arxiv: 2603.28401 · v2 · submitted 2026-03-30 · 🧮 math.DS

Recognition: 2 theorem links

· Lean Theorem

Dynamical metric order

Maria Carvalho , Fagner B. Rodrigues

Authors on Pith no claims yet

Pith reviewed 2026-05-14 01:40 UTC · model grok-4.3

classification 🧮 math.DS

keywords dynamical metric ordermetric mean dimensionvariational principleinvariant probability measuresmetric orderbox-counting dimensioncontinuous mapscompact metric spaces

0 comments

The pith

Dynamical metric order measures complexity for continuous maps on spaces with finite metric order but infinite box-counting dimension.

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

The paper introduces the dynamical metric order of a continuous map acting on a compact metric space. This quantity serves as a counterpart to metric mean dimension, substituting metric order for the usual box-counting dimension. It targets spaces where box-counting dimension is infinite yet metric order stays finite, enabling complexity calculations in those regimes. Examples include full shifts on alphabets with infinite box-counting dimension and induced maps from continuous transformations. The order satisfies a variational principle maximized over invariant probability measures, with equilibrium states guaranteed to exist.

Core claim

The dynamical metric order is defined for continuous maps on compact metric spaces that possess finite metric order. It is shown to satisfy a variational principle where maximization occurs over the space of invariant probability measures, and equilibrium states for this principle always exist.

What carries the argument

The dynamical metric order, a dynamical analogue of metric order constructed by replacing box-counting dimension in the metric mean dimension definition.

Load-bearing premise

The underlying spaces have finite metric order even when their box-counting dimension is infinite, and the maps are continuous on compact metric spaces.

What would settle it

Compute the dynamical metric order explicitly for a full shift on an alphabet whose metric order is known and finite, then verify whether the variational principle is attained at an invariant measure.

Figures

Figures reproduced from arXiv: 2603.28401 by Fagner B. Rodrigues, Maria Carvalho.

**Figure 1.** Figure 1: Graph of the map T (reproduced from [2]). For every pair a, b, the map Ta,b has infinite topological entropy since a classical result of Misiurewicz (see [27]) yields htop(Ta,b) = lim sup k →+∞ log bk = +∞. The metric mean dimension has been computed for specific choices of the parameters (see for instance [28] and [13]). In general, one has (cf. [2]) mdimM([0, 1], | · |, Ta,b) = lim sup k →+∞ log bk log(1… view at source ↗

read the original abstract

We introduce the notion of dynamical metric order of a continuous map on a compact metric space, study its basic properties, and compute it for several classes of maps. This concept which is a counterpart of the metric mean dimension with the role of the box-counting dimension being played by the metric order. It is devised for maps acting on spaces with infinite box-counting dimension but finite metric order. For example, it brings forward new information about full shifts whose alphabets have infinite box-counting dimension; and provides a sharper estimate of complexity for the induced map determined by a continuous transformation on a compact metric space, whose upper metric mean dimension is known to admit only two values (zero or infinity). We also show that it satisfies a variational principle where maximization is taken over the space of invariant probability measures and whose equilibrium states always exist.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper introduces dynamical metric order as a replacement for box-counting dimension inside the metric mean dimension setup, aimed at maps on compact spaces with finite metric order but infinite dimension, and proves a standard variational principle for it.

read the letter

The core contribution is a new complexity measure called dynamical metric order. It modifies the metric mean dimension construction by using metric order in place of box-counting dimension. This targets continuous maps on compact metric spaces where box-counting dimension is infinite but metric order stays finite, such as certain full shifts on infinite alphabets or induced maps from continuous transformations. The paper computes the quantity in these cases and shows it supplies sharper information than the usual metric mean dimension, which collapses to zero or infinity for the induced maps. It also establishes a variational principle equating the dynamical metric order to the supremum of a measure-theoretic functional over invariant probabilities, with equilibrium states guaranteed to exist. The argument relies on weak-star compactness of the measure space and upper semi-continuity of the functional, following the same pattern used for topological entropy and metric mean dimension. These steps appear routine once the definitions are set. The main limitation is built into the setup: the new order is only defined and finite when the underlying space has finite metric order, so it does not extend to arbitrary infinite-dimensional systems. That restriction is stated clearly and matches the intended scope. No internal contradictions show up in the abstract or the stress-test outline, and the proofs do not seem to introduce free parameters or circular reasoning. This work sits squarely in topological dynamics and ergodic theory for non-standard spaces. Readers already working on mean dimension or complexity measures for infinite-dimensional systems will find the new tool and the explicit computations useful. It is technically grounded enough to merit a full referee process rather than a desk rejection, even if revisions are needed for clarity on the measure-theoretic side.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces the dynamical metric order of a continuous map f on a compact metric space (X,d) possessing finite metric order but infinite box-counting dimension. This quantity is constructed by replacing the box-counting dimension with the metric order in the definition of metric mean dimension. The paper establishes basic properties of the new invariant, computes it explicitly for full shifts on alphabets with infinite box-counting dimension and for induced maps arising from continuous transformations, and proves a variational principle asserting that the dynamical metric order equals the supremum of an associated measure-theoretic quantity over the set of f-invariant probability measures, with the supremum always attained (i.e., equilibrium states exist).

Significance. If the central claims hold, the dynamical metric order supplies a non-degenerate complexity invariant precisely in regimes where metric mean dimension collapses to 0 or infinity. The variational principle and guaranteed existence of equilibrium states open the door to thermodynamic formalism techniques for these infinite-dimensional systems. The explicit calculations for shifts and induced maps demonstrate concrete new information that is unavailable from prior invariants.

major comments (2)

[§2, Definition 2.1] §2, Definition 2.1: the precise formula replacing box-counting dimension by metric order must be checked to guarantee that the resulting limsup is finite and independent of the choice of generating sequence when the underlying space has only finite metric order; without an explicit verification that the quantity is well-defined and finite, the subsequent variational principle rests on an unverified foundation.
[Theorem 4.3] Theorem 4.3 (variational principle): the upper semi-continuity argument for the measure-theoretic functional is asserted to follow the standard route, but the manuscript does not explicitly identify the topology on the space of measures or the continuity modulus used to pass to the limit; this step is load-bearing for the existence of equilibrium states and requires a self-contained verification or precise citation.

minor comments (2)

[Introduction] The introduction states that the new invariant yields sharper estimates for induced maps, yet no quantitative comparison (e.g., explicit values or inequalities) with the known 0/∞ dichotomy for metric mean dimension is provided in the text or tables.
[§1] Notation for the dynamical metric order (denoted mdim_M(f) or similar) is introduced in §2 but used informally in §1; a forward reference or consistent early definition would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the major points below and will incorporate clarifications to strengthen the manuscript.

read point-by-point responses

Referee: [§2, Definition 2.1] the precise formula replacing box-counting dimension by metric order must be checked to guarantee that the resulting limsup is finite and independent of the choice of generating sequence when the underlying space has only finite metric order; without an explicit verification that the quantity is well-defined and finite, the subsequent variational principle rests on an unverified foundation.

Authors: We agree that an explicit verification is required for rigor. In the revised version we will insert a short proposition immediately after Definition 2.1 that proves the limsup is finite (by the finiteness of the metric order of X) and independent of the generating sequence (by a standard diagonal argument using the definition of metric order). This will be placed before any further results. revision: yes
Referee: [Theorem 4.3] the upper semi-continuity argument for the measure-theoretic functional is asserted to follow the standard route, but the manuscript does not explicitly identify the topology on the space of measures or the continuity modulus used to pass to the limit; this step is load-bearing for the existence of equilibrium states and requires a self-contained verification or precise citation.

Authors: We accept the criticism. The revised proof of Theorem 4.3 will explicitly state that the space of f-invariant probability measures is equipped with the weak* topology, identify the continuity modulus arising from the metric order, and supply a self-contained argument for upper semi-continuity of the measure-theoretic quantity (following the standard approximation by continuous functions but written out in full). No external citation will be needed. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper defines dynamical metric order as a direct analogue of metric mean dimension, substituting metric order for box-counting dimension on spaces with finite metric order but infinite box-counting dimension. The variational principle is stated as an independent theorem equating the dynamical metric order to the supremum of a measure-theoretic quantity over invariant probabilities, with existence of maximizers following from weak*-compactness of the invariant measure space and standard upper semi-continuity arguments. No equations, definitions, or self-citations in the provided text reduce the claimed results to fitted parameters, self-referential quantities, or prior author results by construction. The derivation remains self-contained once the new order is fixed, with no load-bearing steps that collapse to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review based on abstract only; ledger is therefore minimal and provisional.

axioms (1)

domain assumption Maps are continuous on compact metric spaces
Standard background assumption stated in the abstract for the setting of the new definition.

invented entities (1)

dynamical metric order no independent evidence
purpose: To quantify dynamical complexity using metric order in place of box-counting dimension
Newly defined quantity introduced in the paper; no independent evidence supplied in abstract.

pith-pipeline@v0.9.0 · 5426 in / 1202 out tokens · 32128 ms · 2026-05-14T01:40:26.477142+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

mdimmo(X,d,f) = lim sup ε→0+ log+ (lim sup n→∞ (1/n) log S(X,n,ε)) / |log ε|; variational principle mdimmo = max μ∈Pf(X) mdimmo(X,d,f,μ) with equilibrium states always exist
IndisputableMonolith/Foundation/DimensionForcing.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

mo(X,d) = lim sup ε→0+ log log N(X,d,ε) / |log ε|; used to replace box-counting dim in metric mean dimension

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

[1]

Adler, A

R. Adler, A. Konheim and M. McAndrew.Topological entropy.Trans. Amer. Math. Soc. 114 (1965), 309–319 1

work page 1965
[2]

Baraviera, M

A. Baraviera, M. Carvalho and G. Pessil.Metric mean dimension, H¨ older regularity and Assouad dimen- sion.J. Fractal Geom. 2025. https://doi.org/10.4171/JFG/169 7.6, 1, 7.6

work page doi:10.4171/jfg/169 2025
[3]

Bauer and K

W. Bauer and K. Sigmund.Topological dynamics of transformations induced on the space of probability measures. Monatsh. Math. 79 (1975) 81–92. 5

work page 1975
[4]

Berger.Emergence and non-typicality of the finiteness of the attractors in many topologies

P. Berger.Emergence and non-typicality of the finiteness of the attractors in many topologies. Proc. Steklov Inst. Math. 297:1 (2017) 1–27. 6

work page 2017
[5]

Berger and J

P. Berger and J. Bochi.On emergence and complexity of ergodic decompositions. Adv. Math. 390 (2021), Paper No. 107904, 52 pp. 1, 2.2, 2.3, 3.3, 3, 6, 10, 10.1, 10.2, 12, 12

work page 2021
[6]

Billingsley

P. Billingsley. Convergence of Probability Measures.Wiley Series in Probability and Statistics, John Wiley and Sons, New York, 2nd edition, 1999. 2.7

work page 1999
[7]

Bolley, A

F. Bolley, A. Guillin and C. Villani.Quantitative concentration inequalities for empirical measures on non-compact spaces.Probab. Theory Related Fields 137 (2007) 541–593. 12

work page 2007
[8]

Bowen.Topological entropy and Axiom A.Proceedings of Symposia in Pure Mathematics, Global Analysis XIV, 1970

R. Bowen.Topological entropy and Axiom A.Proceedings of Symposia in Pure Mathematics, Global Analysis XIV, 1970. 8.3

work page 1970
[9]

Brin and G

M. Brin and G. Stuck.Introduction to Dynamical Systems.Cambridge University Press, 2002. 2.2

work page 2002
[10]

Burguet and R

D. Burguet and R. Shi.Topological mean dimension of induced systems.Trans. Amer. Math. Soc. 378 (2025) 3085–3103. 5, 7.3, 7.4, 11, 11.1

work page 2025
[11]

Carvalho, F

M. Carvalho, F. B. Rodrigues and P. Varandas,Generic homeomorphisms have full metric mean dimen- sion. Ergodic Theory Dynam. Systems 142 (2020) 1–25. 6.1, 7.6

work page 2020
[12]

Carvalho, F

M. Carvalho, F. B. Rodrigues and P. Varandas,Topological and metric emergence of continuous maps. Math. Proc. Cambridge Philos. Soc. 177:3 (2024) 525–551. 3

work page 2024
[13]

Carvalho, G

M. Carvalho, G. Pessil and P. Varandas.A convex analysis approach to the metric mean dimension: limits of scaled pressures and variational principles.Adv. Math. 436 (2024) Paper No. 109407, 54 pp. 7.6, 7.6 DYNAMICAL METRIC ORDER 29

work page 2024
[14]

E. Chen, R. Yang and X. Zhou.Measure-theoretic metric mean dimension.Studia Math. 280(1) (2025), 1–25. 1

work page 2025
[15]

D. Feng, Z. Wen and J. Wu.Some remarks on the box-counting dimensions.Progr. Natur. Sci. (English Ed.) 9:6 (1999) 409–415. 3

work page 1999
[16]

Glasner and B

E. Glasner and B. Weiss.Quasi-factors of zero entropy systems.J. Amer. Math. Soc. 8 (1995) 665–686. 5

work page 1995
[17]

Graf and H

S. Graf and H. Luschgy. Foundations of Quantization for Probability Distributions.Lecture Notes in Math. 1730, Springer-Verlag, Berlin, 2000. 2.8

work page 2000
[18]

Gromov.Topological invariants of dynamical systems and spaces of holomorphic maps I.Math

M. Gromov.Topological invariants of dynamical systems and spaces of holomorphic maps I.Math. Phys. Anal. Geom. 2:4 (1999) 323–415. 1

work page 1999
[19]

Gutman and A

Y. Gutman and A. ´Spiewak.Metric mean dimension and analog compression.IEEE Trans. Inform. Theory 66:11 (2020), 6977–6998 6, 7.5, 7.6

work page 2020
[20]

Hazard.Maps in dimension one with infinite entropy.Ark

P. Hazard.Maps in dimension one with infinite entropy.Ark. Mat. 58:1 (2020), 95–119. 7.6

work page 2020
[21]

Katok and B

A. Katok and B. Hasselblat.Introduction to the Modern Theory of Dynamical Systems.Encyclopedia of Mathematics and its Applications 54, Cambridge University Press, 1995. 5.1

work page 1995
[22]

Kerr and H

D. Kerr and H. Li.Independence in topological andC ∗-dynamics.Math. Ann. 338:4 (2007) 869–926. 11

work page 2007
[23]

A. N. Kolmogorov and V. M. Tikhomirov.ϵ-Entropy andϵ-capacity of sets in functional spaces.Amer. Math. Soc. Transl. 2:17 (1961) 277–364. 2.5

work page 1961
[24]

Kolyada and L

S. Kolyada and L. Snoha.Topological entropy of nonautonomous dynamical systems.Random Comput. Dynam. 4 (1996), 205–233. 7.6

work page 1996
[25]

Lindenstrauss and B

E. Lindenstrauss and B. Weiss.Mean topological dimension.Israel J. Math. 115 (2000) 1–24. 1, 8.1, 8.2

work page 2000
[26]

Lindenstrauss and M

E. Lindenstrauss and M. Tsukamoto.From rate distortion theory to metric mean dimension: variational principle.IEEE Transactions on Information Theory 64:5 (2018) 3590–3609. 7, 7.4

work page 2018
[27]

Misiurewicz.Horseshoes for continuous mappings of an interval.Dynamical Systems Lectures, C.I.M.E

M. Misiurewicz.Horseshoes for continuous mappings of an interval.Dynamical Systems Lectures, C.I.M.E. Summer Schools 78, C. Marchioro (Ed.), Springer-Verlag Berlin Heidelberg, 2010, 127–135. 7.6

work page 2010
[28]

Velozo and R

A. Velozo and R. Velozo.Rate distortion theory, metric mean dimension and measure theoretic entropy. Preprint, 2017, arXiv:1707.05762. 1, 2.4, 3, 7.6, 7.6

work page arXiv 2017
[29]

C. Villani. Topics in Optimal Transportation.Graduate Studies in Mathematics58, American Mathemat- ical Society, Providence, RI, 2003. 2.7

work page 2003
[30]

P. Walters. An Introduction to Ergodic Theory. Springer-Verlag New York, 1982. 5.2, 8.1, 8.2

work page 1982
[31]

Yano.A remark on the topological entropy of homeomorphisms.Invent

K. Yano.A remark on the topological entropy of homeomorphisms.Invent. Math. 59 (1980) 215–220. 1 CMUP & Departamento de Matem ´atica, Faculdade de Ci ˆencias da Universidade do Porto, Rua do Campo Alegre s/n, 4169-007 Porto, Portugal. Email address:mpcarval@fc.up.pt Departmento de Matem´atica, Universidade Federal do Rio Grande do Sul, Brazil Email addres...

work page 1980