Wandering Flows on the Plane

Both texts establish the same classification: two regular flows on R^2 are topologically equivalent iff there is a homeomorphism between their orbit (leaf) spaces that preserves the precedence relation ≻ on inseparable separatrices. In the paper, this appears as Theorem 3 and is proved via Kaplan’s chordal systems: preservation of ≻ determines the chordal relations (Lemmas 4.24 and 4.27) and then Theorem J′ lifts a chordal (anti-)isomorphism to a plane homeomorphism carrying oriented leaves to oriented leaves . The necessity direction (flow equivalence induces a homeomorphism of orbit spaces that preserves ≻) is also shown in the paper’s proof of Theorem 3 . The candidate solution proves the same theorem but via Haefliger–Reeb’s classification of oriented foliations by their non-Hausdorff 1-manifold leaf spaces plus the intrinsic left/right order; it identifies ≻ with the right/left order and then applies the lifting theorem. The only notable gap in the model is that the identification “≻ equals ‘right’ non-separation” is stated with a brief sketch rather than a fully rigorous derivation; the paper circumvents this by a detailed chordal-relations development. Otherwise, the arguments align in substance.