{"id":"e60028e2-de44-4512-8de7-9a0a3d793b73","arxiv_id":"1909.00920","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For countable abelian group actions, Banach-, Weyl- and Besicovitch-mean equicontinuity are equivalent, and transitive almost Banach-mean equicontinuous systems have zero topological entropy.","lead":"This paper introduces Banach-mean equicontinuity for group actions and proves that for abelian groups it is equivalent to two older averaging notions. It then shows that transitive systems with this stability have zero topological entropy, a measure of dynamical complexity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1 is false as stated: a transitive subshift with one isolated point and a full binary shift as limit set has positive entropy, and the isolated point is Banach-mean equicontinuous.","rationale":"The reader's weakest assumption correctly located the no-isolated-points condition. My stress test goes one step further: the condition is not merely unproven in Theorem 1.1, it is necessary. Every isolated point is trivially Banach-mean equicontinuous, so any transitive system with an isolated point is automatically almost Banach-mean equicontinuous. The explicit subshift construction shows that such a system can also have positive topological entropy, since the full binary shift appears as the limit set. Thus the abstract/introduction Theorem 1.1 is false as stated. The paper does contain a substantial valid core: the equivalence of Banach-, Weyl-, and Besicovitch-mean equicontinuity for abelian actions and Theorem 6.6 under the no-isolated-points hypothesis appear coherent. The problem is that the advertised main theorem overstates what is proved and is contradicted by a simple example. The verdict should therefore be REJECT for the current manuscript, with the expectation that a revised version restating Theorem 1.1 with the no-isolated-points hypothesis (or an equivalent condition on the equicontinuous point) could be salvageable.","tokens_in":18737,"tokens_out":25072,"duration_ms":285969,"concrete_test":"Implement the construction explicitly. Pick a two-sided binary sequence y whose positive orbit is dense in the full shift, and form the subshift X generated by x with x_0=* and x_n=y_n for n≠0. Then verify three facts: (a) the cylinder [* at 0] contains only x, so x is isolated; (b) every binary sequence is a coordinatewise limit of shifts of x as the * escapes to infinity, so the full binary shift embeds in X as an invariant subset; (c) the topological entropy of X is therefore at least log 2, for instance by counting the number of n-blocks of the embedded full shift. If (a)-(c) hold, Theorem 1.1 as stated is refuted.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim fails in the stated generality, not just in proof coverage. Let A={0,1,*}, let y in {0,1}^Z be a point whose forward orbit is dense in the full two-shift, and define x in A^Z by x_0=* and x_n=y_n for n≠0. Put X = closure of {σ^n x : n in Z}, a compact metric Z-system. The action is transitive by construction, and x is isolated in X because the cylinder [* at 0] intersects X only in {x}. Since an isolated point is automatically a Banach-mean equicontinuous point (choose δ so B(x,δ)={x}), the system is almost Banach-mean equicontinuous. Yet X contains the full shift {0,1}^Z as a closed invariant subset: for any w in the full shift, choose n_k→∞ with σ^{n_k} y→w; then σ^{n_k} x→w because the unique * escapes to -∞. Hence htop(X) ≥ log 2 > 0. This contradicts Theorem 1.1. The proof of Theorem 1.1 invokes Theorem 6.6, which explicitly assumes X has no isolated points, via Proposition 2.4. The counterexample shows that this assumption cannot be dropped; it is essential to the truth of the theorem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines a Banach-mean pseudometric D for actions of countable discrete groups on compact metric spaces and studies the corresponding equicontinuity notions. For countable amenable group actions it proves D equals the Weyl-mean pseudometric (Theorem 4.3), and for abelian group actions it asserts the equivalence of Banach-, Weyl-, and Besicovitch-mean equicontinuity (Theorem 4.5). The main advertised result is Theorem 1.1: every transitive, almost Banach-mean equicontinuous action of a countably infinite abelian group has zero topological entropy. Section 6 contains the proof via IE-pairs, Følner independence, and a dichotomy between almost Weyl-mean equicontinuity and Weyl-mean sensitivity; Theorem 1.2 derives the zero-entropy conclusion for globally Banach-mean equicontinuous actions using the variational principle.","tokens_in":19040,"tokens_out":17444,"duration_ms":174375,"significance":"If the results were correct in the stated generality, the paper would give a useful extension of mean-equicontinuity/zero-entropy results from Z-actions to abelian group actions. The equality D=D̄ in Theorem 4.3 is a clean observation, and the IE-pair argument in Theorem 6.6 is nontrivial and appears largely coherent for systems without isolated points. However, Theorem 1.1 is false as stated: the proof requires a no-isolated-points hypothesis, and a simple counterexample with an isolated equicontinuous point and positive entropy shows the omission is essential. The headline claim is therefore not valid.","major_comments":[{"comment":"Theorem 1.1 is false as stated because the proof goes through Theorem 6.6, which assumes X has no isolated points, and that hypothesis is essential. Let A={0,1,*}, choose y∈{0,1}^Z with dense forward orbit in the full two-shift, and define x∈A^Z by x_0=* and x_n=y_n for n≠0. Let X be the closure of {σ^n x: n∈Z}. The Z-action on X is transitive (x is a transitive point). The cylinder [* at 0] meets X only in {x}, so x is isolated; hence for δ small enough B(x,δ)={x} and D(x,y)=0<ε for every y∈B(x,δ), making x a Banach-mean equicontinuous point. Thus (X,Z) is almost Banach-mean equicontinuous. On the other hand, for every w∈{0,1}^Z there is a sequence n_k→∞ with σ^{n_k}y→w, and then σ^{n_k}x→w, so X contains the full two-shift as a closed invariant subset and h_top(X,Z)≥log2>0. This contradicts Theorem 1.1 exactly in the omitted isolated-points case.","section":"Theorem 1.1 (Abstract and Introduction); Theorem 6.6"},{"comment":"The proof begins with the stronger hypothesis that every open set U contains u,v with D(u,v)>2δ, whereas the proposition is stated with >δ. As written the proof does not establish the stated implication. The gap is readily fixable: the stated hypothesis with constant δ gives Weyl-mean sensitivity with sensitivity constant δ/2 by the same triangle inequality used in the proof. Still, the mismatch must be corrected because Proposition 5.3 is used in the proof of Theorem 5.5 and hence in Theorem 6.6.","section":"Proposition 5.3"}],"minor_comments":[{"comment":"The definition of a transitive point as Gx=X should read overline{Gx}=X (dense orbit), consistent with Proposition 2.2 and with the later usage in the paper.","section":"Definition 2.1"},{"comment":"There are typos in the abstract and introduction: 'Bnanach', 'Wely', 'abelain', and 'alelian'; also 'for for every' appears in Definition 4.1 and Definition 4.2.","section":"Abstract and Definitions 4.1, 4.2"},{"comment":"The invoked theorem [11, Theorem 1.3, p.6] should be stated explicitly, since Theorem 4.5 is one of the paper's main equivalences and the reader cannot otherwise verify it.","section":"Theorem 4.5"},{"comment":"After proving J1∩J2=∅, the text says 'Hence, J1∩J2 ⁄= ∅'; this should be '=∅'.","section":"Section 6, proof of Theorem 6.6"},{"comment":"The one-line proof of Theorem 1.1 should be expanded to name the results that convert almost Banach-mean equicontinuity to almost Weyl-mean equicontinuity (Theorem 4.3/Corollary 4.4) and to flag the no-isolated-points hypothesis from Theorem 6.6.","section":"Section 6, proof of Theorem 1.1"}],"recommendation":"reject","confidential_remarks":"The counterexample in my report is elementary, which makes the false statement of Theorem 1.1 difficult to repair without changing the advertised claim. The authors should consider resubmitting a version in which Theorem 1.1 is stated with the no-isolated-points hypothesis and the abstract is adjusted accordingly; the Følner/IE-pair material in Sections 5-6 may then be publishable. My recommendation is based on the manuscript as submitted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the paper proves something true and interesting—Banach-mean equicontinuity collapses into Weyl- and Besicovitch-mean equicontinuity for abelian group actions—but the headline theorem as stated is false. There is a simple counterexample with an isolated point. That is not a minor gap.\n\nThe new notion (Definition 4.1) is a natural Banach-density analog of existing mean metrics. The proof of Theorem 4.3 that the Banach pseudometric D equals the Weyl pseudometric D looks correct: the Følner-shift argument is standard, and the contradiction step is sound. Theorem 4.5, which adds Besicovitch into the equivalence, is essentially quoted from Fuhrmann–Gröger–Lenz [11]; the paper should say so more explicitly, but the dependency is legitimate. The IE-pair route to zero entropy in Theorem 6.6 is also coherent: choose a transitive recurrent point, use the independence set of an IE-pair, and produce two points in a small ball with Weyl-mean distance bounded away from zero. Under the explicit assumption that X has no isolated points, that argument works.\n\nThe problem is Theorem 1.1. The abstract and introduction state it for every compact metric transitive system. The proof imports Theorem 6.6, which assumes no isolated points. The stress-test counterexample is exactly what one would fear: take a transitive point in the full 2-shift, insert a single star at time 0, and take the orbit closure. The star point is isolated, hence trivially Banach-mean equicontinuous, while the closure contains the entire 2-shift, so entropy is log 2. So Theorem 1.1 is not merely unproven in the isolated-points case; it is false. The paper already contains the right fix—state the theorem for systems without isolated points—but the current wording is a genuine mathematical error.\n\nThere are smaller issues: Proposition 5.3 assumes points with D(u,v)>2δ to conclude Weyl-mean sensitivity with δ; the δ/2δ mismatch should be cleaned up. The abstract has a typo ('Bnanach'). These are trivial. The citation pattern is healthy; the paper builds on Kerr–Li and Li–Tu–Ye without obvious overselling.\n\nWho should read this: people working on mean equicontinuity and the structure of abelian actions will want the equivalence result and the corrected zero-entropy theorem. The paper is not publishable as is, but a revision that restricts Theorem 1.1 to no-isolated-points and adjusts the abstract and introduction would make it a useful contribution. I would send it to a referee; the core mathematics is worth checking carefully, and the counterexample to the stated theorem is the kind of thing a referee is supposed to catch.","headline":"A worthwhile equivalence theorem and a coherent no-isolated-points entropy proof, but the headline zero-entropy theorem is false as stated because of a simple isolated-point counterexample.","tokens_in":19557,"tokens_out":3392,"would_cite":false,"duration_ms":34667,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37A35","37B40"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a transitive abelian group action, a single almost Banach-mean equicontinuous point forces zero topological entropy.","keywords":["Banach-mean equicontinuity","Weyl-mean equicontinuity","Besicovitch-mean equicontinuity","topological entropy","IE-pair","independence density","abelian group action","mean sensitivity"],"falsifier":"Build a compact metric space with an isolated point carrying a transitive action of a countably infinite abelian group that is almost Banach-mean equicontinuous and has positive topological entropy; such a system would refute Theorem 1.1 as stated. If instead every transitive abelian group action on a space with isolated points has zero entropy, then the missing case closes and the statement survives.","tokens_in":18555,"feed_emoji":"0️⃣","tokens_out":11489,"duration_ms":95162,"temperature":0.7,"pith_summary":"This paper introduces Banach-mean equicontinuity for group actions on compact metric spaces, defined through the upper Banach density of the times at which two nearby orbits remain close, and proves that for countable abelian group actions this notion coincides with Weyl-mean and Besicovitch-mean equicontinuity. Its main theorem states that if a countably infinite abelian group acts transitively and the action has at least one Banach-mean equicontinuous point, then the topological entropy of the action is zero. The proof establishes the contrapositive through a dichotomy: a transitive action is either almost Weyl-mean equicontinuous or Weyl-mean sensitive, and positive entropy forces the latter through an independence-pair argument inside the independence-density framework. As a corollary, every Banach-mean equicontinuous action of a countable abelian group on a compact metric space has zero entropy. The result matters because it carries the zero-entropy consequences of mean equicontinuity, previously developed for integer actions, to all countable abelian group actions using combinatorial independence rather than invariant measures.","feed_headline":"Zero entropy for almost Banach-mean equicontinuous actions","feed_subtitle":"One almost equicontinuous point is enough to kill entropy in transitive abelian actions.","key_machinery":"The central object is the Banach-mean pseudometric $D(x,y)=\\inf_{F\\in\\operatorname{Fin}(G)}\\sup_{g\\in G}\\frac{1}{|F|}\\sum_{t\\in Fg} d(tx,ty)$, which measures how far two orbits diverge when averaged over right translates of a finite window; the proof uses the theorem that for amenable groups this equals the Weyl-mean pseudometric $D(x,y)$. The combinatorial engine is the IE-pair: a pair of distinct points $x_1,x_2$ such that every product neighbourhood $(U_1,U_2)$ has positive independence density, together with the theorem that positive topological entropy implies the existence of an IE-pair. Given an IE-pair, the argument finds a recurrent transitive point $x_0$, picks two return times $l_1,l_2$ whose shifted independence sets remain large in Banach density, and forces $l_1x_0$ and $l_2x_0$ to be close in the metric but far apart in the Weyl pseudometric, contradicting almost Weyl-mean equicontinuity. This turns positive entropy into Weyl-mean sensitivity.","core_discovery":"The central claim is Theorem 1.1: let $G$ be a countably infinite abelian group, $X$ a compact metric space, and $G \\curvearrowright X$ a transitive continuous action. If the action is almost Banach-mean equicontinuous, meaning some point $x$ has the property that for every $\\varepsilon>0$ there is $\\delta>0$ such that every $y$ within $\\delta$ of $x$ satisfies the Banach-mean bound $D(x,y)<\\varepsilon$, then $h_{\\mathrm{top}}(X,G)=0$. The proof runs through Theorem 6.6, which shows the contrapositive for compact metric spaces without isolated points: positive topological entropy implies the action is Weyl-mean sensitive. Theorem 4.3 identifies the Banach-mean and Weyl-mean pseudometrics for amenable groups, so the almost equicontinuity assumption places the system on the equicontinuous side of the dichotomy in Theorem 5.5, leaving no room for positive entropy.","pith_inferences":["Because the dichotomy and the Banach/Weyl pseudometric equality rely only on amenability, the same zero-entropy argument may extend to countable amenable groups beyond the abelian case, provided the Besicovitch/Weyl equivalence and the IE-pair machinery hold there; the paper proves the equivalence only for abelian groups.","A testable extension is whether almost Banach-mean equicontinuity without transitivity already forces zero entropy; the paper's Theorem 1.2 covers only the fully equicontinuous case in the non-transitive setting, leaving the almost case open.","Settling Theorem 1.1 for spaces with isolated points would close the gap between the theorem's statement and Theorem 6.6's no-isolated-points hypothesis; a transitive system with an isolated point and positive entropy would disprove the theorem as stated.","The independence-density mechanism—an IE-pair yielding two disjoint shifted independence sets that separate nearby points in the mean pseudometric—could be adapted to other sensitivity notions, such as Besicovitch sensitivity, wherever a Furstenberg correspondence principle is available."],"forward_implications":["A transitive action of a countably infinite abelian group on a compact metric space that is almost Banach-mean equicontinuous has zero topological entropy (Theorem 1.1).","For countable abelian group actions, Banach-, Weyl-, and Besicovitch-$F$-mean equicontinuity are the same property, so each of them forces zero entropy in the transitive setting.","Every Banach-mean equicontinuous action of a countable abelian group on a compact metric space, transitive or not, has zero topological entropy (Theorem 1.2).","A transitive action of a countable amenable group is either almost Weyl-mean equicontinuous or Weyl-mean sensitive; positive entropy puts it in the sensitive branch whenever the space has no isolated points (Theorems 5.5 and 6.6)."],"supporting_citations":[{"why":"Supplies the transitivity and recurrence facts, the independence-density machinery, and the IE-pair theorem linking positive entropy to IE-pairs.","marker":"[20]"},{"why":"Establishes the structure of mean equicontinuous group actions and the equivalence of Weyl- and Besicovitch-mean equicontinuity used in Theorem 4.5.","marker":"[11]"},{"why":"Introduced Weyl- and Besicovitch-mean equicontinuity and provides Proposition 6.1, the positive-measure intersection lemma used in the correspondence argument.","marker":"[17]"},{"why":"Provides the theorem that every abelian group is amenable, which brings the Følner-sequence and Banach-density machinery into play.","marker":"[6]"},{"why":"Supplies the variational principle for amenable group actions, used in Theorem 1.2 to move from supports of ergodic measures to topological entropy.","marker":"[19]"},{"why":"Source of the correspondence principle used to prove Proposition 6.4 about intersections of positive-density sets.","marker":"[10]"},{"why":"Gives the Følner-sequence formula for upper Banach density used inside the proof of the correspondence proposition.","marker":"[8]"},{"why":"Provides equation (3.1) relating upper Banach density to the supremum over Følner sequences.","marker":"[4]"}],"fun_headline_variants":["Zero entropy from almost Banach-mean equicontinuity","Almost equicontinuous transitive actions have zero entropy","One equicontinuous point gives zero entropy for abelian actions","Transitive and almost equicontinuous implies entropy zero"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof needs a starting point whose orbit is dense and which returns infinitely often to every neighbourhood of itself; the paper obtains such a point only when the space has no isolated points, a condition Theorem 1.1 does not state.","fun_headline_variants_meta":{"raw":{"variants":["Zero entropy from almost Banach-mean equicontinuity","Almost equicontinuous transitive actions have zero entropy","One equicontinuous point gives zero entropy for abelian actions","Transitive and almost equicontinuous implies entropy zero"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000254,"raw_usage":{"total_tokens":1524,"prompt_tokens":854,"completion_tokens":670,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":470,"completion_tokens_details":{"reasoning_tokens":600}},"tokens_in":470,"tokens_out":670,"duration_ms":269934,"temperature":1.0,"reasoning_tokens":600,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T05:33:01.598602+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build a compact metric space with an isolated point carrying a transitive action of a countably infinite abelian group that is almost Banach-mean equicontinuous and has positive topological entropy; such a system would refute Theorem 1.1 as stated. If instead every transitive abelian group action on a space with isolated points has zero entropy, then the missing case closes and the statement survives.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the transitivity and recurrence facts, the independence-density machinery, and the IE-pair theorem linking positive entropy to IE-pairs."},{"cited_title":"The structure of mean equicontinuous group actions","cited_arxiv_id":"1812.10219","evidence_quote":"Establishes the structure of mean equicontinuous group actions and the equivalence of Weyl- and Besicovitch-mean equicontinuity used in Theorem 4.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduced Weyl- and Besicovitch-mean equicontinuity and provides Proposition 6.1, the positive-measure intersection lemma used in the correspondence argument."},{"cited_title":"Ceccherini-Silberstein, M","cited_arxiv_id":null,"evidence_quote":"Provides the theorem that every abelian group is amenable, which brings the Følner-sequence and Banach-density machinery into play."},{"cited_title":"Huang, X","cited_arxiv_id":null,"evidence_quote":"Supplies the variational principle for amenable group actions, used in Theorem 1.2 to move from supports of ergodic measures to topological entropy."},{"cited_title":"Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theo ry","cited_arxiv_id":null,"evidence_quote":"Source of the correspondence principle used to prove Proposition 6.4 about intersections of positive-density sets."},{"cited_title":"Downarowicz, D","cited_arxiv_id":null,"evidence_quote":"Gives the Følner-sequence formula for upper Banach density used inside the proof of the correspondence proposition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides equation (3.1) relating upper Banach density to the supremum over Følner sequences."}],"review_version":1}