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

• 成果类型: Article
• 署名作者: Matheus, Carlos; Moeller, Martin; Yoccoz, Jean-Christophe
• 署名单位: Universite Paris 13; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Goethe University Frankfurt; Universite PSL; College de France
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-014-0565-5
• 发表日期: 2015
• 页码: 333-425
• 关键词: teichmuller-curves; abelian differentials; MODULI SPACES; transformations; deviation; exponents

摘要

We present a Galois-theoretical criterion for the simplicity of the Lyapunov spectrum of the Kontsevich-Zorich cocycle over the Teichmuller flow on the -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 . We apply our criterion to square-tiled surfaces of genus 3 with one single zero. Conditionally to a conjecture of Delecroix and LeliSvre, we prove with the aid of Siegel's theorem (on integral points on algebraic curves of genus ) that all but finitely many such square-tiled surfaces have simple Lyapunov spectrum.