Oscillations and the Kontsevich-Zorich cocycle

Al-Saqban, Hamid

doi:10.48550/arxiv.2103.15126

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“…This note builds on and extends unpublished thesis work of H. A. [AS21a] (written under the direction of G. F.), which dealt with the genus 2 case and more generally the top exterior power of the Kontsevich-Zorich cocycle, but left unanswered the positivity of the deterministic variance question and extensions to lower exterior powers that are addressed in the present paper. H. A. is grateful to G. F. for suggesting this fruitful direction of research, for his patience and encouragement, and for invaluable conversations and ideas on this subject.…”

Section: Acknowledgementsmentioning

confidence: 84%

A Central Limit Theorem for the Kontsevich-Zorich Cocycle

Al-Saqban¹,

Forni²

2022

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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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Section: Acknowledgementsmentioning

confidence: 84%