ACCESSIBILITY FOR DYNAMICALLY COHERENT PARTIALLY HYPERBOLIC DIFFEOMORPHISMS WITH 2D CENTER

The paper proves Theorem A: for r ≥ 2, stable accessibility is C^r-dense among dynamically coherent, plaque-expansive, strongly bunched partially hyperbolic diffeomorphisms with two-dimensional center, including the volume-preserving case (see the statement of Theorem A and its roadmap in Sections 3 and 8) . The proof hinges on: (i) strong bunching implying C^1 center (un)stable holonomies with uniform C^1 dependence (Lemma 4.1, relying on Obata) ; (ii) a detailed analysis of center accessibility classes (Theorem 2.10) ; (iii) constructing adapted su-loops and using random-perturbation-based deformations to obtain a submersion from perturbation space to the phase space (Section 6) ; and (iv) spanning c-families to globalize local accessibility (Section 8) . By contrast, the model's write-up is circular: it “reduces” the problem to the very theorem proved in the paper (Leguil–Piñeyrúa 2021) rather than providing an independent argument, so it does not solve the problem. It also overstates the regularity of holonomies (claiming C^{1+θ} where the paper requires and proves C^1 under strong bunching) and omits the paper’s dimension restriction d ≥ 4, which is explicitly assumed in the main results . The conservative case in the paper is handled via volume-preserving deformations built from divergence-free flows (Remark 6.3), not by invoking an external pasting lemma, though the latter could be an alternative tool; the model’s reliance on that external lemma is unnecessary here .