Zero Lyapunov exponents of the Hodge bundle

Forni, Giovanni; Matheus, Carlos; Zorich, Anton

doi:10.4171/cmh/325

Cited by 23 publications

(29 citation statements)

References 22 publications

(33 reference statements)

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…• K = H and G is a quaternionic unitary group U H (p, q), p + q = d . Our main motivation to consider these cases come from the recent works [MYZ], [FMZ2] and [AMY] where several examples of cocycles with values in these matrix groups appear naturally as "blocks" of the Kontsevich-Zorich cocycle over the closure of SL 2 (R)-orbits of certain "symmetric" translation surfaces.…”

Section: 2mentioning

confidence: 99%

See 1 more Smart Citation

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

2014

Self Cite

View full text Add to dashboard Cite

ABSTRACT. We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle over the Teichmüller flow on the SL 2 (R)-orbit of a square-tiled surface. The simplicity of the Lyapunov spectrum has been proved by A. Avila and M.Viana with respect to the so-called Masur-Veech measures associated to connected components of moduli spaces of translation surfaces, but is not always true for square-tiled surfaces of genus 3. We apply our criterion to square-tiled surfaces of genus 3 with one single zero. Conditionally to a conjecture of Delecroix and Lelièvre, we prove with the aid of Siegel's theorem (on integral points on algebraic curves of genus > 0) that all but finitely many such square-tiled surfaces have simple Lyapunov spectrum.

show abstract

Section: 2mentioning

confidence: 99%

“…The matrix group G determines a priori constraints for the Lyapunov exponents (see, e.g., [MYZ] and [FMZ2]):…”

Section: For a Locally Constant Cocycle The Integrability Condition mentioning

confidence: 99%

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

2014

Self Cite

View full text Add to dashboard Cite

show abstract

“…Specifically, a connection between the curvature of the Hodge bundle and the neutral Oseledec bundle was studied in [FMZ14a]. In [FMZ14b], it was conjectured that there are exactly two mechanisms that produce zero Lyapunov exponents. The Forni subspace mechanism defined below will be the only one relevant to this paper.…”

Section: Aulicinomentioning

confidence: 99%

Affine invariant submanifolds with completely degenerate Kontsevich–Zorich spectrum

Aulicino¹

2016

Ergod. Th. Dynam. Sys.

View full text Add to dashboard Cite

Link to this article: http://journals.cambridge.org/abstract_S0143385716000262How to cite this article: DAVID AULICINO Afne invariant submanifolds with completely degenerate Kontsevich-Zorich spectrum.Abstract. We prove that if the Lyapunov spectrum of the Kontsevich-Zorich cocycle over an affine SL 2 (R)-invariant submanifold is completely degenerate, i.e. if λ 2 = · · · = λ g = 0, then the submanifold must be an arithmetic Teichmüller curve in the moduli space of Abelian differentials over surfaces of genus three, four, or five. As a corollary, we prove that there is at most a finite number of such Teichmüller curves. IntroductionThe Lyapunov exponents of the Kontsevich-Zorich (KZ) cocycle give information about the dynamics of numerous systems such as billiards in polygons with rational angles, interval exchange transformations and the Teichmüller geodesic flow on the moduli space of Abelian differentials on Riemann surfaces. More precisely, we consider the Lyapunov exponents of the KZ-cocycle on the absolute cohomology bundle over affine SL 2 (R)-invariant submanifolds of the moduli space of Abelian differentials on genus g surfaces. These exponents were studied extensively in [For02], [AV07] and [EKZ14]. In [For02], it was proved that the smallest non-negative exponent is always positive with respect to the canonical measures on strata of the moduli space of Abelian differentials. In [AV07], the KZ conjecture was verified using techniques independent of [For02], i.e. when the spectrum of 2g exponents of the cocycle is simple with respect to the canonical measures on strata. Explicit formulae for sums of the positive Lyapunov exponents of the KZ-cocycle were given in [EKZ14]. On the contrary, there are examples of orbit closures where the Lyapunov exponents are zero. An affine invariant submanifold with λ 2 = · · · = λ g = 0 is said to have completely degenerate KZ-spectrum. In this paper, we prove the following theorem.

show abstract

“…Our work is inspired in part by the questions raised by Forni, Matheus and Zorich [17,18]. A question they asked concerning zero Lyapunov exponents is addressed, using again techniques from Hodge theory, in [15].…”

Section: Remarks and Referencesmentioning

confidence: 99%

“…Example 6.11 Let us start with a situation which has been considered many times (e.g. [17,18]). Suppose that E R is an irreducible real piece of the cocycle, and suppose that its complexification splits as E C := E R ⊗ C = E 1 ⊕ E 1 .…”

Section: Remark 69mentioning

confidence: 99%

Semisimplicity and rigidity of the Kontsevich-Zorich cocycle

Filip

2015

Invent. math.

View full text Add to dashboard Cite

We prove that invariant subbundles of the Kontsevich-Zorich cocycle respect the Hodge structure. In particular, we establish a version of Deligne semisimplicity in this context. This implies that invariant subbundles must vary polynomially on affine manifolds. All results apply to tensor powers of the cocycle and this implies that the measurable and real-analytic algebraic hulls coincide. We also prove that affine manifolds parametrize Jacobians with non-trivial endomorphisms. Typically a factor has real multiplication. The tools involve curvature properties of the Hodge bundles and estimates from random walks. In the appendix, we explain how methods from ergodic theory imply some of the global consequences of Schmid's work on variations of Hodge structures. We also derive the Kontsevich-Forni formula using differential geometry.Comment: Two appendices, 42 page

show abstract

Zero Lyapunov exponents of the Hodge bundle

Cited by 23 publications

References 22 publications

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

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

Affine invariant submanifolds with completely degenerate Kontsevich–Zorich spectrum

Semisimplicity and rigidity of the Kontsevich-Zorich cocycle

Contact Info

Product

Resources

About