SMOOTH MODELS FOR CERTAIN FIBERED PARTIALLY HYPERBOLIC SYSTEMS

The paper’s Theorem A states exactly the target claim and provides a complete proof: it shows the quotient dynamics is an Anosov homeomorphism (via Bohnet–Bonatti), upgrades it to a hyperbolic nilmanifold automorphism through a new Theorem D for Anosov homeomorphisms on nilmanifolds, and then constructs a smooth bundle with structure group Isom(F), lifts the base automorphism, and proves partial hyperbolicity using a graph-transform argument, yielding a leaf conjugacy and properties (1)–(4) . By contrast, the model assumes the induced base map is an Anosov diffeomorphism and invokes Franks–Manning directly; the paper avoids this regularity pitfall by working with Anosov homeomorphisms and proving the needed conjugacy (Theorem D) on nilmanifolds. The model also asserts that Dg splits block-diagonally as DA ⊕ (fiber isometry); in general there is an off-diagonal term from the b-dependence of the fiber isometries, and the paper addresses this via a linear graph transform to build invariant Es, Eu and then verify domination/partial hyperbolicity. These are substantive gaps in the model’s proof, even though its high-level outline aligns with the paper’s structure. Hence, the paper is correct; the model’s solution is not rigorous in the stated generality.