# Parabolic degrees and Lyapunov exponents for hypergeometric local systems

## TL;DR

This paper investigates the relationship between parabolic degrees and Lyapunov exponents for hypergeometric local systems on the punctured sphere, providing explicit computations and numerical analysis of their dependence.

## Contribution

It introduces a method to compute parabolic degrees of subbundles and explores how these degrees influence Lyapunov exponents in hypergeometric local systems.

## Key findings

- Explicit formulas for parabolic degrees of subbundles.
- Numerical evidence linking degrees to Lyapunov exponents.
- Insights into the variation of exponents with geometric data.

## Abstract

Consider the flat bundle on $\mathrm{CP}^1 - \{0,1,\infty \}$ corresponding to solutions of the hypergeometric differential equation $ \prod_{i=1}^h (\mathrm D - \alpha_i) - z \prod_{j=1}^h (\mathrm D - \beta_j) = 0$ where $\mathrm D = z \frac {d}{dz}$. For $\alpha_i$ and $\beta_j$ distinct real numbers, this bundle is known to underlie a complex polarized variation of Hodge structure. Setting the complete hyperbolic metric on $\mathrm{CP}^1 - \{0,1,\infty \}$, we associate $n$ Lyapunov exponents to this bundle. We compute the parabolic degrees of the holomorphic subbundles induced by the variation of Hodge structure and study the dependence of the Lyapunov exponents in terms of these degrees by means of numerical simulations.