Oscillations and the Kontsevich-Zorich cocycle

TL;DR

This paper investigates oscillations in the Teichmüller geodesic flow's lift to the Hodge bundle, revealing new mechanisms and applying them to specific strata in genus 4 with variable Lyapunov exponents.

Contribution

It introduces a novel mechanism for oscillations in the Hodge bundle under Teichmüller flow and applies it to complex strata in genus 4, expanding understanding of Lyapunov exponents.

Findings

Oscillations occur along the lift of Teichmüller flow due to unipotent deformations.

The mechanism applies to all but finitely many genus 4 strata with variable Lyapunov exponents.

New insights into the dynamics of the Kontsevich-Zorich cocycle and Lyapunov spectrum.

Abstract

We present a mechanism for producing oscillations along the lift of the Teichm\"uller geodesic flow to the (real) Hodge bundle, as the basepoint surface is deformed by a unipotent element of $\text{SL}_2(\mathbb{R})$. Invoking Chen-M\"oller, we apply our methods to all but finitely many strata in genus $4$, those exhibiting a varying Lyapunov-exponents phenomenon.