Gap between Lyapunov exponents for Hitchin representations

TL;DR

This paper investigates the Lyapunov exponents of Hitchin representations over hyperbolic curves, establishing a gap between consecutive exponents and characterizing the extremal case of the uniformizing representation.

Contribution

It introduces a novel analysis linking Lyapunov exponents to geometric invariants and proves a gap theorem for these exponents in the context of Hitchin representations.

Findings

A proven gap between consecutive Lyapunov exponents.

Characterization of the uniformizing representation as having extremal gaps.

Relation of Lyapunov exponents to foliated and thermodynamic invariants.

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 characterize the uniformizing representation of the Riemann surface as the one with the extremal gaps. The strategy of the proof is to relate Lyapunov exponents in the case of Anosov representations to other invariants, where the gap result is already available or where we can directly show it. In particular, firstly we relate Lyapunov exponents to a foliated Lyapunov exponent associated to a foliation H\"older isomorphic to the unstable foliation on the unitary tangent bundle of a Riemann surface. Secondly, we relate them to the renormalized intersection product in the setting of the thermodynamic formalism developed by Bridgeman, Canary, Labourie and Sambarino.