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

By Carlos Matheus, Martin Möller, J. Yoccoz. 2013

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We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich–Zorich cocycle over the Teichmüller flow on the $${\mathrm {SL}}_2(\mathbb {R})$$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 $${\geqslant }3$$⩾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 $${>}0$$>0) that all but finitely many such square-tiled surfaces have simple Lyapunov spectrum.

Subfields: Geometry and Topology, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/6944407558bb3d186f2498501c07508f719c8d69