GAP BETWEEN LYAPUNOV EXPONENTS FOR HITCHIN REPRESENTATIONS

The paper proves the gap λ_i(X,ρ) − λ_{i+1}(X,ρ) ≥ 1 for all i for Hitchin representations and characterizes equality via Fuchsian uniformization (Theorem 1.1). It does so by (i) relating λ_1−λ_2 to a transverse Lyapunov exponent and proving a sharp bound via a Jensen-type argument (Theorem 2.5), and (ii) alternatively identifying λ_1−λ_2 with the renormalized intersection J(1,f_ρ) and invoking the BCLS inequality J ≥ 1 with equality characterized by Livšic cohomology (Sections 3 and 5). Wedge powers then give all gaps, and the equality case is finished using a Hitchin rigidity input (Section 5.3). These ingredients appear correct and complete in the text . By contrast, the model’s primary route hinges on an unjustified “uniform e^t domination” between consecutive line bundles that would imply a pointwise inequality φ_i ≥ 1. Labourie’s dominated splitting yields an exponential gap with some rate a>0, not the specific rate 1 tied to geodesic time; the constant 1 requires the paper’s more delicate transverse/pressure arguments, not a direct pointwise norm estimate (compare the paper’s use of Theorem 5.1 and Proposition 3.1 rather than a pointwise bound) . The model’s optional, entropy/pressure-based route aligns in spirit with the paper’s second proof, but it is only sketched and conflates singular-value and eigenvalue data at the periodic level, which the paper treats carefully via reparametrized flows and eigenvalue ratios (Remark 5.6 and the subsequent rigidity step) .