Lyapunov Exponents of Rank 2-Variations of Hodge Structures and Modular Embeddings

TL;DR

This paper derives an explicit formula for the non-negative Lyapunov exponent associated with rank two subspaces in variations of Hodge structures over hyperbolic curves, focusing on cases with discrete monodromy.

Contribution

It provides a new explicit formula for Lyapunov exponents in rank 2 variations of Hodge structures with discrete monodromy, linking monodromy representation to Lyapunov exponents.

Findings

Explicit formula for Lyapunov exponent derived

Applicable to cases with discrete monodromy

Connects monodromy representation to dynamical invariants

Abstract

If the monodromy representation of a VHS over a hyperbolic curve stabilizes a rank two subspace, there is a single non-negative Lyapunov exponent associated with it. We derive an explicit formula using only the representation in the case when the monodromy is discrete.