Lyapunov exponents, holomorphic flat bundles and de Rham moduli space

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, Möller and Zorich showing that the sum of the first k exponents is greater than or equal to 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.