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A note on multivariate diam mean equicontinuity and frequent stability

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Pith's one-line read Theorem 5.2 equates m-regularity of the maximal equicontinuous factor, weakly mean-sensitive m-tuples, and the diam-mean sensitivity threshold for minimal locally Bronstein systems.

desk verdict New and correct Auslander-Yorke dichotomies for amenable groups; the tuple characterization of m-regularity is not self-contained and the note needs cleanup. read the letter →

arxiv 2506.23313 v1 pith:5LDMGV4X submitted 2025-06-29 math.DS

Lino Haupt , Tobias Jäger , Chunlin Liu This is my paper · ORCID
classification math.DS MSC 37B0537A0537A15
keywords topologicaldynamicsdiam-meanequicontinuityfrequentstabilityAuslander-Yorkedichotomyweaklymean-sensitivetuplesmaximalequicontinuousfactoramenablegroupactionslocalBronsteincondition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes exact either/or laws for two multivariate notions of stability in topological dynamics. For a minimal action of a countable discrete amenable group, the system is either diam-mean $m$-equicontinuous or diam-mean $m$-sensitive, and either frequently $m$-stable or strongly $m$-spreading; there is no mixed behaviour at any finite order $m$. The paper then shows that when the system satisfies the local Bronstein condition—a mild approximation property automatic for actions of virtually nilpotent groups, including all abelian groups—these thresholds carry structural meaning: the first order at which diam-mean sensitivity appears is exactly the size of almost every fiber of the maximal equicontinuous factor. This equivalence is certified locally by weakly mean-sensitive tuples: an essential $m$-tuple whose product neighbourhoods are visited by every open set with positive density, together with the absence of such an $(m+1)$-tuple, is equivalent to the factor map being regular $m$-to-one.

What carries the argument

The proof has two workhorses. One is the set of diam-mean $m$-equicontinuity points $DE^m$ and the set of frequent $m$-stability points $FS^m$: for each sensitivity scale $\varepsilon$ the approximating sets are open, invariant, and $G_\delta$, and their complements are closed and invariant. Invariance plus a transitive point forces the whole space to lie on one side of the dichotomy (Propositions 3.2 and 3.4). The other workhorse is the local Bronstein condition, which lets the paper invoke an imported structural result from reference [26]: absence of essential weakly $(m+1)$-sensitive tuples forces the maximal equicontinuous factor map to be regular $m'$-to-one with $m'\le m$, and regular fibers contain essential weakly $m'$-sensitive tuples. Weakly mean-sensitive tuples are the local certificates that connect the two: an essential $m$-tuple has positive upper Banach density of visits to product neighbourhoods of any open set, which is exactly what turns tuple sensitivity into diam-mean $m$-sensitivity (Theorem 5.1).

What would settle it

Find a minimal abelian (or virtually nilpotent) system whose first diam-mean-sensitive order differs from the almost-everywhere fiber size of its maximal equicontinuous factor; for example, a Toeplitz subshift with known maximal-factor fiber statistics where an essential weakly 3-sensitive tuple appears although the fibers have size 2, or where the fibers have size 3 but the first sensitive order is 2. Either outcome would falsify Theorem 5.2. A more targeted check is to verify whether implication $(ii)\Rightarrow(i)$ holds when the imported result from reference [26] is replaced by a direct calculation of maximal-factor fibers.

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Extended reading notes

Core claim

The central claim is Theorem 5.2. It says that for a minimal topological dynamical system satisfying the local Bronstein condition, the following are equivalent for each positive integer $m$: the map to the maximal equicontinuous factor is regular $m$-to-one; there exists an essential weakly mean-sensitive $m$-tuple and no essential weakly mean-sensitive $(m+1)$-tuple; every fiber of the maximal equicontinuous factor contains an essential weakly $m$-sensitive tuple and none of size $m+1$; the system is diam-mean $(m+1)$-equicontinuous but not diam-mean $m$-equicontinuous; and the system is not diam-mean $(m+1)$-sensitive but is diam-mean $m$-sensitive. Theorems 1.1 and 1.2 provide the underlying dichotomies for transitive and minimal systems, respectively, for both diam-mean equicontinuity/sensitivity and frequent stability/strong spreading. In the locally Bronstein setting, the equivalence pins down the integer $m$ at which mean sensitivity begins as the regular fiber cardinality of the maximal equicontinuous factor.

Load-bearing premise

The argument leans on an imported, as-yet-unpublished result from reference [26]: if a minimal system has no essential weakly mean-sensitive $(m+1)$-tuple, then the map to its maximal equicontinuous factor must be regular $m'$-to-one for some $m'\le m$, and each generic fiber must contain an essential weakly $m'$-sensitive tuple; should that theorem fail, the $(ii)\Rightarrow(i)$ direction of Theorem 5.2 collapses.

Editorial extensions

If this is right

  • In every minimal locally Bronstein system, the smallest $m$ for which an essential weakly mean-sensitive $m$-tuple exists equals the smallest $m$ for which diam-mean $m$-sensitivity holds, and both equal the almost-everywhere fiber size of the maximal equicontinuous factor.
  • Diam-mean $(m+1)$-equicontinuity without diam-mean $m$-equicontinuity is equivalent to diam-mean $m$-sensitivity without diam-mean $(m+1)$-sensitivity, so the two notions record the same threshold from opposite sides.
  • For minimal actions of abelian groups—where the local Bronstein condition is automatic—the equivalence gives a purely dynamical criterion for $m$-regularity: count the first order at which sensitivity appears, without constructing the maximal equicontinuous factor.
  • The dichotomy theorems imply that almost diam-mean $m$-equicontinuity and diam-mean $m$-sensitivity exhaust the possibilities for transitive systems, and for minimal systems the 'almost' disappears, making the alternative a genuine partition of the space.
  • The frequent-stability dichotomy runs in parallel: minimal systems are either frequently $m$-stable or strongly $m$-spreading, with the same threshold structure under the local Bronstein condition.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same tuple machinery could be pushed further: the existence of weakly mean-sensitive tuples of size exactly $m$ might be decidable for subshifts of finite type by finite-block search, giving a numerical 'sensitivity spectrum' that predicts the maximal-equicontinuous-factor fiber size without constructing the factor.
  • The local Bronstein assumption in Theorem 5.2 is likely not optimal; the dichotomy theorems need only minimality, so extending the imported result from reference [26] beyond locally Bronstein systems would immediately widen the equivalence to any minimal amenable action.
  • Because Proposition 5.5 shows strong $m$-spreading is a collective, finite-family phenomenon, quantitative versions of the dichotomy—rates of divergence or densities of spreading times—may be accessible even when single tuples fail to certify spreading.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper studies multivariate diam-mean equicontinuity, diam-mean sensitivity, frequent stability, and strong spreading for topological dynamical systems acted on by countable discrete amenable groups. It establishes Auslander-Yorke type dichotomies: for transitive systems, either almost diam-mean m-equicontinuity or diam-mean m-sensitivity (Theorem 1.1), and either almost frequent m-stability or strong m-spreading (Theorem 1.2), with the minimal versions following as corollaries. It also proves a characterization of diam-mean m-sensitivity via essential weakly mean-sensitive m-tuples (Theorem 1.3 / Theorem 5.1) and states a local characterization of m-regular extensions of the maximal equicontinuous factor (Theorem 5.2) under a local Bronstein condition, together with a characterization of strong m-spreading via finite families of tuples (Proposition 5.5).

Significance. If all results hold, the paper would give clean multivariate analogues of classical dichotomies and a local tuple-based description of finite-to-one regularity, extending earlier work in [18] to virtually nilpotent groups and to amenable groups under a local Bronstein condition. The proofs of the dichotomies in Propositions 3.1--3.4 are self-contained and appear correct, and Theorem 5.1 provides a useful and largely elementary tuple characterization. However, the central structural result, Theorem 5.2, is not proven in this manuscript: its decisive implications are imported from an unpublished preprint by one of the authors, and the proof of Theorem 4.1 contains a genuine gap. The significance of the paper is therefore conditional on an external result that is not made available here.

major comments (4)
  1. [§5.1, Theorem 5.2] The equivalence in Theorem 5.2 is not self-contained. In direction (ii)⇒(i), the implication 'no essential weakly (m+1)-sensitive tuple implies π_eq is regular m′-to-one for some m′ ≤ m' is imported from [26, Corollary 5.3], and in direction (i)⇒(iii), the statement that in a regular m-to-one extension every fiber contains an essential weakly m-sensitive tuple is also imported from the same corollary. Neither the precise statement nor the proof of Corollary 5.3 is given in this note, and [26] is an unpublished preprint coauthored by one of the present authors. These steps are load-bearing for the main characterization, so Theorem 5.2 is incomplete as written. The authors should either reproduce the needed arguments or state the exact result they rely on and make it verifiable.
  2. [§4, Theorem 4.1] The proof of Theorem 4.1 establishes the desired bound only for tempered Følner sequences and then concludes 'As any Følner sequence has a tempered Følner subsequence [25], this completes the proof.' This is the wrong direction: passing to a subsequence can only decrease a limsup, so a bound on the limsup along a tempered subsequence does not imply the same bound for the original Følner sequence. In addition, the proof says it will show that any x is a diam-mean m-equicontinuous point, but the target is diam-mean (m+1)-equicontinuity. Thus the implication (i)⇒(iv) lacks a valid proof; a different argument, for instance selecting K with μ(∂K)=0 and using unique ergodicity along all Følner sequences, is needed.
  3. [§2.4, Definition 2.14 and §5.1] The definition of 'locally Bronstein' is garbled: the display in Definition 2.14 uses an m-tuple quantifier '∀i ∈ {1,...,m}' and variables x_i, y_j in what is supposed to be a condition on pairs (x,y), so the displayed formula is not meaningful as written. Moreover, Section 5.1 restates the local Bronstein condition differently, requiring minimal points (x,x′) with π_eq(x)=π_eq(x′) rather than almost periodic pairs satisfying xQy. Since Theorem 5.2 assumes this condition, the manuscript must state one consistent, precise definition and show that the two formulations are equivalent if both are used.
  4. [§5.1, Theorem 5.1] The proof of the converse direction in Theorem 5.1 contains several notational slips that make it hard to verify: after 'for any k ∈ F there exist x^k_1, ..., x^k_m ∈ U', the symbol k is used both as a group element and as an index, and later the finite-family construction refers to diameters shrinking like 1/l but initially uses 'Diam(A^1_i) ≤ 1'. These are local and presumably repairable, but they should be corrected so that the construction of the weakly mean-sensitive tuple is unambiguous.
minor comments (4)
  1. [Abstract and Introduction] The abstract contains typographical errors: 'action of a is a countable discrete infinite group' and 'vial weakly mean sensitive tuples' should be read as 'action of a countable discrete infinite group' and 'via weakly mean sensitive tuples'.
  2. [§3.2, Proposition 3.3] In the proof of invariance, two occurrences of 'x /∈ F Sm_ε' should read 'x /∈ DE^m_ε'; this is presumably a copy-and-paste error.
  3. [§4, Theorem 4.1, equation (4.1)] The display after (4.1) uses an equality where an inequality is needed, and the constant '2ε · μ_eq(K)' should be '(2ε/3) · μ_eq(K)' to match the preceding bound Diam_{m+1}(gB(x,η)) < 2ε/3.
  4. [§2.4, Definition 2.10] In the definition of the regionally proximal m-relation, the expression 'd(g_n(x'_{i,n}), g_n(y'_{j,n}))' should presumably be 'd(g_n(x'_{i,n}), g_n(x'_{j,n}))' since only the x' sequences are defined.

Circularity Check

1 steps flagged · score 6.0 of 10

Theorem 5.2 imports the decisive tuple-to-regularity implications from [26, Cor. 5.3], an unpublished preprint coauthored by present author Chunlin Liu.

  1. self citation load bearing [Section 5.1, Theorem 5.2 proof]
    "The first statement has been proven in [26, Corollary 5.3]. ... (ii) ⇒ (i). By Corollary 5.3 in [26], as (X, G) has no essential weakly (m + 1)-sensitive tuple, we know that πeq is regular m′-to-one for some 0 < m′ ≤ m."

    Reference [26] is Liu, Xu and Zhang, 'Independence and sensitivity for group actions', preprint (2025), coauthored by Chunlin Liu, one of the authors of the present note. The two decisive tuple-to-regularity implications—that regularity forces an essential weakly m-sensitive tuple in every fiber, and that the absence of an essential weakly (m+1)-sensitive tuple forces a regular m′-to-one MEF with m′ ≤ m—are not proved here; the proof of Theorem 5.2 imports them verbatim from [26]. The equivalence in Theorem 5.2 therefore rests on an unpublished same-author result rather than on a derivation contained in the note, so the central claim is load-bearing on a self-citation.

full rationale

The Auslander-Yorke dichotomies (Theorems 1.1, 1.2, Propositions 3.2 and 3.4) and the tuple-to-sensitivity direction (Theorem 5.1) are self-contained and do not depend on [26], so the paper is not globally circular. However, the advertised local characterisation of m-regularity in Theorem 5.2 is not self-contained: its (i)⇒(iii) first part and the whole (ii)⇒(i) direction are delegated to Corollary 5.3 of [26], an unpublished preprint sharing author Chunlin Liu. Since the present note provides no proof or independent verification of that corollary, the main equivalence is partly forced by a self-citation chain rather than by the paper's own arguments. This warrants score 6 rather than a higher score because the dichotomy theorems and the new (v)⇒(ii) direction are independent content. A separate proof gap in Theorem 4.1 (using a tempered Følner subsequence in the wrong direction to bound an arbitrary Følner limsup) is a correctness issue, not a circularity, and was not counted in the score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard facts about amenable group actions, the MEF, and the Lindenstrauss ergodic theorem. The main unverified input is [26], an unpublished preprint by coauthor Chunlin Liu, which supplies the key implication from weak mean-sensitive tuples to m-regularity. No free parameters or invented entities appear.

assumptions (5)
  • domain assumption G is a countable discrete amenable group (for the main theorems; virtually nilpotent or local Bronstein for Theorem 5.2).
    Throughout the paper, the acting group is assumed countable discrete amenable, with a Følner sequence; Theorem 5.2 adds virtually nilpotent (or local Bronstein) so that multivariate regionally proximal relation simplifies and the results of [26] apply.
  • standard math Existence and uniqueness of the maximal equicontinuous factor for any tds (Theorem 2.7, citing [13]).
    The MEF and the factor map pi_eq are used throughout; this is a classical result the paper cites rather than proves.
  • standard math Lindenstrauss pointwise ergodic theorem for tempered Følner sequences of amenable groups.
    Invoked in the proof of Theorem 4.1 to convert pointwise orbit averages to integrals over the uniquely ergodic MEF; cited as [25].
  • ad hoc to paper Corollary 5.3 of [26] (Liu-Xu-Zhang, preprint): absence of essential weakly (m+1)-sensitive tuples forces a regular m'-to-one MEF with m'<=m, and in the regular m-to-one case every fiber contains an essential weakly m-sensitive tuple.
    The load-bearing directions of Theorem 5.2 are imported from this unpublished coauthored preprint; the paper provides no proof of these statements.
  • domain assumption Virtually nilpotent acting groups make minimal systems incontractible, hence locally Bronstein (Lemma 2.15, via [17]).
    This is how the abelian/virtually nilpotent case of Theorem 5.2 is derived; the proof is sketched in Remark 2.16 with references to [16,17].

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Pith. "Pith review of A note on multivariate diam mean equicontinuity and frequent stability." pith.science (2026). https://pith.science/paper/5LDMGV4X

@misc{pith2026250623313,
  author       = {Pith},
  title        = {Pith review of: A note on multivariate diam mean equicontinuity and frequent stability},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5LDMGV4X}},
  note         = {Machine review of arXiv:2506.23313}
}
abstract

Let $(X,G)$ be a topological dynamical system, given by the action of a is a countable discrete infinite group on a compact metric space $X$. We prove that if $(X,G)$ is minimal, then it is either diam-mean $m$-equicontinuious or diam-mean $m$-sensitive. Similarly, $(X,G)$ is either frequently $m$-stable or strongly $m$-spreading. Further, when $G$ is abelian (or, more generally, virtually nilpotent), then the following statements are equivalent: $\bullet$ $(X,G)$ is a regular $m$-to-one extension of its maximal equicontinuous factor; $\bullet$ $(X,G)$ is diam-mean $(m+1)$-equicontinuious, and not diam mean $m$-equicontinuious; $\bullet$ $(X,G)$ is not diam-mean $(m+1)$-sensitive, but diam mean $m$-sensitive; $\bullet$ $(X,G)$ has an essential weakly mean sensitive $m$-tuple but no essential weakly mean sensitive $(m+1)$-tuple. This provides a {\em \enquote*{local}} characterisation of $m$-regularity and mean $m$-sensitivity vial weakly mean sensitive tuples. The same result holds when $G$ is amenable and $(X,G)$ satisfies the local Bronstein condition.

Figures

Figures reproduced from arXiv: 2506.23313 by the authors.

Figure 1
Figure 1. Universal Property of MEF The existence and uniqueness — up to conjugacy — of the MEF is classical [13, Theorem 1]. Theorem 2.7. Any tds (X, G) has a MEF (Y, G). If (Z, G) is another MEF, then there is a conjugacy h : (Y, G) → (Z, G). As the MEF of (X, G) is unique, we may give it a name and call it (Xeq, G). The factor map to the MEF shall be denoted by πeq : (X, G) → (Xeq, G). 2.4. Regionally Proximal Relation. Th… view at source ↗

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