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arXiv · 1305.2033

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

Carlos Matheus·Martin Moeller·Jean-Christophe Yoccoz
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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 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.

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Carlos Matheus, Martin Moeller, Jean-Christophe Yoccoz. 2014-11-10. A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces. https://doi.org/10.1007/s00222-014-0565-5

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