# Chaoticity of the Teichmüller flow

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Artur Avila, Instituto Nacional de Matemática Pura e Aplicada (IMPA) and Clay Mathematics Institute
Fine Hall 314

A non-zero Abelian differential on a compact Riemann surface determines an atlas, outside the singularities, whose coordinate changes are translations. The vertical flow with respect to this translation structure generalizes the genus one notion of rational and irrational flows on tori. A fundamental tool in the understanding of the dynamics of vertical flows is the Teichmüller flow (acting on the moduli space of Abelian differentials), regarded as a renormalization operator. The chaotic nature of the dynamics of the Teichmüller flow has been a much researched topic, and currently it is known that it displays exponential decay of correlations. (This is equivalent to the spectral gap for the ambient $SL(2)$ action, a very familiar result in genus $1$.) Even much weaker aspects of the chaoticity of the Teichmüller flow however can be exploited in the description of the dynamics of typical vertical flows. Two such results are the proofs of the Kontsevich-Zorich conjecture and of weak mixing for interval exchange transformations.