Unclear relation between Lyapunov exponents and asymptotic growth of parabolic degrees under covering maps

Investigate whether Lyapunov exponents for the pulled-back hypergeometric variations of Hodge structure on the projective line are related to the asymptotic growth of the parabolic degrees deg_par(E^{3,0}) and deg_par(E^{2,1}) as the covering degree d increases for the maps [t:s] → [t^d:s^d].

Background

The paper computes parabolic degrees of Hodge bundles for variations of Hodge structure obtained by pulling back hypergeometric Picard–Fuchs equations along coverings of P1. These computations yield explicit dependence on the covering degree d.

The authors note uncertainty about a possible connection between their degree computations and Lyapunov exponents. They suggest examining whether the asymptotic behavior of parabolic degrees under increasing covering degree d correlates with Lyapunov exponents, referencing related rank-2 results by Kappes.

References

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 $\deg_{\mathrm{par}} \mathscr{E}{3,0}$ and $\deg_{\mathrm{par}} \mathscr{E}{2,1}$ as the degree $d$ of the covering $[t:s] \mapsto [td: sd]$ grows.

— Hodge Numbers from Picard-Fuchs Equations  (1612.09439 - Doran et al., 2016) in Remark after Table 1, Section 5 (Families of Calabi–Yau threefolds)