YOUNG STRUCTURES FOR PARTIALLY HYPERBOLIC SYSTEMS WITH MOSTLY CONTRACTING CENTRAL DIRECTION

The uploaded paper proves Theorem A: under (H1)–(H3) for partially hyperbolic diffeomorphisms with splitting TM = Ecs ⊕ Euu, there exists a Young structure satisfying (Y1)–(Y5) with exponentially decaying return-time tails on unstable disks. The proof builds an auxiliary partition via a growth–capture–release cycle (Section 5), obtains exponential tail bounds (Section 6), refines to an inducing scheme with stopping times S_j and returns τ* (Section 7), and verifies (Y1)–(Y5), including bounded distortion (Y4) and stable-holonomy regularity (Y5) (Section 8). These steps are explicit and coherent in the paper, culminating in Theorem A with m_γ{τ>n} ≤ Cθ^n for all γ ∈ Γ^u (see the statement of (Y1)–(Y5) and Theorem A, and the proofs of the exponential tails and Y4–Y5: ). The model’s solution sketches broadly the same end goal and references the growth–capture–release idea, but its key step appeals to a “uniform density of c s-hyperbolic times” to guarantee a fixed captured fraction each cycle. That uniform density does not follow from (H3) (which only asserts λ_cs^+(x)<0 on a positive-measure subset of each unstable disk, with no uniform bound away from 0), so the model’s proof relies on an unproven and generally false uniformity assumption. The paper’s argument avoids this by a geometric selection and filtration that yields a uniform selection fraction and exponential tails without requiring uniform hyperbolic-time densities (e.g., Lemma 5.1, Proposition 7.2) . Hence, the paper is correct, while the model’s proof is flawed/incomplete at this crucial step.