2013/05/31 by Carlos Matheus, Martin Möller, Martin Moeller +1
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Geometry #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Pure mathematics #Quantum chaos and dynamical systems #Simplicity #Spectrum (functional analysis) #Square (algebra) #math.DS
paper · pdf · doi:10.1007/s00222-014-0565-5
published as Invent. Math., vol. 202, no. 1, 333-425 (2015) · 69 pages, Appendix C by Samuel Lelièvre. Final version based on the referees' reports. To appear in Invent. Math
arxiv created 2014/11/10 · openalex publication_date 2014/11/26 · arxiv updated 2016/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle over the Teichmueller flow on the SL2(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.