{"id":"697e9bb9-da46-4452-b358-50b904d2de47","arxiv_id":"1908.09536","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A point that is minimally expansive and shadowable is topologically stable and GH-stable; a point that is mu-uniformly expansive and mu-shadowable for a Borel measure is strong mu-topologically stable.","lead":"This math paper proves that if one point of a dynamical system has two special properties, minimal expansivity and shadowing, then that point is stable under small perturbations, both in the usual topological sense and in a Gromov-Hausdorff sense. It also proves a measure-based version for systems studied with Borel measures. Generalists might read it because stability of individual orbits under perturbation is a bridge between numerical simulation and rigorous dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Definition 3.7 of µ-topological stability is vacuous: the empty set-valued map satisfies (i)–(iii), so Theorem 3.9's first assertion holds without µ-uniform expansivity and needs a nonempty/full-measure-domain condition.","rationale":"The reader's conditional verdict is well founded, but the most load-bearing issue is not primarily the orbit-closure notation: it is that the pointwise µ-topological stability notion in Definition 3.7 is vacuous, because the empty set-valued map satisfies all listed conditions. This does not refute the strong µ-topologically stable conclusion of Theorem 3.9, whose proof does force nonempty values on the shadowing set U, but it makes the first half of Theorem 3.9 and the corresponding non-strong assertions in Corollary 3.10 true for trivial reasons independent of expansivity. The fix is definitional, not merely notational. I therefore keep the reader's CONDITIONAL disposition: the paper needs a corrected Definition 3.7 with a nonempty-domain or full-measure condition, and the non-strong claim needs to be either re-proved or withdrawn. The strong stability argument appears largely sound modulo the minor δ<η convention and the intended orbit-closure notation, but the stated weak theorem is currently content-free.","tokens_in":15271,"tokens_out":30507,"duration_ms":308887,"concrete_test":"Check Definition 3.7 against the empty map H(z)=∅ for all z∈\\overline{O_g(x)}. Since Dom(H)=∅ is Borel, H is upper semi-continuous and compact-valued; (i) is vacuous, (ii) holds as ∅⊂B[z,ε], and (iii) holds as f(∅)=∅=H(g(z)). If this check passes, Theorem 3.9's first assertion is true for every point without any expansivity or shadowing assumption. The definition must then be amended—for example, by restoring the global condition µ(X\\Dom(H))=0 or by requiring Dom(H) to contain a full-measure subset of B(x,δ)∩\\overline{O_g(x)}—and Theorem 3.9 must be re-proved under the amended definition, since the current proof does not establish such a condition for the non-strong case.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Definition 3.7 requires an upper semi-continuous compact-valued map H : \\overline{O_g(x)} → 2^X with measurable domain satisfying (i) µ(H(z))=0 on B(x,δ/4)∩\\overline{O_g(x)}, (ii) d(H,Id)≤ε, and (iii) f∘H=H∘g. It does not require nonempty values or any full-measure condition on Dom(H). The empty map H(z)=∅ for every z satisfies all three conditions: Dom(H)=∅ is Borel, ∅ is compact, upper semi-continuity is vacuous, (i) is vacuous, (ii) holds because ∅⊂B[z,ε], and (iii) holds because both compositions are empty. Hence every point, with or without µ-uniform expansivity or µ-shadowing, is µ-topologically stable in the sense of Definition 3.7. This makes the first assertion of Theorem 3.9 content-free, and the sentence 'Former case follows by choosing δ=η in the proof' cannot repair it. The internal inconsistency is apparent from the global measure-stability definition earlier in Section 3, where condition (i) explicitly requires µ(X\\Dom(H))=0; the pointwise Definition 3.7 has dropped that condition. The strong µ-topologically stable statement may still be salvageable, but the paper's measure-stability theorem as stated conflates a vacuous notion with a genuine stability property.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces pointwise versions of expansivity, shadowing, topological stability, and GH-stability for homeomorphisms on compact metric spaces, together with measure-theoretic analogues. The main theorems are Theorem 2.7, asserting that every minimally expansive shadowable point is both topologically stable and GH-stable, and Theorem 3.9, asserting that every μ-uniformly expansive μ-shadowable point is strong μ-topologically stable. The paper also proves invariance and sequence-characterization results and derives corollaries for pointwise and measure-theoretic stability.","tokens_in":15515,"tokens_out":6063,"duration_ms":62732,"significance":"If the results are correct after appropriate repairs, they would extend the classical Walters stability theorem and the Lee–Morales measure-stability theorem from global assumptions to pointwise assumptions, and the GH-stability part would extend the Arbieto–Rojas result to points. The paper is self-contained and proof-based, with no fitted parameters or circular derivations. However, the current manuscript contains a vacuous definition in Section 3 and a load-bearing orbit-versus-orbit-closure ambiguity, so the main theorems as stated are not yet established.","major_comments":[{"comment":"Definition 3.7 of μ-topological stability is vacuous. The empty set-valued map H(z)=∅ for every z∈Og(x) satisfies all three conditions: Dom(H)=∅ is Borel, ∅ is compact and upper semi-continuous, condition (i) holds vacuously, condition (ii) holds because ∅⊂B[z,ε], and condition (iii) holds because both compositions are empty. Consequently every point of every homeomorphism is μ-topologically stable in the sense of Definition 3.7, and the first assertion of Theorem 3.9 is content-free rather than a genuine stability statement. This also conflicts with the global definition earlier in Section 3, which explicitly requires μ(X\\Dom(H))=0. The proof of Theorem 3.9 cannot repair this by saying 'Former case follows by choosing δ=η', because without a nonempty-domain or full-measure-domain condition there is nothing to prove. The definition needs to be repaired, for example by adding μ(X\\Dom(H))=0 or a nonempty-valued/full-measure condition, and then the first assertion of Theorem 3.9 needs a new proof.","section":"Definition 3.7 and Theorem 3.9"},{"comment":"The proof states 'Since Og(x) is closed, z∈Og(x)' when a sequence xk∈Dom(H) converges to z. The literal orbit Og(x) need not be closed in a compact metric space, e.g. for an irrational rotation on the circle. If the intended set is the orbit closure \\overline{Og(x)}, the notation must be changed consistently throughout Definition 3.7 and Theorem 3.9, and the text should explicitly say that H is defined on the closed invariant set \\overline{Og(x)} rather than on the raw orbit. This is not a presentation quibble: the closedness of the domain is used to conclude that the limit z lies in the domain and to make the compactness argument for measurability of Dom(H) valid. As written, the proof fails at this step if Og(x) denotes the literal orbit.","section":"Theorem 3.9, proof of Dom(H) closed"},{"comment":"The proof of Theorem 2.7 only treats the GH-stable case; the topological-stability half is deferred with 'Proof of first case can be done on similar lines.' This is a central theorem of the paper, so the omitted half should be written out in full. Moreover, the GH-stable proof contains an extension step whose wording is problematic: h is defined on the orbit Og(y) and shown uniformly continuous, and then the text says 'Since Y is compact and dX(j(y1),j(y2))<δ+dY(y1,y2) ... we can extend h continuously to a function H:Og(y)→X.' If Og(y) is the literal orbit, no extension is needed; if it is the orbit closure, the extension follows from uniform continuity and completeness of X, not from the displayed inequality involving j. The manuscript should clarify the intended domain in Definition 2.5 and give the extension argument explicitly.","section":"Theorem 2.7"}],"minor_comments":[{"comment":"The notation Og(x) is used inconsistently for the orbit and the orbit closure. The paper should define both O_g(x) and \\overline{O_g(x)} and use them carefully; in particular Definition 2.5, Definition 3.7, and Theorem 3.9 all need this distinction.","section":"Throughout"},{"comment":"The statement of item (2) has a grammatical/formal issue: 'x∈Mf(X) if and only if there exists a δ>0 such that for each y∈B(x,δ), if for every pair of distinct points u,v∈Of(y), ∪...≠∅' contains an extra 'if' and is hard to parse. It should be rephrased, e.g. '... such that for each y∈B(x,δ) and every pair of distinct points u,v∈Of(y), we have ...'.","section":"Theorem 2.4(2)"},{"comment":"In the proof of item (6), the line 'f^{n_i^u}∘h^{-1}(w)→h^{-1}(u) and f^{n_i^u}∘h^{-1}(w)→h^{-1}(u)' should presumably read '...→h^{-1}(v)' in the second convergence. This is a typographical error but should be corrected.","section":"Proposition 2.2(6)"},{"comment":"The term 'Borelian' is used without definition; it should say 'Borel set' or define 'Borelian' explicitly.","section":"Section 3, Definition 3.2 and Proposition 3.4"},{"comment":"In the statement of Theorem 3.6, the set Cz is written as 'Cz={y∈B(x,δ): ∪...=φ}=0', which mixes the set definition with the measure-zero assertion. It should be defined as a set and then asserted to have μ(Cz)=0.","section":"Theorem 3.6"}],"recommendation":"major_revision","confidential_remarks":"The vacuous definition in Section 3 is a serious but local defect; adding a full-measure/nonempty-domain condition is a natural repair. The orbit-versus-orbit-closure ambiguity also appears fixable by a consistent notation change. The missing topological-stability half of Theorem 2.7 is a matter of completeness rather than an irreparable error. I therefore recommend major revision rather than rejection. The authors should also state more precisely how their pointwise results relate to the earlier pointwise results of Koo–Lee–Morales and to the measure-stability theorem of Lee–Morales."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read this one if you want a case study in how a vacuous definition can sink a theorem. The main measure-theoretic result, Theorem 3.9, is content-free as stated: Definition 3.7 of a µ-topologically stable point is satisfied by the empty set-valued map. For any x, any g within δ, take H(z)=∅ for all z. Conditions (i)–(iii) hold vacuously. So every point is µ-topologically stable, without any µ-uniform expansivity or µ-shadowing. The \"strong\" variant adds condition (iv), which rules out the empty map on a full-measure set, but the first assertion of Theorem 3.9 is just true by definition.\n\nThat said, the paper isn't worthless. The pointwise notions—minimally expansive points, GH-stable points—are natural, and the proof of the GH-stability half of Theorem 2.7 is mostly coherent. The discussion around minimally expansive points (rational rotation example) is informative, and the idea of extending Walters and Lee-Morales to pointwise assumptions is reasonable. The Section 2 machinery (Lemma 2.6, the uniform convergence theorem) looks plausible.\n\nThe soft spots beyond the vacuity: (1) The topological stability half of Theorem 2.7 is not proved; \"proof of first case can be done on similar lines\" is not acceptable for a central result. (2) In the proof of Theorem 3.9, O_g(x) is treated as closed. A literal orbit isn't closed (irrational rotation). The intended domain is presumably the orbit closure; Definition 3.7 and the proof need to be consistent on this. (3) Minor notational confusions—e.g., T S in Prop 3.8(i) vs Ts elsewhere.\n\nWho is this for? People working in shadowing and pointwise stability. They might find the minimally expansive point notion useful, and the GH-stability proof is worth a look. But as it stands, the main advertised result (ii) is vacuous and the other main theorem is half-proved. I wouldn't cite this in its current form.\n\nRecommendation: send it to a serious referee rather than desk-reject, but expect heavy revision. A referee should catch the vacuous definition and the missing proof; if the authors fix both, there may be a decent paper here.","headline":"The measure-stability theorem is vacuous as stated—the empty set-valued map satisfies Definition 3.7—so the paper needs major revision, though the pointwise GH-stability half has some substance.","tokens_in":16067,"tokens_out":3988,"would_cite":false,"duration_ms":37761,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["54H20","37C75","37C50"],"pacs":[],"model":"deepseek-v4-flash","headline":"A minimally expansive shadowable point of a homeomorphism is topologically stable and GH-stable.","keywords":["minimally expansive point","GH-stable point","shadowable point","topological stability","Borel measure expansivity","measure shadowing","strong topological stability","pointwise dynamics"],"falsifier":"Construct a homeomorphism on a compact metric space with a point $x$ whose orbit is not closed but which is minimally expansive and shadowable, and test whether the conclusion holds. A concrete test is a saddle-connection type system in which the orbit of $x$ accumulates on a periodic orbit without reaching it; if the domain of the set-valued map $H$ in Theorem 3.9 has a sequence $x_k$ with limit in $\\overline{O_g(x)} \\setminus O_g(x)$, the proof's closedness assertion fails, and whether $x$ is actually strong $\\mu$-topologically stable would decide the theorem.","tokens_in":15025,"feed_emoji":"🔄","tokens_out":17059,"duration_ms":140891,"temperature":0.7,"pith_summary":"This paper proves that two classical stability theorems survive when their hypotheses are weakened from a whole system to a single point. It introduces minimally expansive points and GH (Gromov–Hausdorff) stable points for homeomorphisms, together with $\\mu$-uniformly expansive, $\\mu$-shadowable, and strong $\\mu$-topologically stable points for Borel measures. The main results are that on a compact metric space, a minimally expansive shadowable point of a homeomorphism is topologically stable and GH-stable, and a $\\mu$-uniformly expansive $\\mu$-shadowable point is strong $\\mu$-topologically stable. If correct, these theorems extend the classical topological stability theorem for expansive homeomorphisms with shadowing, and its measure analogue, to the pointwise setting: a system need not be globally expansive or globally shadowing for one orbit to be stable.","feed_headline":"Shadowable expansive points are stable, point by point","feed_subtitle":"Paper's pointwise theorems extend global stability results to single orbits, with a measure version.","key_machinery":"The load-bearing construction is a semiconjugacy obtained by tracing pseudo-orbits: for a perturbation $g$ of $f$, start with a pseudo-orbit through $x$, let $z$ be the point that shadows it, and define $h(g^n(y))=f^n(z)$. Minimal expansivity makes $h$ well defined: if two iterates of $y$ under $g$ agree, the shadowing inequality and the common expansivity constant force the corresponding iterates of $z$ to agree. Lemma 2.6 supplies the compactness step that turns 'close for all times along a long segment' into 'close at time zero', giving uniform continuity of $h$ and permitting extension to the orbit closure. In the measure case, the same tracing is packaged as the set-valued map $H(u)=\\bigcap_{n\\in\\mathbb{Z}} f^{-n}(B[g^n(u),\\eta])$, whose nonempty fibers are the traced points; the proof shows its domain is closed, hence measurable, and that its fibers have $\\mu$-measure zero.","core_discovery":"The central discovery is that expansivity and shadowing can be localized to a point and still imply stability. A point $x$ is minimally expansive when some $c>0$ makes the homeomorphism expansive on the orbit closure of every $y$ within distance $c$ of $x$, with the same constant $c$; $x$ is shadowable when every sufficiently small pseudo-orbit that starts at $x$ is traced by a real orbit. Theorem 2.7 asserts that such a point is topologically stable and GH-stable: every homeomorphism $g$ sufficiently close to $f$ in the relevant distance admits a continuous semiconjugacy $h$ defined on the orbit (or orbit closure) of $x$ under $g$, with $h$ close to the identity. Theorem 3.9 moves the same conclusion to Borel measures: a $\\mu$-uniformly expansive $\\mu$-shadowable point is strong $\\mu$-topologically stable, and the $\\mu$-shadowability hypothesis can be dropped if one only wants $\\mu$-topological stability. These are the pointwise versions of the global stability theorems, and they reduce to the classical statements when every point of the space satisfies the pointwise hypotheses.","pith_inferences":["Because stability of a point is certified by data on a single orbit and its neighborhood, numerical shadowing of a finite pseudo-orbit could be used to certify topological stability of that orbit in simulations, without verifying the shadowing property on the whole space.","The measure version suggests an almost-everywhere stability: the semiconjugacy's domain carries full μ-measure on the relevant orbit fragment, so nearby systems could be compared for μ-almost every point rather than for every point.","The definitions are stated for homeomorphisms; a natural test is whether the same pointwise implications hold for continuous non-invertible maps, where pseudo-orbits are one-sided and orbit closures behave differently.","The GH-stability condition at a point depends on the geometry of the orbit through the epsilon-isometry; one could try to localize the Gromov-Hausdorff distance to the pair of orbit closures, which may remove the compactness of the whole space from the hypothesis."],"forward_implications":["On a compact metric space, one minimally expansive shadowable point of a homeomorphism is enough to make that point topologically stable and GH-stable; the whole system need not be expansive or shadowing.","If the homeomorphism itself is expansive, every shadowable point is topologically stable and GH-stable, recovering the pointwise form of the classical stability theorem.","On compact manifolds of dimension at least two, a minimally expansive point is shadowable if and only if it is topologically stable.","For every Borel measure μ, a μ-uniformly expansive point is μ-topologically stable, and adding μ-shadowability upgrades it to strong μ-topologically stable.","When the measure is non-atomic, any topologically stable point is automatically strong μ-topologically stable."],"supporting_citations":[{"why":"Supplies the global theorem that expansive homeomorphisms with shadowing on compact metric spaces are topologically stable, which the pointwise theorem localizes.","marker":"[19, Theorem 4]"},{"why":"Gives the predecessor result that every shadowable point of an expansive homeomorphism is topologically stable; Theorem 2.7 weakens expansivity to minimal expansivity.","marker":"[7, Corollary 3.16]"},{"why":"Establishes the global GH-stability theorem for expansive homeomorphisms with shadowing that the pointwise GH-stability result mirrors.","marker":"[3, Theorem 4]"},{"why":"Introduces measure shadowing and topological stability for Borel measures and proves the global measure stability theorem that Theorem 3.9 localizes.","marker":"[11]"},{"why":"Defines shadowable points and provides the lemma that shadowing at x can be localized to pseudo-orbits through B(x, δ), used in both main proofs.","marker":"[14]"},{"why":"Introduces expansive points for Borel measures and specification, supplying the measure-expansivity framework and the Γ-fibers used in Section 3.","marker":"[5]"}],"fun_headline_variants":["Pointwise shadowing and expansivity imply point stability","Minimal expansivity plus shadowing gives GH-stability","Measure version: μ-expansive shadowable points are stable","Stability at a point from shadowing and expansivity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the orbit of the perturbed point can be treated as closed (or replaced by its closure), because the proof writes $O_g(x)$ and then uses 'since $O_g(x)$ is closed' to extend the stability map; if a literal orbit is meant, that step is unjustified for points whose orbits are not closed.","fun_headline_variants_meta":{"raw":{"variants":["Pointwise shadowing and expansivity imply point stability","Minimal expansivity plus shadowing gives GH-stability","Measure version: μ-expansive shadowable points are stable","Stability at a point from shadowing and expansivity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001116,"raw_usage":{"total_tokens":4617,"prompt_tokens":883,"completion_tokens":3734,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":499,"completion_tokens_details":{"reasoning_tokens":3667}},"tokens_in":499,"tokens_out":3734,"duration_ms":31305,"temperature":1.0,"reasoning_tokens":3667,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:08:46.156901+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a homeomorphism on a compact metric space with a point $x$ whose orbit is not closed but which is minimally expansive and shadowable, and test whether the conclusion holds. A concrete test is a saddle-connection type system in which the orbit of $x$ accumulates on a periodic orbit without reaching it; if the domain of the set-valued map $H$ in Theorem 3.9 has a sequence $x_k$ with limit in $\\overline{O_g(x)} \\setminus O_g(x)$, the proof's closedness assertion fails, and whether $x$ is actually strong $\\mu$-topologically stable would decide the theorem.","supporting_citations":[{"cited_title":"Differential Equations, 262, 3467-3487(2017)","cited_arxiv_id":null,"evidence_quote":"Introduces measure shadowing and topological stability for Borel measures and proves the global measure stability theorem that Theorem 3.9 localizes."},{"cited_title":"Bulletin of the Brazilian Mathematical Society, New Series(2019)","cited_arxiv_id":null,"evidence_quote":"Introduces expansive points for Borel measures and specification, supplying the measure-expansivity framework and the Γ-fibers used in Section 3."}],"review_version":1}