{"id":"c15ddc57-af2d-468d-8c4a-fca2fe384c6e","arxiv_id":"1908.05299","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper claims that C0-stable actions of the free group on two generators on compact manifolds of dimension at least two have the shadowing property, with a constructed non-expansive stable action on the two-sphere.","lead":"The paper claims that every C0-stable action of the two-generator free group on a compact manifold of dimension at least two has the shadowing property, and it constructs a C0 and C1-stable action on the two-sphere. The result would extend a classical theorem for single maps to group actions, but the proof has significant gaps.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem A asserts without argument that y_{g0} can be chosen so the finite ~x_g are pairwise distinct; this is essential for Lemma 4.2 but is not established (though a Baire argument may repair it).","rationale":"The reader's weakest assumption identifies the same point I would stress-test: the pairwise distinctness of the ~x_g is asserted without proof and is essential for Lemma 4.2. I agree with that identification, but only partially: I do not think the statement is false, because Remark 5 together with Baire category is likely sufficient to prove it; what is missing is the argument itself. The reader's separate criticism of Definition 2.1 property 4 is not valid: for homeomorphisms of a compact metric space, uniform convergence does imply convergence of inverses, so property 4 can be enforced by taking delta sufficiently small. The rest of the proof of Theorem A is coherent: Lemma 4.3 reduces C0-stability of the action to C0-stability of T = Phi_{b^{-1}}Phi_a; Lemma 4.4 reduces the pseudotrajectory to a pseudotrajectory for T; and the construction of ~Phi via Lemma 4.2 would finish the proof if the distinctness issue were settled. Because the missing distinctness argument is likely repairable, outright rejection on the basis of a counterexample is not warranted. The appropriate disposition is CONDITIONAL: the authors should supply a rigorous selection lemma for the y_{g0}. Minor issues, such as the S1/S2 inconsistency in Section 2 and the off-by-one indexing in the application of Lemma 4.2, should also be corrected in a revision.","tokens_in":9616,"tokens_out":23414,"duration_ms":243422,"concrete_test":"Write the missing selection lemma rigorously: for each coset C with finite index set I_C, define S_C = intersection over n in I_C of T^{-n}(B(z_n, eta)). Prove S_C is open and nonempty, then prove by induction over the finitely many cosets that one can choose y_C in S_C avoiding union over 1 <= |d| <= N of Fix(T^d) and also avoiding the finite set of preimages T^{-n}(w) of previously chosen ~x-values. If this proof cannot be completed, the perturbation argument in Theorem A fails; if it can be completed, the gap is a missing detail rather than a counterexample.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The most load-bearing concern is in Section 4, in the proof of Theorem A, just before equation (4.1). The authors need to apply Lemma 4.2 to build the perturbing diffeomorphism f, and this requires both the domain points Phi_a(~x_{a^{-1}g}) and the target points ~x_g to be pairwise distinct. They write: 'by Remark 5, it is possible to take y_{g0} such that ~x_g != ~x_{g'}' for distinct g,g' with |g|,|g'| <= n. No proof is given. This assertion must rule out two separate sources of collision: (i) periodic repetition inside one coset of <b^{-1}a>, i.e. T^{n-m}(y_{g0}) = y_{g0} for T = Phi_{b^{-1}}Phi_a; and (ii) collisions between different cosets, where T^n(y_{g0}) equals T^m(y_{g0'}) for two independently chosen shadowing points. Remark 5, which excludes an open set of points with one fixed period, is relevant to (i) but the reduction via Baire category is absent; case (ii) is not addressed at all. The gap is central because without distinctness the perturbation f of Lemma 4.2 cannot be constructed. It is plausible the gap is repairable: the finite shadowing sets S_c are open and nonempty, so a Baire/nowhere-dense argument might select y_{g0} avoiding all bad sets. But as written, the proof of Theorem A is incomplete at exactly the point where the perturbation is produced.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper has two main parts. First, it constructs an action Φ of the free group F2 on the two-sphere, claimed to be both C0-stable and C1-stable and to have a Cantor minimal set, thereby showing that expansivity is not necessary for stability of actions. Second, it proves Theorem A: every C0-stable action of F2 on a compact manifold of dimension at least two has the shadowing property. The proof reduces the action to a single homeomorphism T = Φ_{b^{-1}}Φ_a, uses Walters' theorem that C0-stable homeomorphisms have shadowing to correct each equivalence class of a pseudotrajectory, then constructs a C0-close action via the Nitecki–Shub perturbation lemma and uses a semiconjugacy to produce a shadowing point.","tokens_in":9965,"tokens_out":25209,"duration_ms":226419,"significance":"If the proof can be completed, Theorem A is a genuine extension of Walters' classical result from Z-actions to actions of a free group, and the example would be a useful counterpoint to the fact that expansive actions with shadowing are C0-stable. The strategy is natural and the use of the Nitecki–Shub lemma is apt. However, the manuscript in its present form contains several unproved assertions at load-bearing points, so the significance is conditional on repairs.","major_comments":[{"comment":"The assertion that y_{g0} can be chosen so that all ~x_g for |g| ≤ n are pairwise distinct is not proved and is load-bearing. It must rule out two sources of collision: within a single coset, where T^{n-m}(y_{g0}) = y_{g0} for the map T = Φ_{b^{-1}}Φ_a, and between different cosets, where T^n(y_{g0}) = T^m(y_{g0'}) for two independently chosen shadowing points. Remark 5 only excludes an open set of points with one fixed period and does not by itself supply the needed Baire/nowhere-dense argument; cross-coset collisions are not addressed at all. Moreover, because equation (4.1) involves ~x_{a^{-1}g} and ~x_{b^{-1}g}, whose lengths can be n+1, distinctness is actually needed for indices of length up to n+1, not only up to n. Without distinctness of both the domain points and the target points, Lemma 4.2 cannot be applied and the perturbation f cannot be constructed. The gap appears repairable via a finite Baire-category argument, but the proof must be written.","section":"Section 4, proof of Theorem A, before (4.1)"},{"comment":"The proof invokes 'As the map Φ_{b^{-1}}Φ_a is C0-stable', but Lemma 4.3 proves only that Φ_{a^{-1}}Φ_b is C0-stable. C0-stability of a homeomorphism is not automatically preserved under taking inverses, because inversion is not continuous in the C0 topology. A separate argument is required, for example the same proof as Lemma 4.3 with the roles of a and b exchanged. This is a missing step in the derivation, not an outright falsehood, but it must be supplied.","section":"Lemma 4.4 and proof of Theorem A"},{"comment":"Property 4 asserts that for d0_S(~Φ, Φ) < δ one has d(~Φ_s^{-1}Φ_s(x), x) < ε/2 and d(Φ_s^{-1}~Φ_s(x), x) < ε/2. This does not follow from C0-closeness alone, since the map f ↦ f^{-1} is not continuous in the uniform topology on the space of homeomorphisms of a compact space. The existence of a δ satisfying all five properties, including property 4, is asserted without proof. The authors should prove that for the specific Φ constructed in Section 2, and with the contraction structure, such a δ exists; as written this is an unproved assertion in the C0/C1-stability example.","section":"Section 2, Definition 2.1, property 4"},{"comment":"The paper states without proof that Φ_a and Φ_b can be constructed to satisfy simultaneously the five listed contraction and intersection properties in the paragraph after the definition of Φ_b. While plausible from the figure, the existence of such maps with the required derivative bounds and the disjointness of the intervals is asserted rather than demonstrated. Since the stable-action example is a stated contribution, this construction should be made precise or at least a proof sketch should be given.","section":"Section 2, construction of Φ_a and Φ_b"}],"minor_comments":[{"comment":"The abstract and the introduction say the example is a C0- and C1-stable action on S1, but Section 2 constructs the action on the two-sphere S2. The target space should be made consistent throughout.","section":"Abstract and Introduction"},{"comment":"Lemma 4.4 is stated for a full δ-pseudotrajectory, but the proof of Theorem A speaks of applying it to a finite δ-n pseudotrajectory. The text should clarify that Lemma 4.4 is applied to the full pseudotrajectory and the result is then restricted to |g| ≤ n.","section":"Section 4, proof of Theorem A"},{"comment":"Equation (4.1) is written as 'f(Φ_a(~x_{a^{-1}g})) = f(Φ_b(~x_{b^{-1}g})) = ~x_g', which is confusing because it suggests f is applied to two different inputs. Since Φ_a(~x_{a^{-1}g}) and Φ_b(~x_{b^{-1}g}) coincide by Lemma 4.4, the notation should be clarified, for instance by introducing p_g as the common point.","section":"Equation (4.1)"},{"comment":"There are numerous typographical and grammatical issues, including 'Considerer', 'diffeomorphisms', 'parte', inconsistent spacing in d0_S(~Φ,Φ), and the undefined notation X1\\X1 in Lemma 2.5. A careful proofreading pass is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is likely true and the proof strategy is sound, but the current version is not complete. The distinctness gap in the proof of Theorem A and the missing proof of C0-stability for Φ_{b^{-1}}Φ_a are repairable in a revision. The example section would benefit from a more rigorous treatment of the contraction properties and of Definition 2.1, property 4. The S1/S2 discrepancy indicates that the manuscript needs careful proofreading before it can be considered for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe main new thing here is a group-action version of Walters' theorem: C0-stable F2 actions on compact manifolds of dimension at least two have the shadowing property. That is a natural question and, as far as I know, genuinely open in the literature. The paper also constructs a C0- and C1-stable non-expansive action on S^1 with a Cantor minimal set. The construction is substantial, and the extension lemma (Lemma 2.5) for semiconjugacies on compacta is a useful tool in its own right. I want to give credit for that.\n\nI disagree with one of the reader's main doubts. Definition 2.1 property 4 asks that for a small C0 perturbation the inverses are pointwise close to the original inverses. On a compact manifold this is not a problem: the homeomorphism group is a topological group in the uniform topology, so a small C0 neighborhood of Phi can be chosen to satisfy those inequalities. That particular concern should not be a reason for rejection.\n\nThe real problem is in the proof of Theorem A, before equation (4.1). To apply Lemma 4.2 you need both the domain points Phi_a(~x_{a^{-1}g}) and the target points ~x_g to be pairwise distinct over the finite set of g with |g| <= n. The authors assert that by Remark 5 one can choose y_{g0} to make the ~x_g distinct. Remark 5 only says there is no open set of periodic points all with the same period; it does not, by itself, produce a shadowing point avoiding the finite collection of coincidence conditions. The candidate shadowing sets are G_delta, not open, and collisions can occur both within a single coset (periodicity of T = Phi_{b^{-1}}Phi_a) and between different cosets with independently chosen y_{g0}. This is exactly where the perturbation f is constructed, so the gap is load-bearing, not cosmetic. It may well be repairable, perhaps by a Baire category argument or by perturbing the finite pseudo-orbit first, but the paper does not provide it. There is also a smaller technical point: even if the ~x_g are distinct, Lemma 4.2 needs the domain points Phi_a(~x_{a^{-1}g}) distinct as well, which is a separate condition involving the next length level.\n\nMinor issues: the abstract says \"actions\" while the theorem is for F2; and several lemmas are stated with proofs omitted, which is acceptable in a research paper but makes checking harder.\n\nThe paper is worth taking seriously. I would send it to a referee who knows Walters' theorem and group actions, with the expectation that the distinctness step be fixed or explicitly reduced to a known lemma. In its present form the central proof is incomplete, but the idea and the example are valuable.","headline":"A worthwhile but incomplete paper: the F2 version of Walters' theorem is plausible and the stable non-expansive example has real content, but the proof of Theorem A has an unproved distinctness step that is load-bearing.","tokens_in":10470,"tokens_out":9822,"would_cite":false,"duration_ms":101283,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37C50","37C85","37B05"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every C0-stable action of the two-generator free group on a compact manifold of dimension at least two has the shadowing property.","keywords":["C0-stability","shadowing property","free group actions","pseudo-orbit tracing","expansivity","compact manifolds","semiconjugacy"],"falsifier":"The theorem would be refuted by any C0-stable action of F2 on a compact surface and a sequence of δ-pseudotrajectories with δ→0 that are not uniformly close to any true orbit. Short of that, a concrete calculation is to run the proof's construction on the paper's own stable action and test whether the corrected points ~x_g can be kept pairwise distinct for all g of length at most n; failure there would block Lemma 4.2.","tokens_in":9382,"feed_emoji":"🔁","tokens_out":14006,"duration_ms":135883,"temperature":0.7,"pith_summary":"The paper proves Theorem A: for an action of the free group on two generators on a compact manifold of dimension at least two, C0-stability implies the shadowing property. That is, if every sufficiently small perturbation of the action is semiconjugate to it by a near-identity continuous surjection, then every sufficiently fine approximate orbit lies uniformly close to a genuine orbit. This extends to group actions a classical property of single homeomorphisms, and it gives a necessary condition that any stable group action must satisfy. Along the way the authors construct a non-expansive action that is both C0- and C1-stable, showing that expansivity is not required for stability.","feed_headline":"Stable group actions must shadow their approximate orbits","feed_subtitle":"On compact manifolds of dimension ≥2, stable two-generator free-group actions always shadow approximate orbits.","key_machinery":"The load-bearing mechanism is the coset reduction of a group-action pseudotrajectory to pseudotrajectories of one homeomorphism. Writing the free group F2 with generators a and b, each index g is equivalent to g′ when g=($b^{{-1}}$a)^n g′; along such a class the action's δ-error becomes a δ+δ1-error for Φ_{$b^{{-1}}$}Φ_a, transferring the classical shadowing theorem for single maps into the group setting. The second mechanism is the interpolation lemma: on a compact manifold of dimension at least two, a small diffeomorphism can send finitely many prescribed points to nearby prescribed points, provided the points are pairwise distinct; this is what allows the corrected points to be woven into a perturbed action. The third mechanism is an extension lemma ensuring that a semiconjugacy built on a dense subset extends to the whole space.","core_discovery":"On its own terms, the paper's discovery is the implication C0-stability implies shadowing for actions of the free group F2 on compact manifolds of dimension at least two, stated as Theorem A. The proof shows that a δ-pseudotrajectory of the action can be organized along the cosets of the cyclic subgroup generated by the group element $b^{{-1}}$a: for each coset, the restricted sequence is a δ+δ1-pseudotrajectory of the single homeomorphism Φ_{$b^{{-1}}$}Φ_a. Since C0-stability of the action makes that homeomorphism C0-stable, and C0-stable homeomorphisms on compact manifolds of dimension at least two have the shadowing property, each coset can be approximated by a true orbit of Φ_{$b^{{-1}}$}Φ_a. A point-interpolation lemma then builds a small diffeomorphism f so that the action generated by fΦ_a and fΦ_b is C0-close to Φ; the near-identity semiconjugacy supplied by C0-stability converts the corrected finite pseudo-orbit into a genuine orbit of Φ, giving the desired ε-shadowing.","pith_inferences":["The coset-reduction argument should extend to free groups on k generators: pick any two generators to define the cosets and treat the remaining generators as perturbations, likely yielding the same implication for F_k-actions.","The interpolation step is the only place where the manifold hypothesis really enters, so a similar theorem may hold on any compact space that admits a finite point-placement lemma of the same kind.","Combining the theorem with the reverse implication, that expansive actions with shadowing are C0-stable, points toward a two-sided characterization of C0-stability for F2-actions on compact manifolds of dimension at least two, possibly with a weaker recurrence condition in place of full expansivity."],"forward_implications":["Every C0-stable F2-action on a compact manifold of dimension at least two has the shadowing property, so shadowing belongs to the list of necessary conditions for stability of such actions.","The single homeomorphism Φ_{b^{-1}}Φ_a is C0-stable whenever the action is, so one generator combination carries the shadowing information for the whole action.","Expansivity is not a necessary condition for stability, because the constructed action is C0- and C1-stable and non-expansive.","A finite pseudo-orbit of bounded word length can be ε-shadowed, and the finite-to-infinite passage in Lemma 4.1 upgrades this to the full shadowing property.","The proof identifies a concrete route from stability of group actions to shadowing: reduce to a cyclic subgroup, shadow a single homeomorphism, and interpolate the corrected points by a small diffeomorphism."],"supporting_citations":[{"why":"Provides the general definitions of δ-pseudotrajectories and shadowing for actions of finitely generated groups, plus the generator-independence result used to fix the metric.","marker":"[OT]"},{"why":"Supplies the interpolation lemma (Lemma 4.2): a small C0 perturbation can send finitely many pairwise distinct points to nearby prescribed points, which builds the perturbed action.","marker":"[NS, Lemma 13]"},{"why":"Establishes that a C0-stable homeomorphism on a compact manifold of dimension at least two has the shadowing property, applied here to Φ_{b^{-1}}Φ_a.","marker":"[W]"},{"why":"Proves that for actions on compact spaces, C0-stability is well-defined independently of the generating set and that expansive actions with shadowing are C0-stable, framing the stability notion used in the paper.","marker":"[CL]"}],"fun_headline_variants":["C0-stable actions on dim≥2 manifolds shadow approximate orbits","Stability of F2 actions on compact manifolds implies shadowing","C0-stability forces shadowing for actions on dim≥2 manifolds","C0-stable group actions on dim≥2 manifolds have shadowing","Shadowing from C0-stability on compact manifolds dim≥2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes that the shadowing point chosen for each group element class can be picked so that the finitely many corrected points are all different from each other; the paper gives no argument for this, and the interpolation lemma requires it.","fun_headline_variants_meta":{"raw":{"variants":["C0-stable actions on dim≥2 manifolds shadow approximate orbits","Stability of F2 actions on compact manifolds implies shadowing","C0-stability forces shadowing for actions on dim≥2 manifolds","C0-stable group actions on dim≥2 manifolds have shadowing","Shadowing from C0-stability on compact manifolds dim≥2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000762,"raw_usage":{"total_tokens":3305,"prompt_tokens":789,"completion_tokens":2516,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":405,"completion_tokens_details":{"reasoning_tokens":2419}},"tokens_in":405,"tokens_out":2516,"duration_ms":18549,"temperature":1.0,"reasoning_tokens":2419,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:18:16.133737+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The theorem would be refuted by any C0-stable action of F2 on a compact surface and a sequence of δ-pseudotrajectories with δ→0 that are not uniformly close to any true orbit. Short of that, a concrete calculation is to run the proof's construction on the paper's own stable action and test whether the corrected points ~x_g can be kept pairwise distinct for all g of length at most n; failure there would block Lemma 4.2.","supporting_citations":[],"review_version":1}