Characteristic space of orbits of Morse–Smale diffeomorphisms on surfaces

The paper’s Theorem 1 precisely asks for examples on surfaces where, once at least one of the conditions (no heteroclinic points, orientable surface, orientation-preserving map) is violated, the characteristic orbit space is disconnected; the authors give constructive models and prove all three cases via Lemmas 4.1–4.3, using the orbit-space toolkit (Propositions 2.1–2.3 and their corollary) to track connected components (for example, Klein bottle versus torus components) of V̂_Σ for all admissible Σ. This statement and its scope are clearly formulated and proved in the PDF (see the theorem statement and setup around the definition of V_Σ and V̂_Σ; cf. Theorem 1 and the supporting propositions) . By contrast, the candidate solution hinges on a “common lemma” asserting that the complement of the separatrix skeleton S is a finite disjoint union of f-invariant open bands, each homeomorphic to an open annulus or Möbius band, and that every such band lies inside V_Σ for all admissible Σ. This claim is not established and, as stated, is generally false for Morse–Smale dynamics on surfaces: the complement M\S can have simply connected faces (e.g., polygonal regions), so it is not a priori a union solely of annuli/Möbius bands. In addition, in the orientable case the argument that composing the time–one map of a gradient(-like) flow with an orientation-reversing involution τ yields two disjoint f-invariant bands U_i (claimed τ-invariant merely from h∘τ=h) is not justified; τ can permute complementary regions, and extra structure is needed to ensure the specific f-invariance of the chosen U_i. Finally, while the perturbation argument to create a heteroclinic intersection is plausible, it relies on keeping the bands invariant without a precise, localized persistence argument. In short, the model’s conclusion might be salvageable with a different route (e.g., working directly with the punctured stable basins V_ω whose orbit-space images are torus/Klein-bottle components, as in Proposition 2.1), but as written it depends on a faulty lemma and unproven invariance claims. The paper’s construction-based proof is sound and matches the problem’s requirements, establishing disconnection for every admissible Σ in each of the three requested cases (see Lemmas 4.1–4.3) .