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Flow, topology and applications to Lyapunov exponents

  • Vaibhav Gadre (University of Glasgow)
A3 01 (Sophus-Lie room)

Abstract

Given a flow on a manifold, we may define the flow group to be the subgroup of the fundamental group generated by the almost flow loops, namely by using based loops that are obtained by coning to a base-point flow segments that start and end in a fixed contractible neighbourhood of that base-point. Under mild hypothesis, we prove that the flow group equals the fundamental group. As a special case, the flow group result holds for the diagonal (Teichmüller) flow on moduli spaces of abelian and quadratic differentials. As an application, we prove simplicity of Lyapunov exponents for the plus and minus Kontsevich—Zorich cocycles, a generalisation both in statement and approach of the result for abelian strata by Avila—Viana. In the process, we also answer affirmatively several questions by Yoccoz regarding Rauzy diagrams for interval exchange maps. This is (variously) joint work with Arana-Herrera, Bell, Delecroix, Gutierrez-Romo and Schleimer.

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