Upper semi-continuity of metric entropy for C^{1,α} diffeomorphisms

The paper proves exactly the target statement (Theorem A): for C^{1,α} diffeomorphisms on manifolds of dimension ≤ 3, if (f_n, μ_n) → (f, μ) and the sum of positive Lyapunov exponents is continuous along the sequence, then limsup h_{μ_n}(f_n) ≤ h_μ(f) . The core ingredients are (i) a quantitative partition-level entropy estimate for measures with exactly one positive Lyapunov exponent (Theorem 2.1) and its symmetric ‘one negative exponent’ version (Theorem 2.3) derived via a C^{r,α} reparametrization lemma for 1D curves, and (ii) a three-way measure decomposition in dimension 3 that isolates the ‘one positive’, ‘one negative’, and ‘other’ cases (the last having zero entropy by Ruelle’s inequality), followed by a continuity argument for the Lyapunov sums and a 1/q approximation lemma (Lemma 3.1) that controls the gap between 1/q ∫ log||D f_n^q|| dμ_n and λ^+(μ_n) . The model’s solution sketches a similar high-level route (Ledrappier–Young along unstable leaves + 1D reparametrization), but the claimed key step that a nonnegative ‘defect’ term in a partition-entropy inequality is lower semicontinuous in (g, ν) is not established and is nontrivial; the paper does not rely on such a lower-semicontinuity assertion. Instead, the paper uses Lemma 3.1 to make the defect small by passing to large q, together with a careful 3D ergodic-measure decomposition and symmetric treatment of the “two-positive/one-negative” case via Theorem 2.3, to conclude the desired inequality . Consequently, while the model points in the right direction, its proof relies on an unproven and likely false continuity claim for the ‘defect’ term, so it is not a correct proof.