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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.
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Marcelo Viana, Artur Avila, Simplicity of Lyapunov spectra: a sufficient criterion. Port. Math. 64 (2007), no. 3, pp. 311–376DOI 10.4171/PM/1789