We consider Lyapunov exponents for flat bundles over hyperbolic curves defined via parallel transport over the geodesic flow. We refine a lower bound obtained by Eskin, Kontsevich, Moeller and Zorich showing that the sum of the first k exponents is greater or equal than the sum of the degree of any rank k holomorphic subbundle of the flat bundle and the asymptotic degree of its equivariant developing map. We also show that this inequality is an equality if the base curve is compact. We moreover relate the asymptotic degree to the dynamical degree defined by Daniel and Deroin. We then use the previous results to study properties of Lyapunov exponents on variations of Hodge structures and on Shatz strata of the de Rham moduli space. In particular we show that the top Lyapunov exponent function is unbounded on the maximal Shatz stratum, the oper locus. In the final part of the work we specialize to the rank two case, generalizing a result of Deroin and Dujardin about Lyapunov exponents of holonomies of projective structures.

中文翻译：

---

Lyapunov 指数、全纯平丛和 de Rham 模空间

我们考虑通过测地线流上的平行输运定义的双曲线上扁平丛的 Lyapunov 指数。我们改进了由 Eskin、Kontsevich、Moeller 和 Zorich 获得的下界，表明前 k 个指数的总和大于或等于扁平丛的任何 k 阶全纯子丛的度数与其等变项的渐近度数之和发展地图。我们还表明，如果基曲线是紧凑的，则这种不等式是一个等式。此外，我们将渐近度与 Daniel 和 Deroin 定义的动态度联系起来。然后，我们使用先前的结果来研究 Lyapunov 指数在 Hodge 结构变化和 de Rham 模空间的 Shatz 地层上的性质。特别地，我们证明了在最大 Shatz 层上的顶部 Lyapunov 指数函数是无界的，oper 基因座。在工作的最后部分，我们专门研究等级二的情况，概括了 Deroin 和 Dujardin 关于射影结构完整度的 Lyapunov 指数的结果。