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

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.

Hodge Numbers from Picard-Fuchs Equations

“…It is not clear whether there is any relation between the computations presented here and Lyapunov exponents; however, it would be interesting to see whether there is any relation between Lyapunov exponents and the asymptotic growth of deg par E 3,0 and deg par E 2,1 as the degree d of the covering [t : s] → [t d : s d ] grows. Such growth has been investigated in the rank 2 case by Kappes [24].…”