LYAPUNOV SPECTRUM VIA BOUNDARY THEORY I - FRAMEWORK

The paper establishes (1) simplicity (ΛF ∈ a++) for Zariski-dense representations and (2) continuity of Λ under uniform integrability with complete, self-contained proofs in the Apafic Greg framework, including an explicit Iwasawa-cocycle formula and a tight continuity argument (Proposition 6.5 and the convergence estimate (6.11)) . However, for (3) positivity of the drift, the case where the Zariski closure H of ρ(Γ) is disconnected relies on a “cocycle trick” explicitly marked ‘to be justified in the updated version with cocycles,’ leaving a gap in the present manuscript . The model’s solution captures the right high-level strategy and gives correct conclusions for (1)–(2) in spirit, but it glosses over key hypotheses: it appeals to continuity-of-stationary-measures results that typically require stronger (e.g., exponential) moment bounds than the paper assumes, and it asserts exact functoriality of the Cartan projection under homomorphisms in (3), which is too strong as stated. Hence both are incomplete: the paper on (3), the model on technical hypotheses and rigor in (2)–(3).