Saddle-Connections in Translation Surfaces and Eskin-Kontsevich-Zorich Regularity Conjecture

Carlos Santos
Centre National de la Recherche Scientifique (CNRS)

After the works of A. Zorich and G. Forni, it is known that the Lyapunov exponents of the so-called Kontsevich-Zorich cocycle are interesting quantities related to the deviations of ergodic averages of interval exchange transformations and translation flows. In a recent breakthrough article, A. Eskin, M. Kontsevich and A. Zorich obtained a formula for the sum of non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle with respect to ergodic SL(2,R)-invariant measures satisfying a certain regularity condition. Concerning the hypothesis for the validity of Eskin-Kontsevich-Zorich, the regularity condition is always satisfied in all known examples of ergodic SL(2,R)-invariant measures. In particular, partly motivated by this scenario, A. Eskin, M. Kontsevich and A. Zorich conjectured that all ergodic SL(2,R)-invariant measures are regular. In this talk, we will discuss our joint work with A. Avila and J.-C. Yoccoz where the Eskin-Kontsevich-Zorich regularity conjecture is confirmed.

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