Hyperbolicity and Rigidity for Fibred Partially Hyperbolic Systems

The uploaded paper proves a precise trichotomy (Theorem A) for accessible, C^r (r≥2), centre‑bunched, volume-preserving fibred partially hyperbolic diffeomorphisms with 2D centre: (1) distinct centre Lyapunov exponents with continuity at f; or (2) S=T^2 and invariant continuous line field(s) in Ec; or (3) S∈{S^2,T^2} and a continuous conformal structure invariant under f and its holonomies. The proof uses holonomy-invariant projective cocycles, an invariance principle yielding su-states when the extremal centre exponents coincide, an atomic vs non-atomic disintegration dichotomy to obtain line fields vs conformal structures, and a topological classification limiting the fibres to T^2 or S^2. This is all explicit in the paper’s Theorem A and its proof, including continuity in case (1) via the absence of su-states and a continuity criterion (LMY18/AV10) . The candidate solution reproduces the same trichotomy with a slightly different route: it invokes Avila–Viana’s invariance principle, a dimension‑two su-state structure (citing Kalinin–Sadovskaya) to obtain continuous line fields or a conformal structure, and the same topological restrictions; it also argues continuity in the simple‑spectrum case. One step in the model’s continuity argument is heuristic (gap persistence via a limiting su-state), whereas the paper uses a standard continuity criterion; nonetheless, the model’s overall conclusions match the paper’s result.