DEHN-FRIED SURGERIES ON NON-TRANSITIVE EXPANSIVE FLOWS

The paper’s Theorem 1.1 states precisely that after a Dehn–Fried surgery along a periodic orbit with solid-torus tubular neighborhoods, the resulting flow Ψ is expansive if and only if its stable and unstable foliations admit no circle 1‑prong singularities, with the foliations F^s_Ψ, F^u_Ψ produced by Proposition–Definition 3.11 . The “only if” direction is immediate from the structure theorem of Inaba–Matsumoto and Oka (Theorem 2.5: expansive flows in dimension three are topological pseudo‑Anosov) . For the “if” direction, the paper proves a contradiction statement using blow‑up/blow‑down, good foliations on the boundary (Lemma 3.6), local pseudo‑hyperbolic models (Proposition 2.9), and a key closeness‑along‑leaves result (Proposition 3.5), transferring hypothetical non‑expansive behavior back to the original expansive flow; this is carried out in Section 4.4 with Proposition 4.1 and Proposition 3.13 providing the necessary local models for the surgically obtained flow . The candidate model gives the same equivalence and correctly invokes the no‑1‑prong obstruction in the “only if” direction. For the “if” direction, the model argues via standard local product boxes, including singular p‑prong boxes (p ≥ 3), to produce a uniform expansivity constant in the Bowen–Walters sense. This differs from the paper’s global contradiction strategy but is mathematically sound as a classical product‑structure proof that topological pseudo‑Anosov flows are expansive (cf. Theorem 2.6) . Minor gaps in the model’s write‑up (e.g., it assumes without proof that absence of 1‑prongs after surgery yields a full topological pseudo‑Anosov structure, which in the paper is established via Proposition–Definition 3.11 together with the almost pseudo‑Anosov properties) are addressable with the cited constructions. Overall, both are correct, using different proof strategies.