We present a mechanism for producing oscillations along the lift of the Teichmüller geodesic flow to the (real) Hodge bundle, as the basepoint surface is deformed by a unipotent element of SL 2 (R). Invoking Chen-Möller [CM13], we apply our methods to all but finitely many strata in genus 4, those exhibiting a varying Lyapunov-exponents phenomenon.
We prove that for every flat surface ω, the Hausdorff dimension of the set of directions in which Teichmüller geodesics starting from ω exhibit a definite amount of deviation from the correct limit in Birkhoff's and Oseledets' Theorems is strictly less than 1. This theorem extends a result by Chaika and Eskin where they proved that such sets have measure 0. We also prove that the Hausdorff dimension of the directions in which Teichmüller geodesics diverge on average in a stratum is bounded above by 1/2, strengthening a classical result due to Masur. Moreover, we show that the Hausdorff codimension of the set of non-weakly mixing IETs with permutation (d, d − 1, . . . , 1), where d is an odd number, is exactly 1/2 and strengthen a result by Avila and Leguil.
In this note, we show that a central limit theorem holds for exterior powers of the Kontsevich-Zorich (KZ) cocycle. In particular, we show that, under the hypothesis that the top Lyapunov exponent on the exterior power is simple, a central limit theorem holds for the lift of the (leafwise) hyperbolic Brownian motion to any strongly irreducible, symplectic, SL(2, R)-invariant subbundle, that is moreover symplectic-orthogonal to the so-called tautological subbundle. We then show that this implies that a central limit theorem holds for the lift of the Teichmüller geodesic flow to the same bundle.For the random cocycle over the hyperbolic Brownian motion, we are able to prove under the same hypotheses that the variance of the top exponent is strictly positive. For the deterministic cocycle over the Teichmüller geodesic flow we can prove that the variance is strictly positive only for the top exponent of the first exterior power (the KZ cocycle itself) under the hypothesis that its Lyapunov spectrum is simple.
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