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

Abstract

We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle over the Teichmueller flow on the $S{L}_2(R)$-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 $\ge 3$. 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.

69 pages, Appendix C by Samuel Lelièvre. Final version based on the referees' reports. To appear in Invent. Math

References in corpus (3)

• Invariant and stationary measures for the SL(2,R) action on Moduli space
• Square-tiled cyclic covers
• Complete periodicity of Prym eigenforms

Cited by in corpus (13)

• Diffusion for chaotic plane sections of 3-periodic surfaces
• The Deligne-Mumford and the Incidence Variety Compactifications of the Strata of $\Omega {\mathcal{M}}_g$
• Simplicity of Lyapunov spectrum for linear cocycles over non-uniformly hyperbolic systems
• Non-Veech surfaces in H^hyp(4) are generic
• Eigenvalues of Curvature, Lyapunov exponents and Harder-Narasimhan filtrations
• Teichmüller dynamics in the eyes of an algebraic geometer
• SL(2,R)-invariant probability measures on the moduli spaces of translation surfaces are regular
• Simplicity of spectrum for certain multidimensional continued fraction algorithms
• Systolic geometry of translation surfaces
• Convergence of Siegel-Veech constants
• A coding-free simplicity criterion for the Lyapunov exponents of Teichmueller curves
• Quelques contributions à la théorie de l'action de SL(2,R) sur les espaces de modules de surfaces plates
• Diagonal flow detects topology of strata