Oseledets subspaces {E^{\pm}_{\mu_{MV}}} have dimension {1} (and corresponding exponents {\pm4/9}), and so, if {E^0_{\mu_{MV}}(\zeta_6^4)\cap\Sigma} were non-trivial, we would get a subspace {(E^0_{\mu_{MV}}(\zeta_6^4)\cap\Sigma)\oplus E^-_{\mu_{MV}}\subset \Sigma} of dimension at least {2} inside the light-cone {\Sigma} of an Hermitian form of signature {(3,1)}, a contradiction with the lemma above. In other words, the light-cone is a geometric mechanism of production of neutral Oseledets subbundles with isometric behavior genuinely different from the also geometric method of using the annihilator of the second fundamental form of Gauss-Manin connection of the Hodge bundle.

Heuristically, one strategy to “prove” that {E^0_{\mu_{MV}}(\zeta_6^4)} is not very smooth goes as follows: as it is indicated in this previous post here, the Lyapunov exponents of the Teichmüller flow can be deduced from the ones of the KZ cocycle by shifting them by {\pm1}; in this way, it is possible to check that the smallest non-negative Lyapunov exponent of the Teichmüller flow is {5/9=1-4/9}; therefore, the generic points tend to be separated by Teichmüller flow by {\geq e^{5t/9}} after time {t\in\mathbb{R}}; on the other hand, the largest Lyapunov exponent on the fiber {H^1_{\zeta_6^4}(C_6,\mathbb{C})} is {4/9}, so that the angle between the neutral Oseledets bundle over two generic points grows by {\leq e^{4t/9}} after time {t\in\mathbb{R}}; hence, in general, one can’t expect the neutral Oseledets bundle to be better than {\alpha=(4/9)/(5/9)=4/5} Hölder continuous.

Of course, there are several details missing in this heuristic, and currently I don’t know how to render it into a formal argument. However, in a recent work still in progress, A. Avila, J.-C. Yoccoz and I prove (among other things) that {E^0_{\mu_{MV}}(\zeta_6^4)} is not continuous at all (and hence only measurable by Oseledets theorem). The next section contains a brief sketch of this proof of the non-continuity of {E^0_{\mu_{MV}}(\zeta_6^4)}.

-Coding of the Kontsevich-Zorich cocycle over {\mathcal{Z}}-

The Teichmüller flow and the Kontsevich-Zorich cocycle over (connected components of) strata can be efficiently coded by means of the so-called Rauzy-Veech induction. Roughly speaking, given a (connected component of a) stratrum {\mathcal{C}} of Abelian differentials of genus {g\geq 1}, the Rauzy-Veech induction associates the following objects: a finite oriented graph {\mathcal{G}(\mathcal{C})}, a finite collection of simplices (“Rauzy-Veech boxes”) and a finite number of copies of a Euclidean space {\mathbb{C}^{2g}} over each vertex of {\mathcal{G}(\mathcal{C})}, and, for each arrow of {\mathcal{G}(\mathcal{C})}, a (expanding) projective map between (parts of) the simplices over the vertices connected by this arrow, and a matrix between the copies of {\mathbb{C}^{2g}} over the vertices connected by this arrow. (I strongly recommend J.-C. Yoccoz’s survey for more details on the Rauzy-Veech induction).

In this language, the simplices (Rauzy-Veech boxes) over the vertices of this graph represent admissible paramaters determining translations surfaces (Abelian differentials on Riemann surfaces {M}) in {\mathcal{C}}, the (expanding) projective map between (parts of) the simplices (associated to vertices connected by a given arrow) correspond to the action of the Teichmüller flow on the parameter space (after running this flow for an adequate amount of time), and the matrices (attached to the arrows) on {\mathbb{C}^{2g}} are the action of the Kontsevich-Zorich cocycle on the first cohomology group {H^1(M,\mathbb{C})\simeq\mathbb{C}^{2g}}.

Among the main properties of the Rauzy-Veech induction, we can highlight the fact that it permits to “simulate” almost every (with respect to Masur-Veech measure) orbit of Teichmüller flow on on {\mathcal{C}} in the sense that these trajectories correspond to (certain) infinite paths on the graph {\mathcal{G}(\mathcal{C})}. In order words, the Rauzy-Veech induction allows to code the Teichmüller flow as a subshift of a Markov shift on countably many symbols (as one can use loops on {\mathcal{G}(\mathcal{C})} based on an arbitrarily fixed vertex as basic symbols / letters of the alphabet of our Markov subshift). Moreover, the KZ cocycle over these trajectories of Teichmüller flow can be computed by simply multiplying the matrices attached to the arrows one sees while following the corresponding infinite path on {\mathcal{G}(\mathcal{C})}. Equivalently, we can think the KZ cocycle as a monoid of (countably many) matrices (as we can only multiply the matrices precisely when our oriented arrows can be concatened, but in principle we don’t dispose of the inverses of our matrices because we don’t have the right to “revert” the orientation of the arrows).

In the particular case of {\mathcal{H}(2)}, the associated graph {\mathcal{G}(\mathcal{H}(2))} is depicted below:

Rauzy diagram associated to {\mathcal{H}(2)}. The letters near the arrows are not important here, only the {7} vertices (black dots) and the arrows between them.

Now, we observe that {\mathcal{Z}} was defined by taking certain triple covers of Abelian differentials of {\mathcal{H}(2)}, so that it is also possible to code the Teichmüller flow and KZ cocycle on {\mathcal{Z}} by the same graph and the same simplices over its vertices, but by changing the matrices attached to the arrows: in the case of {\mathcal{H}(2)}, these matrices acted on {\mathbb{C}^4}, but in the case of {\mathcal{Z}} they act on {\mathbb{C}^{20}} and they contain the matrices of the case of {\mathcal{H}(2)} as a block.

At this stage, one can prove non-continuity of {E^0_{\mu_{MV}}(\zeta_6^4)} as follows.

Firstly, one computes the restriction of KZ cocycle (or rather the matrices of the monoid) to {E^0_{\mu_{MV}}(\zeta_6^4)} on certain “elementary” loops and one checks that they have finite order. In particular, every time we can get the inverses of the matrices associated to these elementary loops by simply repeating these loops an appropriate number of times (namely, the order of the matrix minus 1). On the other hand, since these elementary loops are set up so that any infinite path (coding a Teichmüller flow orbit) is a concatenation of elementary loops, one conclude that the action (on {E^0_{\mu_{MV}}(\zeta_6^4)\subset\mathbb{C}^{20}}) of our monoid of matrices is through a group! In particular, given any loop {\gamma} (not necessarily an elementary one), we can find another loop {\delta} such that the matrix attached to {\delta} (i.e., the matrix obtained by multiplying the matrices attached to the arrows forming {\delta} “in the order they show up” with respect to their natural orientation of {\delta}) is exactly the inverse of the matrix attached to {\gamma}.

Secondly, by computing with a pair of “sufficiently random” loops {\gamma_A} and {\gamma_B}, it is not hard to see that we can choose such that their attached matrices {A} and {B} have distinct and/or transverse central eigenspaces {E^0_A} and {E^0_B} (associated to eigenvalues of modulus {1}).

In this way, the periodic orbits (pseudo-Anosov orbits) of the Teichmüller flow coded by the infinite paths {\dots\gamma_A\gamma_A\gamma_A\dots} and {\dots\gamma_B\gamma_B\gamma_B\dots} obtained by infinite concatenation of the loops {\gamma_A} and {\gamma_B} have distinct and/or transverse neutral Oseldets bundle, but this is no contradiction to continuity since the base points of these periodic orbits are not very close. However, we can use {\gamma_A} and {\gamma_B} to produce a contradiction as follows. Let {k\gg 1} a large integer. Since our monoid acts by a group, we can find a loop {\gamma_{C,k}} such that the matrix attached to it is {A^{-k}}. It follows that the matrix attached to the loop {\gamma_{A,B,k}:=\underbrace{\gamma_A\dots\gamma_A}_{k}\gamma_{C,k}\gamma_B} is {A^k\cdot A^{-k}\cdot B}, i.e., {B}. Therefore, the infinite paths {\dots\gamma_A\gamma_A\gamma_A\dots} and {\dots\gamma_{A,B,k}\gamma_{A,B,k}\gamma_{A,B,k}\dots} produce periodic orbits whose neutral Oseledets bundle still are {E^0_A} and {E^0_B} (and hence, distinct and/or transverse), but this time their basepoints are arbitrarily close (as {k\rightarrow\infty}) because the first {k} “symbols” (loops) of the paths coding them are equal (to {\gamma_A}).

Remark 7 Actually, this argument is part of more general considerations (in the forthcoming paper by A. Avila, J.-C. Yoccoz and myself) on certain cyclic covers obtained by taking {2n} copies of a regular polygon with {m} sides, and cyclically gluing the sides of these polygons in such a way that their middle points become ramification points: indeed, {\mathcal{Z}} corresponds to the case {n=3} and {m=5} of this construction.

Remark 8 It is interesting to notice that the real version of Kontsevich-Zorich cocycle over {\mathcal{Z}} on {(H^1_{\zeta_6^2}(C_6,\mathbb{C})\oplus H^1_{\zeta_6^4}(C_6,\mathbb{C}))\cap H^1(C_6,\mathbb{R})} is a irreducible symplectic cocycle with non-continuous neutral Oseledets bundle. In principle, this irreducibility at the real level makes it difficult to see the presence of zero exponents, so that the passage to its complex version (where we can decompose it as a sum of two complex conjugated monodromy representations by matrices in {U(1,3)} and {U(3,1)}) reveals a “hidden truth” not immediately detectable from the real point of view (thus confirming the famous quotation of J. Hadamard: “the shortest route between two truths in the real domain passes through the complex domain”). I believe this example has some independent interest because, to the best of my knowledge, most examples of symplectic cocycles and/or diffeomorphisms exhibiting some zero Lyapunov exponents usually have smooth neutral Oseldets bundle due to some sort of “invariance principle” (see this article of A. Avila and M. Viana for some illustrations of this).

Closing today’s post, we state in the next (final) section two “optimistic guesses” (part of a forthcoming paper by G. Forni, A. Zorich and myself) on the features of the KZ cocycle over the support of general {SL(2,\mathbb{R})}-invariant probabilities. Notice that we call these “optimistic guesses” instead of “conjectures” because we think they’re shared (to some extent) by others working with Lyapunov exponents of KZ cocycle (and so it would be unfair to state them as “our” conjectures).

-Two optimistic guesses-

Optimistic Guess 1. Let {\mu} be a {SL(2,\mathbb{R})}-invariant probability in some connected component of a stratrum of Abelian differentials and denote by {\mathcal{L}} its support. Then, there exists a finite (ramified) cover {\widehat{\mathcal{L}}} such that (the lift of) the Hodge bundle {H^1_{\mathbb{C}}} over {\widehat{\mathcal{L}}} can be decomposed into a direct sum of continuous

\displaystyle H^1_{\mathbb{C}} = L\oplus (A_1\otimes W_1 \oplus\dots\oplus A_m\otimes W_m) \oplus (B_1\otimes(U_1\oplus \overline{U_1})\oplus\dots\oplus B_1\otimes(U_n\oplus \overline{U_n}))

where {W_1,\dots,W_m}, {U_1,\dots, U_n} are distinct {SL(2,\mathbb{R})}-irreducible representations admiting Hodge filtrations {W_i=W_i^{1,0}\oplus W_i^{0,1}}, {U_j= U_j^{1,0}\oplus U_j^{0,1}} such that {W_i=\overline{W_i}}, {U_j\cap\overline{U_j}=\{0\}}, {A_i}, {B_j} are complex vector spaces (taking into account the multiplicities of the irreducible factors {W_i}, {U_j}), and {L} is the tautological bundle {L=L^{1,0}\oplus L^{0,1}}, {L^{1,0}=\mathbb{C}\omega}, {L^{0,1}=\mathbb{C}\overline{\omega}}, {\omega\in\widehat{\mathcal{L}}}. Moreover, this decomposition is unique and it can’t be further refined after passing to any further finite cover.

Remark 9 When {\mu} is the (unique) {SL(2,\mathbb{R})}-invariant probability supported on a Teichmüller cover {\mathcal{L}} (i.e., a closed {SL(2,\mathbb{R})}-orbit), the Optimistic Guess 1 is a consequence of Deligne’s semisimplicity theorem.

Optimistic Guess 2. In the setting of Optimistic Guess 1, denote by

\displaystyle p_i=q_i=r_i=\textrm{dim}_{\mathbb{C}}W_i^{1,0}=\textrm{dim}_{\mathbb{C}}W_i^{0,1}

and

\displaystyle p_j=\textrm{dim}_{\mathbb{C}}U_j^{1,0}, q_j =\textrm{dim}_{\mathbb{C}}W_j^{0,1}, r_j=\min\{p_j,q_j\}

Then, the Lyapunov spectrum of the KZ cocycle on {W_i} is simple, i.e.,

\displaystyle \lambda_{i,1}>\dots>\lambda_{i,r_i}>-\lambda_{i,r_i}>\dots>-\lambda_{i,1}

and the Lyapunov spectrum of the KZ cocycle on {U_j} is “as simple as possible”, i.e.,

\displaystyle \lambda_{j,1}>\dots>\lambda_{j,r_j}>\underbrace{0=\dots=0}_{|q_j-p_j|}>-\lambda_{j,r_j}>\dots>-\lambda_{j,1}

Remark 10 This “guess” is based on the general philosophy (supported by works as the ones of A. Raugi and Y. Guivarch, and I. Goldscheid and G. Margulis) that, after reducing our cocycle to irreducible pieces, if the cocycle restricted to such a piece is “sufficiently generic” inside a certain Lie group of matrices {G}, then the Lyapunov spectrum on this piece should look like the “Lyapunov spectrum” (i.e., collection of the logarithms of the norms of eigenvalues) of the “generic” matrix of {G}. For instance, since a generic matrix inside the group {U(p,q)} has spectrum

\displaystyle \lambda_1>\dots>\lambda_r>\underbrace{0=\dots=0}_r>-\lambda_r>\dots>-\lambda_1

where {r=\min\{p,q\}}, the above guess essentially claims that, once one reduces the KZ cocycle to irreducible pieces, its Lyapunov spectrum on each piece must be as generic as possible.

Remark 11 Notice that the previous guess doesn’t make any attempt to compare Lyapunov exponents within distinct irreducible factors: indeed, in general non-isomorphic representations may lead to the same exponent by “pure chance” (as it happens in the case of certain genus {5} Abelian differentials associated to the “wind-tree model”).


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