CONJUGACY OF FREE MAPPINGS EMBEDDED IN A FLOW

The paper’s main theorem (Theorem 3.1) states that for free plane homeomorphisms f and g with finitely many fundamental regions, embedded in flows, f is conjugate to g or g^{-1} in Homeo^+(R^2) if and only if their oriented plane foliations are equivalent; the authors prove this via Haefliger–Reeb leaf spaces, showing R^2/⟨f⟩ and R^2/⟨g⟩ are S^1-bundles over contractible bases (V_f and V_g) and hence trivial, and then applying a covering-space argument (Lemmas 3.2 and 3.6, Proposition 3.8, and the concluding argument in the proof of Theorem 3.1) . By contrast, the candidate solution attempts a direct foliation-box/gluing proof and crucially asserts that “finitely many fundamental regions” forces a decomposition of the plane into finitely many open Reeb strips bounded by pairs of leaves (plus at most one translation component). This is false in general: already for the standard Reeb flow used in the paper, there are three fundamental regions, two of which are half-planes bounded by a single leaf, not open strips between two boundary leaves . Consequently, the model’s construction of transverse arcs joining “the two boundary leaves” and the subsequent gluing breaks down on such one-sided regions. The paper’s argument does not rely on this incorrect structural claim and is logically complete; the model’s proof is therefore flawed on a key step, even though it correctly observes the easy direction that conjugacy (up to inversion) implies equivalence of oriented foliations (as the paper also shows at the end of the proof of Theorem 3.1) .