{"id":"7f18e1b1-be31-45f4-8a3e-848dc17b0a87","arxiv_id":"2411.15549","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A new relative notion of mean equicontinuity for factor maps between amenable group actions is introduced, characterized, and used to prove unique decompositions into topo-isomorphic and equicontinuous factors under minimality or weak mean equicontinuity.","lead":"This mathematics paper defines a version of 'mean equicontinuity' for maps between dynamical systems, not just for single systems, and proves when such maps are built from simpler pieces. It shows that smoothness-type behavior of a factor map is exactly a combination of mean equicontinuity with either distality or proximality, and gives a unique decomposition into a measure-preserving-like part and a rigid periodic-like part for a large class of systems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Minimal-case decomposition depends on the unproved cited criterion [2, Corollary 7.9]; verify its exact hypotheses before relying on Theorem 10.1.","rationale":"The reader's verdict ACCEPT with moderate confidence is sound. The central Definition 1.1 is tested against natural candidates: property (M) is shown insufficient in Example 6.2, and continuity of D on R(π) is shown too strong even for equicontinuous extensions in Example 6.1. The supplied proofs for Theorems 9.1, 9.2, 8.1, 8.4, 10.3, and 10.5 are checkable and contain no circular argument or fitted assumption. The most delicate structural input is the minimal-case decomposition Theorem 10.1, which uses a cited theorem from [2] rather than a proof. This is a standard way to write a mathematics paper, but it is exactly the kind of step that should be verified when assessing the central claim. The proposed concrete test settles whether the concern lands: if the cited corollary states and supports the relative criterion as used, the paper's central claim is fully supported; if not, the minimal decomposition is unproved, though the weakly mean equicontinuous counterpart still stands. Because the gap is a verification dependency rather than an identified error, the reader's ACCEPT verdict does not need to change, but the confidence should remain moderate until the citation is checked.","tokens_in":20729,"tokens_out":25636,"duration_ms":246946,"concrete_test":"Retrieve the exact statement of [2, Corollary 7.9] from Auslander's book and write out its hypotheses and conclusion. Then instantiate it with π : X → Y, φ : X → X/BP(π), ψ : X/BP(π) → Y, checking three points: (a) X, X/BP(π), and Y are minimal; (b) R(φ) = BP(π) as closed invariant equivalence relations; (c) the corollary's definition of RP(π) agrees with the paper's Definition in Section 2.2 and with the convention that regional proximality is a sequential limit of asymptotically proximal pairs in R(π). If the corollary only states the absolute criterion 'π is equicontinuous iff RP(π) = Δ_X', then prove the relative version for proximal quotients directly; if that proof fails, Theorem 10.1 has a real gap.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is in Theorem 10.1: after constructing the quotient φ : X → X/BP(π), the paper concludes that the induced ψ : X/BP(π) → Y is equicontinuous solely from the cited statement that, for factor maps between minimal flows, ψ is equicontinuous iff RP(π) ⊆ R(φ). This criterion is not stated or proved in the paper, and its hypotheses are not checked. In particular, φ is only shown to be a Banach-proximal (hence proximal) extension; the paper does not verify that [2, Corollary 7.9] applies to quotient maps by BP(π) without additional assumptions such as openness, RIC, or a specific definition of the relative regional proximal relation. If the corollary's sufficiency direction requires a lifting property for proximal extensions that is not automatic, then the minimal-case decomposition would not follow from the supplied argument. This is a citation-gap rather than an internal inconsistency: the weakly mean equicontinuous case (Theorem 10.3) is proved internally and does not rely on this cited criterion.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a relative notion of mean equicontinuity for factor maps between actions of countable amenable groups. Definition 1.1 requires that, for convergent sequences in the fiber relation R(pi), the sequence is asymptotically Banach proximal if and only if its limit is Banach proximal. The paper shows that this definition is strictly stronger than the natural metric condition (M) and is equivalent to (M) together with equality of the relative regional proximal and Banach proximal relations (Proposition 5.1, Theorem 5.2). It then proves that equicontinuous factor maps are exactly the mean equicontinuous distal ones (Theorem 9.1), topo-isomorphic factor maps are exactly the mean equicontinuous proximal ones (Theorem 9.2), and topo-isomorphy coincides with Banach proximality for all factor maps (Theorem 8.1). The main structural results are the decompositions of mean equicontinuous factor maps between minimal actions (Theorem 10.1) and between weakly mean equicontinuous actions (Theorem 10.3) into a topo-isomorphic map followed by an equicontinuous map, with uniqueness up to conjugacy (Theorem 10.5). Section 11 studies composition properties and provides a counterexample showing that the order of composition matters.","tokens_in":20902,"tokens_out":21707,"duration_ms":173728,"significance":"If the results hold, the paper gives a coherent relative generalization of mean equicontinuity that recovers the absolute notion for one-point factors and yields clean structural theorems. The core equivalences (Theorems 8.1, 9.1, 9.2) are proved from the definitions with explicit technical tools (Lemmas 7.2 and 7.3), and the weakly mean equicontinuous decomposition (Theorem 10.3) is proved internally. The paper also contains instructive examples (Examples 6.1, 6.2, 11.3) showing why natural alternative definitions are too weak or too strong. The main external input is the standard regional-proximal criterion [2, Corollary 7.9] used in Theorem 10.1, which is a cited theorem from the established literature rather than a circular or fitted assumption.","major_comments":[],"minor_comments":[{"comment":"Please state explicitly the precise version of [2, Corollary 7.9] used to conclude that psi is equicontinuous, and verify that its hypotheses (minimality of X/BP(pi) and Y, the factorization pi = psi after phi, and the identification R(phi) = BP(pi)) are satisfied.","section":"10.1, proof of Theorem 10.1"},{"comment":"The step 'By considering a subsequence and by a standard Krylov-Bogolyubov argument we can assume w.l.o.g. that x and x' are F-generic' is terse; please spell out how the subsequence is selected so that the inequality D^F_f + epsilon >= D_f remains valid.","section":"8, proof of Theorem 8.1"},{"comment":"The passage from nu_n to nu to the uniform bound '2 sum 2^{-m} ||f_m - h_m after pi||_{L1(nu_n)} <= 3 epsilon' uses a dominated-convergence or finite-tail argument; a short justification would help the reader.","section":"8, proof of Theorem 8.4"},{"comment":"The proof uses implicitly that the quotient X/BP(pi) is weakly mean equicontinuous because X is; please state this explicitly.","section":"10.3, proof of Theorem 10.3"},{"comment":"The proof contains two bullets labelled '(ii) => (iii)'; the second should be labelled '(iii) => (i)'.","section":"9.1, proof of Theorem 9.1"},{"comment":"There are several typos: 'mean equicontinity' in the abstract, 'A extensively studied' in Section 1, 'it's maximal' in Section 4 and the introduction, and 'desintegration' in the proof of Theorem 8.1; these should be corrected.","section":"Throughout"},{"comment":"The claim that D is constantly 1 on R(pi) is made with the phrase 'straightforward to observe'; since this example is a key counterexample, please add a few more details of the computation.","section":"11, Example 11.3"}],"recommendation":"minor_revision","confidential_remarks":"This is a solid contribution to the theory of mean equicontinuity. The only point that gave me pause is the reliance on [2, Corollary 7.9] in the proof of Theorem 10.1; I am satisfied that this is a standard result and appears correctly applied, but the final version should quote the corollary precisely. There is no concern about circularity or fitted parameters. The paper fits well within the scope of a dynamical systems journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: the paper gives a usable relative theory of mean equicontinuity, and the main results look right. It is more than a routine generalization: Definition 1.1 is well targeted, because the two obvious candidates—(M) and continuity of D on R(π)—are respectively too weak and too strong, and the paper shows both with examples.\n\nWhat is actually new: the relative notion itself, the equivalence of Banach proximality and topo-isomorphy for arbitrary factor maps (Theorem 8.1), the relative versions of the equicontinuity and proximality characterizations (Theorems 9.1 and 9.2), and the decomposition theorems for minimal and weakly mean equicontinuous actions (10.1, 10.3) with uniqueness up to conjugacy (10.5). Theorem 5.2, RP(π)=BP(π) for mean equicontinuous maps, is a clean relative version of the known action-level result. The three examples are genuinely useful: they show why the definition has to be what it is.\n\nI checked the main proofs. Proposition 5.1, Theorem 5.2, Theorem 8.1, and Theorem 8.4 are sound; the terse Krylov–Bogolyubov and dominated-convergence steps are standard. There is a label typo in the proof of Theorem 9.1 (two '(ii)⇒(iii)' arrows, the second should be (iii)⇒(i)), but the argument is clear.\n\nThe real soft spot is Theorem 10.1. The proof of equicontinuity of the induced map ψ depends entirely on the cited [2, Corollary 7.9], which is not stated, proved, or checked for hypotheses. The stress-test note is right: this is a citation gap, not an internal inconsistency. The weakly mean equicontinuous case (Theorem 10.3) is proved internally and does not use that criterion. If [2, Corollary 7.9] applies exactly as cited, the proof is fine; a referee should verify that. This is a minor-to-moderate issue, not a load-bearing flaw in the whole paper.\n\nNo fitted parameters, no circularity. The citation pattern is appropriate; [13] is background only. The open problems are stated honestly.\n\nWho this is for: anyone working on mean equicontinuity or structure theory of extensions. It deserves a serious referee. I would accept for peer review, and I would cite it if I work in this area.","headline":"A genuine relative theory of mean equicontinuity with sound main results; the minimal-case decomposition leans on a cited theorem that a referee should check.","tokens_in":21446,"tokens_out":5161,"would_cite":true,"duration_ms":43170,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37B05","37A15","37A30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proposes a relative notion of mean equicontinuity for factor maps and proves it splits every such map into a topo-isomorphic and an equicontinuous part.","keywords":["mean equicontinuity","factor maps","countable amenable groups","Banach proximality","topo-isomorphic extensions","equicontinuous extensions","regional proximal relation","decomposition theorems"],"falsifier":"Take any candidate factor map $\\pi$ and compute the Weyl pseudometric $D$ on the fibre relation $R(\\pi)$ using a Følner sequence of the acting group. If there is a convergent sequence $(x_n,x'_n)\\in R(\\pi)$ with $D(x_n,x'_n)\\ge c>0$ while $D(\\lim x_n,\\lim x'_n)=0$, then $\\pi$ is not mean equicontinuous, directly by Definition 1.1; the paper's Example 6.2 is exactly such a sequence for a map that satisfies only the weaker property (M). Conversely, a proof that no such sequence exists for a given $\\pi$ verifies mean equicontinuity in practice.","tokens_in":20495,"feed_emoji":"🧩","tokens_out":11988,"duration_ms":93111,"temperature":0.7,"pith_summary":"Mean equicontinuity has been studied as a property of a whole action; this paper asks what the right notion is for a factor map—a continuous equivariant surjection between two actions of a countable amenable group. The proposed definition, given in Definition 1.1, requires that along pairs in the same fibre, a convergent sequence is asymptotically Banach proximal exactly when its limit is Banach proximal. With that definition in hand, the paper proves two clean characterizations: a factor map is equicontinuous precisely when it is mean equicontinuous and distal, and it is topo-isomorphic precisely when it is mean equicontinuous and proximal. For minimal actions and for weakly mean equicontinuous actions, every mean equicontinuous factor map decomposes as a topo-isomorphic factor map followed by an equicontinuous one, and the decomposition is unique up to conjugacy. A direct consequence is that such maps preserve topological entropy.","feed_headline":"Mean equicontinuous maps split uniquely into two rigid factors","feed_subtitle":"Every such map, in minimal or weakly mean equicontinuous systems, is topo-isomorphic then equicontinuous.","key_machinery":"The paper's central object is the Weyl pseudometric $D$, defined on an action of a countable amenable group by $D(x,x')=\\sup_F \\limsup_n |F_n|^{-1} \\sum_{g\\in F_n} d(g.x,g.x')$, with the supremum over all Følner sequences $F=(F_n)$. Pairs with $D(x,x')=0$ are Banach proximal; sequences with $D(x_n,x'_n)\\to 0$ are asymptotically Banach proximal. The key mechanism is Definition 1.1, which asks precisely that, on the fibre relation $R(\\pi)$ of a factor map, convergence of such sequences matches their limits: a convergent fibre-pair sequence is asymptotically Banach proximal if and only if its limit is Banach proximal. The earlier candidate conditions are rejected in the paper by examples: the modulus condition (M) is too weak, while continuity of $D$ on $R(\\pi)$ is too strong. The chosen condition implies $\\mathrm{BP}(\\pi)=\\mathrm{RP}(\\pi)$, which is exactly the identity needed to feed the classical characterization of equicontinuity by regional proximal pairs and to define the quotient $X/\\mathrm{BP}(\\pi)$ used in the decomposition theorems.","core_discovery":"The central discovery is that a single sequential condition on the Weyl pseudometric gives the correct relative version of mean equicontinuity. For an action of a countable amenable group on a compact metric space, pairs with Weyl distance $D(x,x')=0$ are Banach proximal; a factor map $\\pi:X\\to Y$ is declared mean equicontinuous when, for every convergent sequence in the fibre relation $R(\\pi)$, asymptotic Banach proximality of the sequence is equivalent to Banach proximality of its limit. On this definition the paper proves that $\\mathrm{RP}(\\pi)=\\mathrm{BP}(\\pi)$ (Theorem 5.2), that equicontinuity is equivalent to mean equicontinuity plus distality (Theorem 9.1), and that topo-isomorphy is equivalent to mean equicontinuity plus proximality (Theorem 9.2). It also proves that Banach proximality and topo-isomorphy coincide for every factor map (Theorem 8.1). For minimal actions (Theorem 10.1) and for weakly mean equicontinuous actions (Theorem 10.3), any mean equicontinuous factor map factors as $\\pi=\\psi\\circ\\varphi$, with $\\varphi:X\\to X/\\mathrm{BP}(\\pi)$ topo-isomorphic and $\\psi$ equicontinuous; Theorem 10.5 shows the two middle spaces are conjugate when such a decomposition exists.","pith_inferences":["The definition suggests that 'mean equicontinuity' is best read as a regularity property of the fibred Weyl geometry, not of the base system; under that reading, one could test whether the minimal decomposition survives for non-minimal actions by studying whether $\\mathrm{RP}(\\pi)\\subseteq R(\\varphi)$ holds without the minimality hypothesis the cited characterization requires.","The equivalence of topo-isomorphy and Banach proximality for all factor maps (Theorem 8.1) gives a purely dynamical, measure-free characterization of a relation usually defined through invariant measures; this may make topo-isomorphy testable on subshifts by sampling Weyl distances along cylinder pairs.","Example 11.3 indicates that the composition of the two building blocks is order-sensitive; a natural conjecture is that the correct categorical structure of mean equicontinuous extensions is a kind of semidirect product, where equicontinuous maps act on topo-isomorphic ones, rather than a symmetric composition class.","The decomposition theorems reduce the classification of mean equicontinuous factor maps to two cleaner classification problems: classify equicontinuous extensions and classify topo-isomorphic (Banach proximal) extensions; if the paper is right, every mean equicontinuous map is uniquely a pair of such objects."],"forward_implications":["If the central claim is right, every mean equicontinuous factor map between minimal actions, and every one between weakly mean equicontinuous actions, is the composition of a topo-isomorphic map and an equicontinuous map; in particular the intermediate quotient $X/\\mathrm{BP}(\\pi)$ is a canonical invariant of the map.","Equicontinuity, distality, proximality and topo-isomorphy fit together in the relative setting exactly as they do for actions: mean equicontinuity is neutral, and distality versus proximality selects the equicontinuous versus topo-isomorphic component.","Because equicontinuous and topo-isomorphic factor maps each preserve topological entropy, mean equicontinuous factor maps between minimal or weakly mean equicontinuous actions preserve topological entropy (Corollary 1.2).","For weakly mean equicontinuous actions, the class of mean equicontinuous factor maps is exactly the class of compositions of a topo-isomorphic map followed by an equicontinuous map (Corollary 11.2); the reverse order need not preserve mean equicontinuity (Example 11.3).","Uniqueness up to conjugacy means the decomposition is not an arbitrary choice: the middle space $X/\\mathrm{BP}(\\pi)$ is determined by the map itself, and any other decomposition is conjugate through a canonical map."],"supporting_citations":[{"why":"Supplies the classical characterization of equicontinuity by regional proximal pairs and the cited Corollary 7.9 that carries the minimal decomposition.","marker":"[2]"},{"why":"Provides the original mean equicontinuity framework for actions, the RP=BP identification for mean equicontinuous actions, and the Banach proximal/topo-isomorphic argument generalized in Theorem 8.1.","marker":"[24]"},{"why":"Establishes that mean equicontinuous actions are topo-isomorphic extensions of equicontinuous factors, the result the decomposition theorems relativize.","marker":"[14]"},{"why":"Defines the Weyl pseudometric and Følner-sequence machinery that the paper's Definition 1.1 is built on.","marker":"[21]"},{"why":"Shows Banach proximality and topo-isomorphy agree for factor maps onto equicontinuous actions, the special case Theorem 8.1 extends to all factor maps.","marker":"[28]"},{"why":"Introduces weak mean equicontinuity through pointwise unique ergodicity and supplies the example that underlies Example 6.2.","marker":"[9]"},{"why":"Provides weak mean equicontinuity for countable amenable group actions and the implication from mean equicontinuity to weak mean equicontinuity.","marker":"[31]"},{"why":"Supplies the odometer/Toeplitz construction used in Example 11.3 to show the reverse composition order fails.","marker":"[6]"}],"fun_headline_variants":["Relative mean equicontinuity splits into proximal and distal parts","Factor maps decompose: topo-isomorphic then equicontinuous","Mean equicontinuous factor maps: unique two-step decomposition","Mean equicontinuous plus distal equals equicontinuous","Unique decomposition of mean equicontinuous factor maps"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The minimal-action decomposition rests on a previously published characterization—not reproved here—that for minimal flows a quotient map is equicontinuous exactly when every limit of pairs that the group action brings arbitrarily close still lies within a single fibre of the quotient.","fun_headline_variants_meta":{"raw":{"variants":["Relative mean equicontinuity splits into proximal and distal parts","Factor maps decompose: topo-isomorphic then equicontinuous","Mean equicontinuous factor maps: unique two-step decomposition","Mean equicontinuous plus distal equals equicontinuous","Unique decomposition of mean equicontinuous factor maps"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000949,"raw_usage":{"total_tokens":4127,"prompt_tokens":1096,"completion_tokens":3031,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":712,"completion_tokens_details":{"reasoning_tokens":2955}},"tokens_in":712,"tokens_out":3031,"duration_ms":19813,"temperature":1.0,"reasoning_tokens":2955,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:13:01.429770+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take any candidate factor map $\\pi$ and compute the Weyl pseudometric $D$ on the fibre relation $R(\\pi)$ using a Følner sequence of the acting group. If there is a convergent sequence $(x_n,x'_n)\\in R(\\pi)$ with $D(x_n,x'_n)\\ge c>0$ while $D(\\lim x_n,\\lim x'_n)=0$, then $\\pi$ is not mean equicontinuous, directly by Definition 1.1; the paper's Example 6.2 is exactly such a sequence for a map that satisfies only the weaker property (M). Conversely, a proof that no such sequence exists for a given $\\pi$ verifies mean equicontinuity in practice.","supporting_citations":[{"cited_title":"153, Amsterdam etc.: North-Holl and, 1988 (English)","cited_arxiv_id":null,"evidence_quote":"Supplies the classical characterization of equicontinuity by regional proximal pairs and the cited Corollary 7.9 that carries the minimal decomposition."},{"cited_title":"8, 2587–2612 (English)","cited_arxiv_id":null,"evidence_quote":"Provides the original mean equicontinuity framework for actions, the RP=BP identification for mean equicontinuous actions, and the Banach proximal/topo-isomorphic argument generalized in Theorem 8.1."},{"cited_title":"1, 75–123 (English)","cited_arxiv_id":null,"evidence_quote":"Establishes that mean equicontinuous actions are topo-isomorphic extensions of equicontinuous factors, the result the decomposition theorems relativize."},{"cited_title":"Second Series 98 (2018), no","cited_arxiv_id":null,"evidence_quote":"Defines the Weyl pseudometric and Følner-sequence machinery that the paper's Definition 1.1 is built on."},{"cited_title":"1, 101–116 (English)","cited_arxiv_id":null,"evidence_quote":"Shows Banach proximality and topo-isomorphy agree for factor maps onto equicontinuous actions, the special case Theorem 8.1 extends to all factor maps."},{"cited_title":"2, 117–132 (English)","cited_arxiv_id":null,"evidence_quote":"Introduces weak mean equicontinuity through pointwise unique ergodicity and supplies the example that underlies Example 6.2."},{"cited_title":"3, 2415–2428 (English)","cited_arxiv_id":null,"evidence_quote":"Provides weak mean equicontinuity for countable amenable group actions and the implication from mean equicontinuity to weak mean equicontinuity."},{"cited_title":"Proceedings of the confere nce, Bonn, Germany, May 1–July 31, 2004, Providence, RI: American Math emat- ical Society (AMS), 2005, pp","cited_arxiv_id":null,"evidence_quote":"Supplies the odometer/Toeplitz construction used in Example 11.3 to show the reverse composition order fails."}],"review_version":1}