JournalspmVol. 64 / No. 3DOI 10.4171/pm/1789

Simplicity of Lyapunov spectra: a sufficient criterion

  • Marcelo Viana

    IMPA - Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, Brazil
  • Artur Avila

    Université Pierre et Marie Curie, Paris, France
Simplicity of Lyapunov spectra: a sufficient criterion cover

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We exhibit an explicit sufficient condition for the Lyapunov exponents of a linear cocycle over a Markov map to have multiplicity 1. This builds on work of Guivarc'h–Raugi and Gol'dsheid–Margulis, who considered products of random matrices, and of Bonatti–Viana, who dealt with the case when the base dynamics is a subshift of finite type. Here the Markov structure may have infinitely many symbols and the ambient space needs not be compact. As an application, in another paper we prove the Zorich–Kontsevich conjecture on the Lyapunov spectrum of the Teichmüller flow in the space of translation surfaces.