Simple Lyapunov spectrum of partially hyperbolic diffeomorphisms

The paper’s Theorem D states exactly the criterion the solver aims to prove and gives a complete proof by coding the nonuniformly hyperbolic base to a (countable) topological Markov shift with a measure of full support and continuous local product structure, then invoking the Avila–Viana criterion (Theorem 3.3) for pinching and twisting to deduce simplicity, and finally conjugating back to the original cocycle (see the definitions of holonomies and transition map in equations (5)–(6) and the statement of Theorem D; the reduction and verification steps are carried out around equations (10) and the subsequent argument) . By contrast, the model’s outline attempts a direct invariance-principle proof on the diffeomorphism without passing through the symbolic model. It asserts (i) an su-invariance principle for ∧kF over SRB/equilibrium measures in the nonuniform setting and (ii) that the transported fiber measure m_p is invariant under (∧kF_p)^{n(p)}. Neither claim is justified in the outline, and (ii) is needed to deduce the atomicity at the pinching periodic point. The paper avoids these gaps by working on the Markov shift where Theorem 3.3 applies cleanly. Hence the paper’s argument is correct and complete, while the model’s proof has unaddressed gaps in its key steps.