Non-varying sums of Lyapunov exponents of Abelian differentials in low genus

TL;DR

This paper demonstrates that in low genus, the sum of Lyapunov exponents for Abelian differentials remains constant across all Teichmueller curves within certain strata, reflecting a deep geometric invariance.

Contribution

It establishes the invariance of Lyapunov exponent sums across Teichmueller curves in specific low genus strata, linking this to divisors on moduli spaces.

Findings

Sum of Lyapunov exponents is constant across Teichmueller curves in many low genus strata.

This invariance is due to the disjointness of Teichmueller curves from certain divisors.

The result connects geometric properties of moduli spaces with dynamical invariants.

Abstract

We show that for many strata of Abelian differentials in low genus the sum of Lyapunov exponents for the Teichmueller geodesic flow is the same for all Teichmueller curves in that stratum, hence equal to the sum of Lyapunov exponents for the whole stratum. This behavior is due to the disjointness property of Teichmueller curves with various geometrically defined divisors on moduli spaces of curves.