Upper semi-continuity of metric entropy for diffeomorphisms with dominated splitting

The PDF’s Theorem A/B/C state exactly the three claims the candidate solved: (a) existence of a uniform ε_f so that small partitions compute entropy for ergodic ν with λ^max_E(ν) ≤ 0 < λ^min_F(ν); (b) upper semi-continuity along sequences with the same sign condition; (c) upper semi-continuity at a limiting measure µ with that sign condition. These appear verbatim in the paper’s statements of Theorems A, B, and C, respectively . The paper’s proof strategy runs through a partial-entropy version Theorem 4.1 (h^F_ν(f) ≤ h_ν(f,Q), and if λ^max_E ≤ 0 then h^F_ν(f)=h_ν(f)) and a quantitative tail estimate (Proposition 5.1) showing limsup_n (1/n) H_ων(P^n_ν|Q^n)=0 along unstable plaques, delivered using a uniform small scale ε_f obtained from dominated splitting (Section 3) and a subexponential covering of restricted Bowen balls (Section 6) . The candidate’s outline mirrors these ingredients: dominated-splitting geometry to build a uniform ε_f, Pesin local unstable manifolds and subordinate partitions, reduction to a tail conditional-entropy term, and a subexponential covering argument inside unstable plaques. The small technical differences are stylistic: the paper derives Theorem A via Theorem 4.1 plus h^F_ν(f)=h_ν(f) when λ^max_E≤0 , while the candidate phrases it as a direct ‘tail-entropy’ inequality. For Theorems B and C, the paper uses an explicit ergodic-component selection and continuity of H_ν(Q^m) to pass to limits , plus a measure-selection claim controlling Lyapunov exponents in the limit , whereas the candidate describes a closely related selection-on-Pesin-blocks argument. Net: the arguments are substantively the same and correct, with the paper providing the detailed constructions (graph transform estimates in Section 3 and the combinatorial covering in Section 6) that the candidate sketches .