Lyapunov spectrum of square-tiled cyclic covers

TL;DR

This paper derives explicit formulas for the Lyapunov exponents of the Hodge bundle over arithmetic Teichmuller curves arising from square-tiled cyclic covers, linking flat geometry and algebraic dynamics.

Contribution

It provides a novel explicit computation method for Lyapunov exponents in the context of cyclic covers with flat structures, enhancing understanding of their dynamical properties.

Findings

Explicit formulas for all Lyapunov exponents derived

Connection established between deck transformations and Hodge bundle subbundles

Advances in understanding flat structures on cyclic covers

Abstract

A cyclic cover over the Riemann sphere branched at four points inherits a natural flat structure from the "pillow" flat structure on the basic sphere. We give an explicit formula for all individual Lyapunov exponents of the Hodge bundle over the corresponding arithmetic Teichmuller curve. The key technical element is evaluation of degrees of line subbundles of the Hodge bundle, corresponding to eigenspaces of the induced action of deck transformations.