Mean Diameter, Regularity and Diam-Mean Equicontinuity

Till Hauser

Mean Diameter, Regularity and Diam-Mean Equicontinuity

T0 review · 0 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read An action of an amenable locally compact sigma-compact group is diam-mean equicontinuous exactly when the map onto its maximal equicontinuous factor is regular.

desk verdict Solid, honest paper that settles the non-minimal diam-mean equicontinuity characterization for amenable lcσ groups and adds a clean regularity criterion; minor presentation issues only. read the letter →

arxiv 2510.22484 v2 pith:KTTI4TLT submitted 2025-10-26 math.DS

Till Hauser This is my paper · ORCID

classification math.DS MSC 37B0537B2537A05

keywords diam-meanequicontinuityregularfactormapmeandiameteramenablegroupmaximalequicontinuousproximalityFølnersequencenon-minimalactions

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

The paper establishes that diam-mean equicontinuity, a quantitative average version of equicontinuity, is not a separate dynamical phenomenon: for actions of amenable locally compact sigma-compact groups on compact metric spaces, it is equivalent to the factor map onto the maximal equicontinuous factor being regular, meaning its injectivity points carry full measure for every invariant measure. This answers an open question for non-minimal actions and removes minimality assumptions from earlier results. The bridge is a new characterization of regular factor maps as diam-mean proximal maps: every fibre has mean diameter zero. As a consequence, the paper constructs a maximal diam-mean equicontinuous factor for every such action and shows the class is closed under countable products.

What carries the argument

The mean diameter Diam(A) = sup_F limsup_n (1/|F_n|) ∫_{F_n} diam(g.A) dg, taken over all left Følner sequences of the acting group, is the operative quantity. It is shown to equal an infimum over compact sets of a supremum over shifts, and it turns the measure-theoretic notion of regularity into a geometric, Følner-average condition. The proof rests on Theorem 5.2, which equates regularity of a factor map with diam-mean proximality — zero mean diameter of every fibre — together with the ability to lift invariant measures from the quotient, a property provided by amenability.

What would settle it

Take the shift action of the free group on two generators on the Cantor set, which is non-amenable, and compute the mean diameter of the fibres over the maximal equicontinuous factor. If the fibres all have mean diameter zero while some point has no neighbourhood of arbitrarily small mean diameter, the equivalence fails outside the amenable setting. Within the paper's setting, a counterexample would be an amenable-lcσ action whose maximal-equicontinuous-factor map is regular but some ball has mean diameter bounded below.

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Extended reading notes

Core claim

The central claim is Theorem 1.1: an action of an amenable lcσ group on a compact metric space is diam-mean equicontinuous if and only if the factor map π_eq: X → X_eq onto its maximal equicontinuous factor is regular. The paper proves this via Theorem 5.2, which states that a factor map π: X → Y is regular if and only if every fibre π^{-1}(y) has mean diameter zero, equivalently the upper Banach density of times g with diam(g.π^{-1}(y)) > ε is at most ε. This characterization, together with a measure-lifting argument supplied by amenability, yields both directions: diam-mean equicontinuous actions have regular maximal equicontinuous factor, and regular extensions of equicontinuous actions a

Load-bearing premise

The load-bearing premise is that every ergodic invariant measure on the quotient lifts to an invariant measure on the extension; this follows from amenability and is exactly what lets regularity of the factor map be detected from the quotient side, and without it the central equivalence can fail.

Editorial extensions

If this is right

Diam-mean equicontinuity can be verified by inspecting only the maximal equicontinuous factor: one checks whether the fibre diameters shrink to zero in mean.
The characterization covers non-minimal actions and actions of general amenable locally compact sigma-compact groups, not just minimal Z-actions or abelian groups.
Regular factor maps are preserved under countable products and under composition and decomposition, so diam-mean equicontinuity is stable under countable products.
Every action of this class has a unique, up to conjugacy, maximal diam-mean equicontinuous factor, and its maximal equicontinuous factor is the maximal equicontinuous factor of that factor.
Regular Toeplitz-type extensions of odometers become concrete examples of diam-mean equicontinuous actions.

Reading between the lines

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

The equivalence suggests diam-mean equicontinuity is best understood as 'regular extension of an equicontinuous action'; in aperiodic-order settings where actions of R^n appear, this yields a measure-one injectivity criterion that needs no minimality.
The proof's reliance on measure lifting indicates the theorem may be sharp: for non-amenable acting groups, regularity could cease to imply the quotient-side measure condition, so the characterization likely requires a modified notion of regularity outside the amenable class.
The paper leaves open whether F-diam-mean equicontinuity for a single Følner sequence already implies diam-mean equicontinuity in this generality; a positive answer would let the whole theory be checked from one averaging sequence.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main thing to know: this paper proves what was explicitly open — for actions of amenable lcσ groups on compact metric spaces, diam-mean equicontinuity is equivalent to regularity of the factor map onto the maximal equicontinuous factor, with no minimality assumption. It also characterizes regular factor maps as those with zero mean diameter in every fibre (Theorem 5.2) and builds a maximal diam-mean equicontinuous factor. That is a genuine advance over the earlier minimal-Z, Abelian-lcσ, and countable-amenable-with-local-Bronstein results. I read the proof of the core equivalence and it holds up: Proposition 7.9 plus Theorem 7.11, with the mean-diameter machinery in Sections 4–5 doing the heavy lifting. The measure-lifting step in Remark 5.1, which the reader flagged as the weakest point, is in fact justified inside the paper by a standard Krylov–Bogolyubov argument in §2.6; the stress-test note confirms it. So that concern does not land. The paper is well grounded in FGL22 and Lindenstrauss's ergodic theorem, and the few self-citations ([Hau24], [FGH25], [CH26]) are auxiliary, not load-bearing. The 'regular iff diam-mean proximal iff F-diam-mean proximal' theorem is a clean and useful formulation, and the proof via the hyperspace action is elegant. Soft spots are minor: dense notation, a pronoun slip in §7.3, an overly brief unique-ergodicity assertion in Example 8.3, and a small typo in Proposition 2.5(ii)'s epsilon factor. None of these affect the central argument. The one thing I would press on in referee report is the Følner-sequence dependence: Remark 6.6 notes that F-diam-mean equicontinuity for a single sequence does not imply diam-mean equicontinuity, and Proposition 6.9's equivalence uses 'for all sequences'. That is handled correctly, but a reader could easily miss it, so a bit more signposting would help. Who is this for? Anyone working on mean equicontinuity, regular extensions, or non-minimal topological dynamics. It deserves a serious referee — the result is important enough for the subfield, and the proofs are explicit enough to check. My verdict: accept after minor revision. I would cite it.

Referee Report

0 major / 5 minor

Summary. The paper studies the mean diameter of subsets in actions of amenable locally compact sigma-compact groups on compact metric spaces. It defines a Følner-sequence-dependent mean diameter Diam_F and a uniform mean diameter Diam obtained by taking the supremum over all left Følner sequences. The central result, Theorem 5.2, characterizes regular factor maps as exactly those for which every fibre has mean diameter zero, with equivalent formulations in terms of upper Banach density and in terms of a single Følner sequence. Building on this, Theorem 1.1 proves that an action is diam-mean equicontinuous if and only if the factor map onto its maximal equicontinuous factor is regular, with no minimality assumption. The paper then establishes stability of regular factor maps under products and composition, gives a counterexample showing that hyperspace maps do not preserve regularity, and constructs a maximal diam-mean equicontinuous factor.

Significance. If correct, Theorem 1.1 resolves an open question in the area, extending earlier results for minimal actions of Z, abelian lc sigma groups, and countable amenable groups to all actions of amenable lc sigma groups. The proof strategy is genuinely different from earlier approaches, avoiding frequent stability and working directly with mean diameters. The characterization of regular factor maps by vanishing mean diameter of fibres is a clean and useful tool. The paper is careful about the non-minimal setting, and the measure-lifting step, which is the least obvious point, is justified by amenability of the acting group. I found no circularity: the main theorem depends on external results such as [FGL22] and [Lin01], and the self-citations are auxiliary. The proofs contain explicit density estimates, and the construction of the maximal diam-mean equicontinuous factor is self-contained. Overall, this is a substantial and convincing contribution.

minor comments (5)

[§2.7, proof of Prop. 2.5(ii)] The displayed estimate "|F_n Δ K^{-1}F_n| ≤ ε|K|" is followed by "|K^{-1}F_n| ≤ (1+ε)|F_n|". As written, this does not follow directly from the previous display; it holds only for sufficiently large n, using that |F_n| → ∞. The argument is correct, but the inequality should be stated with an explicit "for large n" or with ε|F_n| in place of ε|K|.
[§7.3, Prop. 7.10] The proposition is stated for a "left (or right) Følner sequence", but F-genericity is defined only for left Følner sequences in §2.6. If right Følner sequences are intended, the definition of F-generic should be extended accordingly, or the parenthetical should be removed.
[§7.3, proof of Thm. 7.11(i)] The sentence "From Proposition 7.4 we observe that π(x) is F-generic" should explicitly cite Proposition 7.4(ii), since part (i) concerns points in the maximal support. The application is correct because Y is mean equicontinuous, but the citation is currently ambiguous.
[§2.6] The assertion that for every ergodic ν on Y there exists an invariant μ on X with π_*μ = ν is stated as a "straightforward argument". Given that this step is load-bearing in Remark 5.1, a short proof or a precise reference would improve the exposition.
[§8.3, Example 8.3] The inequality 0 < d_H(A,B) ≤ Diam(H(π)^{-1}(Y)) is true, but a one-line justification would be helpful: since A and B are fixed by the action, every term in the Følner average is at least d_H(A,B).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the main equivalence is derived from independent measure-theoretic and Følner-sequence results.

full rationale

Theorem 1.1 is not circular. Regularity (measure-one injectivity) and diam-mean equicontinuity (mean diameter of small neighbourhoods) are distinct definitions. Theorem 5.2 proves their bridge (diam-mean proximality) using Lindenstrauss's pointwise ergodic theorem, Lemma 3.5, and measure-lifting for amenable lcσ groups; it does not assume Theorem 1.1. The forward direction of Theorem 1.1 (Prop 7.9) uses FGL22's characterization of mean equicontinuity and a new estimate (Lemma 7.8) showing Banach proximal fibres have arbitrarily small mean diameter. The converse (Thm 7.11 via Prop 7.10) uses FGL22's genericity theorem, regularity of the factor to control fibres, and Prop 6.9, which is established through Prop 3.3; the latter cites [FGH25] for a common-subsequence lemma about Følner sequences. That lemma is a parameter-free statement about amenable groups, not about diam-mean equicontinuity, so it is independent support rather than a self-referential premise. The other overlapping-author citations ([Hau24], [CH26]) concern auxiliary Banach-proximal/topo-isomorphic and almost-one-to-one facts not needed for the core equivalence; [Hau24] is explicitly used only in Remark 7.2. No fitted parameter is renamed as a prediction, and no definition covertly contains the target theorem.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no fitted constants and no new dynamical entities. It depends on standard ergodic theorems and on prior results in mean equicontinuity, all cited. The group-amenability assumption is the main domain premise: it supplies Følner sequences, invariant measures, and measure-lifting along factor maps, which are used throughout §5-§7.

assumptions (6)

domain assumption Amenable lcσ groups admit left Følner sequences and every action has an invariant Borel probability measure; every invariant measure on a factor has an invariant lift.
Invoked in §2.2, §2.6 and Remark 5.1; foundational for the definitions of mean diameter and regularity and for the measure-lifting step.
standard math Lindenstrauss pointwise ergodic theorem holds for tempered Følner sequences in amenable lcσ groups.
Used in the (iii)⇒(i) part of Theorem 5.2 to find a point y whose Følner averages along A_ε converge to ν(A_ε).
standard math Mean equicontinuity structure theorem of FGL22: mean equicontinuous iff the factor map to the maximal equicontinuous factor is Banach proximal; a Banach proximal factor map onto an equicontinuous action is the maximal equicontinuous factor.
Used as Proposition 7.3, a cornerstone of the proof of Theorem 1.1.
standard math A countable family of left Følner sequences admits a common subsequence (FGH25 Proposition 2.6).
Used in Proposition 3.3 to pass from pointwise Følner behavior to the equality needed in Proposition 6.9.
standard math Closedness of factor maps and the fibre-neighbourhood selection property [AB06 Theorem 17.7(1)].
Used in Proposition 7.10 to control preimages of small neighbourhoods of a fibre and to relate diameters in X and Y.
standard math Compactness of the hyperspace H(X) with the Hausdorff metric and continuity of the diameter map.
Used throughout §4-§6, especially to define mean diameter on H(X) and to apply Lemma 7.5.

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Cite this review

Pith. "Pith review of Mean Diameter, Regularity and Diam-Mean Equicontinuity." pith.science (2026). https://pith.science/paper/KTTI4TLT

@misc{pith2026251022484,
  author       = {Pith},
  title        = {Pith review of: Mean Diameter, Regularity and Diam-Mean Equicontinuity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KTTI4TLT}},
  note         = {Machine review of arXiv:2510.22484}
}

abstract

In the context of (not necessarily minimal) actions, we consider the mean diameter and use it to characterize regular factor maps. Building on this characterization, we prove that an action is diam-mean equicontinuous if and only if it is a regular extension of its maximal equicontinuous factor. Furthermore, we establish the existence of a maximal diam-mean equicontinuous factor and discuss stability properties of regular factor maps. For this, we work in the context of actions of locally compact and $\sigma$-compact amenable groups.

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Works this paper leans on

29 extracted references · 3 linked inside Pith

[1]

Aliprantis and Kim C

Charalambos D. Aliprantis and Kim C. Border. Infinite dimensional analysis. A hitchhiker's guide. Berlin: Springer, 3rd ed. edition, 2006

2006
[2]

Residual properties and almost equicontinuity

Ethan Akin and Eli Glasner. Residual properties and almost equicontinuity. J. Anal. Math. , 84:243--286, 2001

2001
[3]

Minimal flows and their extensions , volume 153 of North-Holland Math

Joseph Auslander. Minimal flows and their extensions , volume 153 of North-Holland Math. Stud. Amsterdam etc.: North-Holland, 1988

1988
[4]

Aperiodic order

Michael Baake and Uwe Grimm. Aperiodic order. Volume 1. A mathematical invitation , volume 149 of Encycl. Math. Appl. Cambridge: Cambridge University Press, 2013

2013
[5]

Toeplitz flows and model sets

Michael Baake, Tobias J \"a ger, and Daniel Lenz. Toeplitz flows and model sets. Bull. Lond. Math. Soc. , 48(4):691--698, 2016

2016
[6]

A note on the structural stability of almost one-to-one maps

María Isabel Cortez and Till Hauser. A note on the structural stability of almost one-to-one maps. Topology Appl. , 377:109641, 2026

2026
[7]

Invariant measures and orbit equivalence for generalized Toeplitz subshifts

Mar \' a Isabel Cortez and Samuel Petite. Invariant measures and orbit equivalence for generalized Toeplitz subshifts. Groups Geom. Dyn. , 8(4):1007--1045, 2014

2014
[8]

Shearer's inequality and infimum rule for Shannon entropy and topological entropy

Tomasz Downarowicz, Bartosz Frej, and Pierre-Paul Romagnoli. Shearer's inequality and infimum rule for Shannon entropy and topological entropy. In Dynamics and numbers. A special programm: June 1 -- July 31, 2014. International conference: July 21--25, 2014, Max-Planck Institute for Mathematics, Bonn, Germany. Proceedings , pages 63--75. Providence, RI: A...

2014

Show all 29 references

[9]

Isomorphic extensions and applications

Tomasz Downarowicz and Eli Glasner. Isomorphic extensions and applications. Topol. Methods Nonlinear Anal. , 48(1):321--338, 2016

2016
[10]

Tilings of amenable groups

Tomasz Downarowicz, Dawid Huczek, and Guohua Zhang. Tilings of amenable groups. J. Reine Angew. Math. , 747:277--298, 2019

2019
[11]

Survey of odometers and Toeplitz flows

Tomasz Downarowicz. Survey of odometers and Toeplitz flows. In Algebraic and topological dynamics. Proceedings of the conference, Bonn, Germany, May 1--July 31, 2004 , pages 7--37. Providence, RI: American Mathematical Society (AMS), 2005

2004
[12]

Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps

Xiongping Dai and Yuxuan Xie. Characterizations of open and semi-open maps of compact Hausdorff spaces by induced maps. Topology Appl. , 350:10, 2024. Id/No 108921

2024
[13]

Roger H. Farrell. Representation of invariant measures. Ill. J. Math. , 6:447--467, 1962

1962
[14]

On the continuity of F lner averages

Gabriel Fuhrmann, Maik Gr \"o ger, and Till Hauser. On the continuity of F lner averages. J. Funct. Anal. , 289(7):24, 2025. Id/No 111039

2025
[15]

The structure of mean equicontinuous group actions

Gabriel Fuhrmann, Maik Gr \"o ger, and Daniel Lenz. The structure of mean equicontinuous group actions. Isr. J. Math. , 247(1):75--123, 2022

2022
[16]

On dynamical systems with a purely point spectrum

Sergei Fomin. On dynamical systems with a purely point spectrum. In Dokl. Akad. Nauk SSSR , volume 77, pages 22--32, 1951

1951
[17]

Ergodic theory via joinings , volume 101 of Math

Eli Glasner. Ergodic theory via joinings , volume 101 of Math. Surv. Monogr. Providence, RI: American Mathematical Society (AMS), 2003

2003
[18]

Weak forms of topological and measure-theoretical equicontinuity: relationships with discrete spectrum and sequence entropy

Felipe Garc \' a-Ramos. Weak forms of topological and measure-theoretical equicontinuity: relationships with discrete spectrum and sequence entropy. Ergodic Theory Dyn. Syst. , 37(4):1211--1237, 2017

2017
[19]

Mean equicontinuity, almost automorphy and regularity

Felipe Garc \' a-Ramos, Tobias J \"a ger, and Xiangdong Ye. Mean equicontinuity, almost automorphy and regularity. Isr. J. Math. , 243(1):155--183, 2021

2021
[20]

When is a dynamical system mean sensitive? Ergodic Theory Dyn

Felipe Garc \' a-Ramos, Jie Li, and Ruifeng Zhang. When is a dynamical system mean sensitive? Ergodic Theory Dyn. Syst. , 39(6):1608--1636, 2019

2019
[21]

Mean equicontinuous factor maps

Till Hauser. Mean equicontinuous factor maps. Preprint, arXiv :2411.15549 [math. DS ], 2024

2024 arXiv
[22]

Multivariate Frequent Stability and Diam - Mean Equicontinuity

Lino Haupt. Multivariate Frequent Stability and Diam - Mean Equicontinuity . Preprint, arXiv :2501.07038 [math. DS ], 2025

2025 arXiv
[23]

A note on multivariate diam mean equicontinuity and frequent stability

Lino Haupt, Tobias J \"a ger, and Chunlin Liu. A note on multivariate diam mean equicontinuity and frequent stability. Preprint, arXiv :2506.23313 [math. DS ], 2025

2025 arXiv
[24]

Pointwise theorems for amenable groups

Elon Lindenstrauss. Pointwise theorems for amenable groups. Invent. Math. , 146(2):259--295, 2001

2001
[25]

Quasi-uniform convergence in dynamical systems generated by an amenable group action

Martha a cka and Marta Straszak. Quasi-uniform convergence in dynamical systems generated by an amenable group action. J. Lond. Math. Soc., II. Ser. , 98(3):687--707, 2018

2018
[26]

Sam B. jun. Nadler. Continuum theory. An introduction , volume 158 of Pure Appl. Math., Marcel Dekker . New York: Marcel, 1992

1992
[27]

Delone sets and dynamical systems

Boris Solomyak. Delone sets and dynamical systems. In Substitution and tiling dynamics: introduction to self-inducing structures. Lecture notes from the research school on tiling dynamical systems, CIRM Jean-Morlet Chair, Marseille, France, Fall 2017 , pages 1--32. Cham: Sprin...

2017
[28]

Topics in optimal transportation , volume 58 of Grad

C \'e dric Villani. Topics in optimal transportation , volume 58 of Grad. Stud. Math. Providence, RI: American Mathematical Society (AMS), 2003

2003
[29]

General topology

Stephen Willard. General topology . Mineola, NY: Dover Publications, reprint of the 1970 original edition, 2004

1970

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Aliprantis and Kim C

Charalambos D. Aliprantis and Kim C. Border. Infinite dimensional analysis. A hitchhiker's guide. Berlin: Springer, 3rd ed. edition, 2006

2006

[2] [2]

Residual properties and almost equicontinuity

Ethan Akin and Eli Glasner. Residual properties and almost equicontinuity. J. Anal. Math. , 84:243--286, 2001

2001

[3] [3]

Minimal flows and their extensions , volume 153 of North-Holland Math

Joseph Auslander. Minimal flows and their extensions , volume 153 of North-Holland Math. Stud. Amsterdam etc.: North-Holland, 1988

1988

[4] [4]

Aperiodic order

Michael Baake and Uwe Grimm. Aperiodic order. Volume 1. A mathematical invitation , volume 149 of Encycl. Math. Appl. Cambridge: Cambridge University Press, 2013

2013

[5] [5]

Toeplitz flows and model sets

Michael Baake, Tobias J \"a ger, and Daniel Lenz. Toeplitz flows and model sets. Bull. Lond. Math. Soc. , 48(4):691--698, 2016

2016

[6] [6]

A note on the structural stability of almost one-to-one maps

María Isabel Cortez and Till Hauser. A note on the structural stability of almost one-to-one maps. Topology Appl. , 377:109641, 2026

2026

[7] [7]

Invariant measures and orbit equivalence for generalized Toeplitz subshifts

Mar \' a Isabel Cortez and Samuel Petite. Invariant measures and orbit equivalence for generalized Toeplitz subshifts. Groups Geom. Dyn. , 8(4):1007--1045, 2014

2014

[8] [8]

Shearer's inequality and infimum rule for Shannon entropy and topological entropy

Tomasz Downarowicz, Bartosz Frej, and Pierre-Paul Romagnoli. Shearer's inequality and infimum rule for Shannon entropy and topological entropy. In Dynamics and numbers. A special programm: June 1 -- July 31, 2014. International conference: July 21--25, 2014, Max-Planck Institute for Mathematics, Bonn, Germany. Proceedings , pages 63--75. Providence, RI: A...

2014