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

TL;DR

This paper introduces a Galois-theoretical criterion to determine when the Lyapunov spectrum of square-tiled surfaces is simple, providing new results for genus 3 surfaces with one zero, contingent on a conjecture.

Contribution

It offers a novel Galois-theoretical criterion for Lyapunov spectrum simplicity and applies it to genus 3 square-tiled surfaces, extending previous results.

Findings

Most genus 3 square-tiled surfaces with one zero have simple Lyapunov spectrum

The criterion is conditional on a conjecture by Delecroix and Lelièvre

Uses Siegel's theorem to prove finiteness results

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 $SL_2(R)$-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 $\geq 3$. We apply our criterion to square-tiled surfaces of genus 3 with one single zero. Conditionally to a conjecture of Delecroix and Leli\`evre, 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.