Gap Between Lyapunov Exponents for Hitchin Representations

Abstract

Abstract We study Lyapunov exponents for flat bundles over hyperbolic curves defined via parallel transport over the geodesic flow. We consider them as invariants on the space of Hitchin representations and show that there is a gap between any two consecutive Lyapunov exponents. Moreover we show that the minimal possible gap for any two consecutive Lyapunov exponents is achieved if and only if the representation is the one uniformizing the hyperbolic structure of the surface. We give two proofs of the previous fact. In the first one, we relate the Lyapunov exponents to a transverse Lyapunov exponent associated to a deformation of the unstable foliation of the geodesic flow, and we establish a general bound for this quantity. In the second one, we relate Lyapunov exponents to the renormalized intersection product in the setting of the thermodynamic formalism developed by Bridgeman, Canary, Labourie, and Sambarino and we use the already existing bound for such a quantity.