{"id":"1590fe40-a05f-45e2-8448-400920a1cf0b","arxiv_id":"2411.15429","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"C-infinity generic area-preserving diffeomorphisms on finitely punctured surfaces have dense periodic orbits, and on punctured spheres the orbits can be chosen equidistributed.","lead":"The paper proves that a C-infinity closing lemma holds for area-preserving diffeomorphisms on a closed surface with finitely many points removed, so a generic such map has periodic orbits dense in the punctured surface. It also shows that on a finitely punctured sphere, a dense class of area-preserving maps has equidistributed periodic orbits, via a new Periodic Floer homology Weyl law for homeomorphisms.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof assumes every area-preserving diffeomorphism of a punctured surface extends to a homeomorphism of the closed surface; this is false for maps with non-removable winding near a puncture, so Theorem 1.1 is not established for the stated domain.","rationale":"The paper's central claim is Theorem 1.1, that the C^∞ closing lemma holds for every area-preserving diffeomorphism of a closed surface with finitely many points removed, in the strong C^∞ topology. The proof's very first step (Section 3.1.1) requires that the given diffeomorphism of the punctured surface extend to an area-preserving homeomorphism of the closed surface. This is not a harmless technicality: the standard definition of Diff(Σ_P,ω_0) includes diffeomorphisms with non-removable winding near the punctures, and the explicit map F(x,y) = (x + ln y, y) on an annulus (S^2 minus two points) is a C^∞ area-preserving diffeomorphism with no continuous extension. The reader's verdict identified the quantitative closing lemma and the rationalization vector field as the weakest assumptions, but those are external results whose validity is a separate question; the extension premise is an internal, demonstrably false step in the proof as written. If the authors intended the theorem only for extendable diffeomorphisms, that restriction must appear in the statement; without it, the central claim is unproven for a nonempty class. The proposed test with the explicit non-extendable map directly exhibits the failure of the proof's starting point, so it would settle whether the concern lands.","tokens_in":36,"tokens_out":14076,"duration_ms":449168,"concrete_test":"Take F(x,y) = (x + ln y, y) on the annulus S^2 \\ {p,q} with area form dx∧dy. Verify: (i) F is a C^∞ area-preserving diffeomorphism of the punctured surface; (ii) F does not extend continuously to the punctures, since along y → 0 the x-coordinate winds infinitely often. Then attempt to run the proof's steps 3.1.1–3.1.3 for this F: no homeomorphic extension Φ exists, so the argument cannot start. This single example settles whether the proof covers Diff(Σ_P,ω_0) as stated; if the intended domain is only extendable maps, the statement must be amended accordingly.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The proof of Theorem 1.1 begins in Section 3.1.1 with: 'Let Φ ∈ Homeo(Σ,ω) be the extension of φ such that Φ(P)=P.' This requires every φ ∈ Diff(Σ_P,ω_0) to extend continuously to the punctures. That is false under the standard meaning of Diff(Σ_P,ω_0) used in the paper. On S^2 with two punctures, an annulus with coordinates (x,y) ∈ R/Z × (0,1) and area form dx∧dy, the map F(x,y) = (x + ln y, y) is a C^∞ area-preserving diffeomorphism: its inverse is (x − ln y, y), and it preserves dx∧dy. But as y → 0, the x-coordinate winds infinitely many times, so F has no continuous extension to the puncture. Hence F ∈ Diff(Σ_P,ω_0), yet the first step of the proof is unavailable. The theorem statement does not restrict to extendable diffeomorphisms, so the proof leaves a nonempty class of maps untreated. The same extension assumption is used in the proof of Theorem 1.3 via Φ. No argument is supplied that non-extendable maps can be handled by approximation or that the closing lemma holds for them.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the C^∞ closing lemma and generic density of periodic points for area-preserving diffeomorphisms of a closed surface with finitely many punctures. The main results are Theorem 1.1 (C^∞ closing lemma), Corollary 1.2 (C^∞-generic density of periodic points), and Theorem 1.3 (equidistribution of orbit sets for a C^∞_loc-dense set of maps on punctured spheres). The strategy is to extend a map of the punctured surface to a homeomorphism of the closed surface, approximate it by Hamiltonian isotopic diffeomorphisms, rationalize via a compactly supported symplectic vector field, apply the quantitative PFH-based closing lemma, and pass to the limit. For the sphere, a PFH Weyl law variant is proved and used to produce near-equidistributed orbit sets.","tokens_in":19492,"tokens_out":19736,"duration_ms":174213,"significance":"If the results hold, they would extend the C^∞ closing lemma and generic density of periodic points to a non-compact setting, a natural and nontrivial problem. The paper draws on recent advances in periodic Floer homology, including quantitative closing lemmas and spectral invariants, and proves a new Weyl-law variant for S^2 that may be of independent interest. The proof is well-structured and relies on machine-checkable cited results rather than fitting parameters, which is a strength. However, the main theorems are currently proved only for a restricted subclass of the stated domain, which is a serious caveat.","major_comments":[{"comment":"The proof begins with 'Let Φ ∈ Homeo(Σ,ω) be the extension of φ such that Φ(P)=P.' This step is not available for every φ ∈ Diff(Σ_P,ω_0) under the standard meaning of the group. On S^2 with two punctures identified with the annulus (R/Z)×(0,1) with area form dx∧dy, the map F(x,y)=(x+ln y,y) is a C^∞ area-preserving diffeomorphism (the inverse is (x−ln y,y) and the determinant is 1), but as y→0 the x-coordinate winds infinitely often, so F has no continuous extension to the puncture. Thus Theorems 1.1 and 1.2 are proved only for the proper subclass of diffeomorphisms that admit a continuous extension to Σ fixing P. The same extension assumption underlies the proof of Theorem 1.3 (Section 6.3.1). Moreover, this gap cannot be filled by approximation: in the strong C^∞ topology, an extendable map has all derivatives bounded near each puncture, while F has derivatives blowing up like 1/y, so an extendable sequence cannot converge to F in the strong topology. The authors should either restrict the statements to extendable maps or provide a separate argument covering non-extendable ones.","section":"3.1.1"}],"minor_comments":[{"comment":"The notation for the function spaces used in Section 6 is inconsistent: Section 6.1 defines C_c^∞(S^2;P) as smooth functions that are constant near P, while Claim 6.3 refers to C_0^∞(S^2;P) and uses cutoff functions χ_N that should vanish near P. Please clarify the intended spaces (e.g., functions vanishing in a neighborhood of P, or compactly supported in S^2\\P) and use one notation throughout.","section":"6.1 / 6.3"},{"comment":"The statement of Theorem 1.3, 'A C∞_loc-dense element of Diff(S^2_P,ω_0) has an equidistributed sequence of orbit sets,' is grammatically unclear. The proof actually shows that for every φ there is a residual set of Hamiltonian perturbations whose maps have the property, which implies a C∞_loc-dense subset of Diff(S^2_P,ω_0) consists of such maps. The theorem statement should be rephrased accordingly.","section":"1.2"},{"comment":"In Lemma 3.1, the notation φ_{H_n}^1 and φ_{K_n}^2 is confusing because the superscripts 1 and 2 could be mistaken for powers. Consider using subscripts or a different label, e.g., φ^1_n and φ^2_n.","section":"3.1"},{"comment":"The expression φ_{F^{τ_n}}^n is hard to parse. It seems to denote φ_n ∘ ϕ^1_{F^{τ_n}}; please use a more explicit notation, for example by writing the composition directly.","section":"6.2.4"}],"recommendation":"major_revision","confidential_remarks":"The extension gap is the central issue. If the authors cannot prove the results for non-extendable diffeomorphisms, they will need to weaken the main statements, which would materially change the paper's claim. I would recommend that the editor request a substantive revision addressing this point before considering publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my take. The paper genuinely extends the C^∞ closing lemma and generic density from closed surfaces to finitely punctured surfaces, and the S^2 equidistribution result via a PFH Weyl law for homeomorphisms is a real new piece of work. The author is honest about limitations: Remark 6.4 flags that equidistribution holds only in the C^∞_loc sense, not the strong topology, and Section 1.3 openly leaves the higher-genus Weyl law as a question. Credit where it's due. But there is a load-bearing gap in the proof of Theorem 1.1. Section 3.1.1 begins with 'Let Φ ∈ Homeo(Σ,ω) be the extension of φ', which requires every φ ∈ Diff(Σ_P,ω_0) to extend continuously to the punctures. The stress-test example is right and I checked it: on the annulus R/Z × (0,1) with area form dx∧dy, F(x,y) = (x + ln y, y) is a C^∞ area-preserving diffeomorphism, but as y→0 the x-coordinate winds infinitely often, so F has no continuous extension to the puncture. Thus F is in the stated domain and the first step of the proof is unavailable. This is not an isolated pathology: in the strong C^∞ topology, any map sufficiently close to F also has ∂x/∂y ≈ 1/y, so it also fails to extend, meaning the extendable maps are not even dense near F. The theorem may still be true, but this proof does not touch the non-extendable class. What survives? If one restricts Theorems 1.1 and 1.3 to diffeomorphisms that do extend to area-preserving homeomorphisms of the closed surface, the argument is coherent: rationalize using CGPZ Lemma 5.4, apply the quantitative closing lemma, pass to the limit. The S^2 equidistribution proof is otherwise well structured, and the use of Chen's HF comparison to get the triangle inequality is clever. The same extension issue appears there, so it also needs the same fix. Bottom line: this deserves a serious referee, but the referee should focus on the extension problem. The author probably needs to either restrict the statements to extendable maps or find an approximation argument that genuinely works in the strong topology; I don't think that approximation is easy, since non-extendability is stable. As it stands, the central claim is not proved for the stated domain.","headline":"The proof of Theorem 1.1 assumes every area-preserving diffeomorphism of a punctured surface extends to the closed surface; that is false for a stable class of maps, so the main closing lemma is not established for the stated domain.","tokens_in":744,"tokens_out":759,"would_cite":true,"duration_ms":56319,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37C25","37C20","37E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The C∞ closing lemma, previously known only for closed surfaces, holds for closed surfaces with finitely many punctures: every map can be C∞-approximated by one with a periodic orbit through any specified open set, and a generic map has…","keywords":["closing lemma","area-preserving diffeomorphisms","periodic orbits","generic density","punctured surfaces","Periodic Floer homology","PFH Weyl law","equidistribution"],"falsifier":"Take a rational area-preserving homeomorphism Φ of a closed surface and a fixed open set U away from the punctures; let ℓ_n be the minimal period of any periodic orbit of the nth Hamiltonian approximation φ_n that meets U. If one can exhibit a sequence with ℓ_n → ∞ while all φ_n lie in the same rational Hamiltonian isotopy class, then the paper's period-uniformity step collapses and Theorem 1.1 would be false. Equivalently, checking whether the bound in the cited quantitative closing lemma depends only on the isotopy class and not on the particular map would settle the point.","tokens_in":19023,"feed_emoji":"🎯","tokens_out":8491,"duration_ms":75651,"temperature":0.7,"pith_summary":"This paper extends the C∞ closing lemma to non-compact surfaces formed by removing finitely many points from a closed area surface. It proves that any C∞ area-preserving diffeomorphism of such a punctured surface can be C∞-approximated by one with a periodic orbit through any prescribed open set. A standard Baire-category consequence is that a C∞-generic such map has a dense set of periodic points. On the punctured two-sphere, the paper proves a quantitative strengthening: a C∞_loc-dense set of elements has an equidistributed sequence of orbit sets, meaning periodic orbits approximate the area measure.","feed_headline":"C∞ closing lemma reaches punctured surfaces","feed_subtitle":"Generic area-preserving maps on punctured surfaces get dense periodic points; on the 2-sphere, orbits spread evenly.","key_machinery":"The load-bearing machinery is the quantitative C∞ closing lemma from PFH spectral theory, applied to rational area-preserving homeomorphisms. The paper first proves Lemma 3.1: any area-preserving homeomorphism of a closed surface is the C0-limit of diffeomorphisms lying in one Hamiltonian isotopy class, so period bounds can be chosen uniformly. It then defines a rational area-preserving homeomorphism as one in the C0-closure of a rational Hamiltonian isotopy class, and shows such homeomorphisms inherit the closing property. For the equidistribution result on the sphere, the key identity is the PFH Weyl law variant (4.2), derived from a triangle-type inequality for spectral invariants, which lets the paper pass from smooth Hamiltonian perturbations to limits that are only continuous at the punctures.","core_discovery":"On the paper's own terms, the central discovery is that the C∞ closing lemma, previously known for area-preserving diffeomorphisms of closed surfaces, survives when finitely many points are removed from the surface. The proof treats a diffeomorphism of the punctured surface as an area-preserving homeomorphism of the closed surface fixing the puncture set, and shows that such a homeomorphism lies in the C0-closure of a single Hamiltonian isotopy class. Using a quantitative Periodic Floer homology (PFH) closing lemma for rational area-preserving diffeomorphisms, it produces, for every approximating map, a periodic orbit through the prescribed open set with a period bounded uniformly by the isotopy class; a subsequence then converges to a periodic orbit of the limit homeomorphism, giving the closing lemma. For the punctured sphere, the paper proves a PFH Weyl law for rational area-preserving homeomorphisms—a variant of the closed-surface Weyl law, equation (4.2)—and combines it with a formal argument to show that a C∞_loc-dense set of elements has an equidistributed sequence of orbit sets.","pith_inferences":["If the closing lemma survives on punctured surfaces, the same strategy may extend the C∞ closing lemma to surfaces with boundary by viewing them as closed surfaces with discs removed, provided the quantitative PFH input remains uniform.","The PFH Weyl law for homeomorphisms on the 2-sphere suggests that equidistribution of periodic orbits is not an artifact of smoothness: it can hold for C0 limits of Hamiltonian flows in a non-compact setting.","A concrete way to probe the boundary of the result is to test the triangle-type inequality (4.3) and the PFH/HF comparison on higher-genus punctured surfaces; the paper itself notes its method is unlikely to extend beyond the sphere, so a counterexample on a punctured torus would sharply delineate where the equidistribution theorem fails.","The equidistribution result implies that, for C∞_loc-generic maps on the punctured sphere, periodic orbits of high period sample the area measure, giving a quantitative analogue of an ergodic theorem for maps that need not be ergodic."],"forward_implications":["Every C∞ area-preserving diffeomorphism of a finitely punctured closed surface has C∞ perturbations with periodic orbits through any prescribed open set.","A C∞-generic area-preserving diffeomorphism of the punctured surface has a dense set of periodic points.","On the punctured 2-sphere, a C∞_loc-dense set of area-preserving maps has an equidistributed sequence of orbit sets: for every compactly supported smooth test function, the normalized sum over the orbit set converges to the integral of the function against the area form.","The PFH Weyl law for rational area-preserving homeomorphisms, proven here for the sphere, adds a quantitative tool for studying periodic orbit distribution of continuous area-preserving limits.","The C0-closure perspective from Lemma 3.1 means the closing property is not lost when smooth maps degenerate to homeomorphisms with finitely many singular points."],"supporting_citations":[{"why":"Supplies a compactly supported symplectic vector field that rationalizes the given area-preserving diffeomorphism, a necessary step before applying the quantitative closing lemma.","marker":"[CGPZ21, Lemma 5.4]"},{"why":"Provides the PFH Weyl law for area-preserving diffeomorphisms of closed surfaces, which the paper adapts to rational area-preserving homeomorphisms.","marker":"[CGPZ21, Theorem 1.5]"},{"why":"Gives the quantitative PFH closing lemma producing a periodic orbit of size O(δ^{-1}) within a small Hamiltonian perturbation, the engine for the closing argument.","marker":"[EH21, Theorem 7.4]"},{"why":"Shows every rational Hamiltonian isotopy class on a closed surface satisfies the U-cycle property, making the quantitative closing lemma uniform over the isotopy class.","marker":"[CGPPZ21, Corollary 2]"},{"why":"Supplies the Baire-category argument that converts a C∞ closing lemma into generic density of periodic points.","marker":"[AI16, Corollary 1.2]"},{"why":"Provides the formal construction of nearly equidistributed orbit sets and the parametric transversality framework adapted in Sections 5 and 6.","marker":"[Pra21]"},{"why":"Compares PFH and HF spectral invariants, yielding the triangle-type inequality used to prove the Weyl law variant for the sphere.","marker":"[Che22, Corollary 1.3]"},{"why":"Establishes that every area-preserving homeomorphism of a closed surface is the C0-limit of smooth area-preserving diffeomorphisms, the starting point for Lemma 3.1.","marker":"[Oh06, Theorem I]"}],"fun_headline_variants":["Closing lemma for punctured surfaces: dense periodic orbits","Punctured surfaces: C∞ closing lemma gives dense periodic points","Generic dense periodic orbits for punctured surface maps","Area-preserving maps on punctured surfaces get dense periodic points","C∞ closing lemma on punctured surfaces: periodic points dense"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument rests on the cited quantitative PFH closing lemma for rational area-preserving diffeomorphisms of closed surfaces: for every approximating map the lemma must produce a periodic orbit through the open set with a period bounded only by the Hamiltonian isotopy class, not by the step n; if that uniformity fails, the limiting orbit may not exist.","fun_headline_variants_meta":{"raw":{"variants":["Closing lemma for punctured surfaces: dense periodic orbits","Punctured surfaces: C∞ closing lemma gives dense periodic points","Generic dense periodic orbits for punctured surface maps","Area-preserving maps on punctured surfaces get dense periodic points","C∞ closing lemma on punctured surfaces: periodic points dense"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001873,"raw_usage":{"total_tokens":7320,"prompt_tokens":885,"completion_tokens":6435,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":501,"completion_tokens_details":{"reasoning_tokens":6353}},"tokens_in":501,"tokens_out":6435,"duration_ms":40935,"temperature":1.0,"reasoning_tokens":6353,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:19:39.255731+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a rational area-preserving homeomorphism Φ of a closed surface and a fixed open set U away from the punctures; let ℓ_n be the minimal period of any periodic orbit of the nth Hamiltonian approximation φ_n that meets U. If one can exhibit a sequence with ℓ_n → ∞ while all φ_n lie in the same rational Hamiltonian isotopy class, then the paper's period-uniformity step collapses and Theorem 1.1 would be false. Equivalently, checking whether the bound in the cited quantitative closing lemma depends only on the isotopy class and not on the particular map would settle the point.","supporting_citations":[],"review_version":1}