Partially hyperbolic diffeomorphisms with a finite number of measures of maximal entropy

The paper’s Theorem A states exactly the claim under audit: for C^{r} (r>1) diffeomorphisms with dominated splitting Ess ⊕ E1 ⊕ E2 ⊕ Euu (E1,E2 one-dimensional) and an entropy gap h(f) > max{hu(f), hs(f)}, there is a C1-neighborhood in which every C^{1+} map has finitely many MMEs and the number for g is bounded above by that for f. This appears verbatim in the paper and is proved by (i) upgrading the gap to a uniform central hyperbolicity of all MMEs (via Ledrappier–Young and unstable entropy inequalities), (ii) building uniform Pesin blocks with stable/unstable local product structure, and (iii) using the Buzzi–Crovisier–Sarig/Lima–Obata–Poletti homoclinic-class criterion to prevent two distinct hyperbolic MMEs from being homoclinically related, yielding finiteness and the upper-semicontinuity bound on the number of MMEs. The relevant statements are Theorem A and Lemmas 5.3–5.5 together with Theorem 3.1 and Lemma 5.1/Thm. 5.2 (and Proposition 2.8) in the paper. The candidate solution follows the same overall path: it turns the entropy gap into a uniform two-sided central Lyapunov gap for MMEs, constructs uniform Pesin blocks (hence a uniform ‘close implies homoclinically related’ radius), and deduces finiteness and upper bounds on the number of MMEs; it also invokes the same homoclinic-class uniqueness principle. The only difference is a technical detour: the model constructs a hyperbolic horseshoe with entropy beating hu and hs to preserve the gap under perturbation, whereas the paper obtains this openness by combining continuity of topological entropy at hyperbolic-MME points with upper semicontinuity of stable/unstable topological entropy. This is a benign methodological difference, not a logical conflict. Overall, both arguments are coherent and compatible with the paper’s framework and references (e.g., Lemma 5.3 for the central exponents, Theorem 3.1 for uniform Pesin blocks, and the homoclinic-class criterion), so we judge both correct and substantially the same in structure. Key citations from the paper: Theorem A (statement), Lemma 5.3 (central gap from h(f) > max{hu,hs}), Theorem 3.1 and Lemma 3.3 (uniform Pesin blocks), Lemma 5.1/Thm. 5.2 and Proposition 2.8 (homoclinic-class obstruction and finiteness), and the final step in the proof of Theorem A establishing the upper bound on the number of MMEs via homoclinic continuations. See Theorem A and its proof; the ‘Idea of the proofs’ section outlines the same steps. Theorem A: ; idea/criterion: ; Pesin blocks: ; homoclinic contradiction: ; Ledrappier–Young-based central bounds: ; proof of Theorem A finishing (upper bound via homoclinic continuations): .