Measures of maximal entropy for C∞ three-dimensional flows

The uploaded paper (Zang, 2025) proves finiteness of ergodic MMEs for C∞ non‑singular 3D flows with positive entropy (Theorem A) and, more generally, for C^r (r>1) flows under the entropy–dilation threshold h_top(X) > (1/r)·min{λ+(X), λ−(X)} (Theorem B) . Its proof has three pillars: (i) convergence of stable/unstable lifts and continuity of Lyapunov exponents for measures with entropies tending to h_top under the threshold (Theorem 3.1, derived from Theorem 3.3) ; (ii) uniform largeness of weak/strong Pesin sets for MMEs (Theorems 4.3, 5.1, 5.2) ; and (iii) uniqueness of the MME within a homoclinic class for 3D non‑singular flows, via Buzzi–Crovisier–Lima’s coding and Gurevich’s uniqueness for irreducible countable Markov shifts, which then yields finiteness (used explicitly in §5.2) . The candidate solution reproduces this scheme: reduce to the time‑one map, use Burguet‑type entropy splitting to control Lyapunov exponents and obtain convergence of lifts, deduce uniform Pesin blocks, show that nearby MMEs are homoclinically related, and then invoke the coding/uniqueness result to conclude finiteness. The only nuance is that the reduction from flows to the time‑one map requires the standard averaging step (e.g., Lemma 7.5 in the paper) rather than asserting that flow‑MMEs and map‑MMEs literally coincide; Zang handles this explicitly, while the candidate treats it informally. Aside from that minor point, the arguments align closely in content and structure.