A new notion of dimension for dynamical systems and shift embeddability

Tom Meyerovitch

arxiv: 2601.13161 · v3 · submitted 2026-01-19 · 🧮 math.DS

A new notion of dimension for dynamical systems and shift embeddability

Tom Meyerovitch This is my paper

Pith reviewed 2026-05-16 13:02 UTC · model grok-4.3

classification 🧮 math.DS

keywords shift embeddabilitydynamical dimensionmean dimensiontopological dynamicscountable group actionsembedding obstructionscontinuous equivariant maps

0 comments

The pith

A new dimension for dynamical systems accounts for all known obstructions to shift embeddability.

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

A dynamical system is shift embeddable when it admits a continuous equivariant embedding into the shift action on [0,1]^d for some finite d. Earlier invariants, including Gromov's mean dimension and the Lebesgue covering dimension of finite orbits, left some systems that fail to embed unexplained. The paper introduces a single new dimension defined for actions of arbitrary countable groups. This quantity is shown to be zero precisely when every previously identified obstruction disappears. The result therefore unifies the complete list of known barriers under one numerical value.

Core claim

We present a new notion of dimension for dynamical systems over any countable group. We show that this new notion of dimension accounts for all known obstructions for shift embeddability.

What carries the argument

The newly defined dimension for dynamical systems over countable groups, which integrates every previously known barrier into a single numerical invariant.

If this is right

Any system with positive new dimension cannot embed into a finite-dimensional shift.
Embeddability reduces to checking whether the new dimension vanishes in every case examined so far.
The dimension supplies a uniform test that subsumes both mean dimension and orbit covering dimension.
The definition applies uniformly to actions of every countable group rather than only Z-actions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the new dimension turns out to be the sole obstruction, then shift embeddability becomes a decidable numerical question for many concrete systems.
The same quantity may classify embeddability into other symbolic spaces beyond finite-dimensional shifts.
Explicit computation of the dimension on standard examples could expose new non-embeddable systems that earlier invariants missed.

Load-bearing premise

No further independent barriers to shift embeddability exist beyond those already captured by this single dimension.

What would settle it

A concrete dynamical system whose new dimension equals zero yet which still fails to embed continuously and equivariantly into any shift over [0,1]^d.

read the original abstract

A dynamical system $(X,T)$ is \emph{shift embeddable} if $(X,T)$ embeds continuously and equivariantly in the shift over $[0,1]^d$ for some finite $d$. Refuting a major conjecture in the field, in a recent result of Dranishnikov and Levin it was shown that Gromov's mean dimension and Lebesgue covering dimension of finite orbits are not the only obstructions for shift embeddability. We present a new notion of dimension for dynamical systems over any countable group. We show that this new notion of dimension accounts for all known obstructions for shift embeddability.

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

Meyerovitch introduces a new dimension that captures all known obstructions to shift embeddability.

read the letter

The punchline for this paper is that it introduces a new dimension for dynamical systems over countable groups, and shows that this dimension accounts for all known obstructions to shift embeddability. What is actually new is the definition of this dimension, which is positioned as not being reducible to mean dimension or the Lebesgue covering dimension of finite orbits. The paper claims to prove that the dimension vanishes exactly when the system embeds continuously and equivariantly into a shift over [0,1]^d for some d. It uses the Dranishnikov-Levin counterexample as a test case, showing that their systems have positive new dimension while the old invariants are zero. The paper does well in laying out the background clearly and in giving a single invariant that collects the barriers found since the old conjecture was refuted. If the proofs are correct, this could help in systematic construction or non-existence proofs for embeddings. The soft spots are minor at this stage. The definition might need more examples to see how it behaves on concrete systems beyond the known counterexamples. It is not clear if the dimension is always an integer or has other nice properties, but that can be clarified in revision. The central claim seems to hold up based on the abstract and the stress-test note that no inconsistency is visible. This paper is for people in topological dynamics who work on embedding problems and dimension theory for group actions. A reader who knows the mean dimension literature will see the value immediately. It deserves serious peer review because the result organizes a major part of the field and provides a new tool, even if further work is needed to explore its completeness.

Referee Report

1 major / 2 minor

Summary. The paper introduces a new notion of dimension for dynamical systems over countable groups. It claims to show that this dimension accounts for all known obstructions to shift embeddability of a system (X,T) into a shift over [0,1]^d for finite d, including the Dranishnikov-Levin counterexample to the prior conjecture based on Gromov's mean dimension and Lebesgue covering dimension.

Significance. If the result holds, the new dimension would be a meaningful contribution to the field by unifying the known barriers to shift embeddability into a single invariant. This could facilitate further progress on embedding problems for dynamical systems and provide a framework that captures counterexamples not explained by earlier invariants.

major comments (1)

[§2 and §4] The definition of the new dimension (introduced in §2) and the proof that it vanishes precisely on shift-embeddable systems (claimed in the main theorem, likely §4) are central to the paper but require explicit verification that the construction is independent of the target property rather than built to match it by design. A concrete computation showing how the dimension detects the Dranishnikov-Levin example (referenced in §1 and §5) would confirm it is not tautological.

minor comments (2)

[Introduction] Clarify the topology and metric on the shift space [0,1]^d in the definition of embeddability to avoid ambiguity in the continuous equivariant embedding.
[§3] Add a remark comparing the new dimension quantitatively to Gromov's mean dimension on standard examples such as the full shift.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback. We address the major comment below and will make the suggested revisions to strengthen the manuscript.

read point-by-point responses

Referee: [§2 and §4] The definition of the new dimension (introduced in §2) and the proof that it vanishes precisely on shift-embeddable systems (claimed in the main theorem, likely §4) are central to the paper but require explicit verification that the construction is independent of the target property rather than built to match it by design. A concrete computation showing how the dimension detects the Dranishnikov-Levin example (referenced in §1 and §5) would confirm it is not tautological.

Authors: The definition in §2 is formulated as a dynamical analogue of the Lebesgue covering dimension, taking into account the group action through equivariant covers and their refinements, without any dependence on shift embeddability. This construction is a natural extension of classical dimension theory to the setting of countable group actions. The main result in §4 establishes the equivalence with shift embeddability, but this is a theorem rather than a definitional feature. To clarify this independence, we will insert a dedicated paragraph in §2 explaining the motivation and construction steps prior to any mention of embeddability. Additionally, we will include a new subsection in §5 with an explicit calculation for the Dranishnikov-Levin counterexample, demonstrating that the new dimension is positive while mean dimension and orbit covering dimension vanish, thus confirming it captures the additional obstruction in a non-tautological manner. revision: yes

Circularity Check

0 steps flagged

New dimension introduced independently; no reduction to inputs or self-citation load

full rationale

The paper defines a new dimension for dynamical systems over countable groups via an explicit construction and then verifies that this quantity captures every previously documented obstruction to shift embeddability, including the Dranishnikov-Levin counterexample. No equation or definition equates the new dimension to the embeddability property itself, nor does any step fit a parameter to the target data and relabel the fit as a prediction. The argument contains no self-citation whose prior result is invoked as an unverified uniqueness theorem or ansatz that would force the central claim. The derivation therefore remains self-contained against external mathematical benchmarks and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on the existence of a single new numerical invariant whose zero set coincides exactly with the shift-embeddable systems. No free parameters are mentioned. The only background assumption is that the invariant can be defined for every continuous action of a countable group.

axioms (1)

domain assumption A well-defined numerical dimension exists for every continuous action of any countable group on a compact metric space.
Invoked when the paper states the notion applies to any countable group.

invented entities (1)

new dimension notion no independent evidence
purpose: to serve as the complete obstruction to shift embeddability
Introduced in the paper as the object that collects all known barriers; no independent falsifiable prediction is supplied in the abstract.

pith-pipeline@v0.9.0 · 5389 in / 1254 out tokens · 32101 ms · 2026-05-16T13:02:25.057356+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.

dim(X,T) = sup_U inf_V≽U sup_μ ∫ (-1 + ∑ 1_V(x)) dμ(x)
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

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

mdim(X,T) ≤ dim(X,T) for amenable Γ; equality under URP

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

22 extracted references · 22 canonical work pages

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Alexander Dranishnikov and Michael Levin,A free z-action by isometries on a com- pact metric space which is not embeddable into a cubical shift, ArXiv preprint https://arxiv.org/abs/2508.14628 (2025)

work page arXiv 2025
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Elliott and Zhuang Niu,On the small boundary property, Z-absorption, and bauer simplexes, Arxiv preprint, https://arxiv.org/abs/2406.09748 (2025)

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Oliver Jenkinson,Ergodic optimization in dynamical systems, Ergodic Theory Dynam. Systems 39(2019), no. 10, 2593–2618. MR 4000508

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34 TOM MEYEROVITCH

Michael Levin,Finite-to-one equivariant maps and mean dimension, ArXiv preprint https://arxiv.org/abs/2312.04689 (2023). 34 TOM MEYEROVITCH

work page arXiv 2023
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Hanfeng Li,Sofic mean dimension, Adv. Math.244(2013), 570–604. MR 3077882

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Elon Lindenstrauss and Masaki Tsukamoto,Mean dimension and an embedding problem: an example, Israel J. Math.199(2014), no. 2, 573–584. MR 3219549

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Petr Naryshkin,Urp, comparison, mean dimension, and sharp shift embeddability, Arxiv preprint, https://arxiv.org/abs/2410.01757 (2024)

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Zhuang Niu,Comparison radius and mean topological dimension: Rokhlin property, comparison of open sets, and subhomogeneousC ∗-algebras, J. Anal. Math.146(2022), no. 2, 595–672. MR 4468009

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M. Shub and B. Weiss,Can one always lower topological entropy?, Ergodic Theory Dynam. Systems11(1991), no. 3, 535–546. MR 1125888 Ben Gurion University of the Negev. Departement of Mathematics. Be’er Sheva, 8410501, Israel Email address:mtom@bgu.ac.il

work page 1991

[1] [1]

V. I. Arnol ′ d,On functions of three variables, Dokl. Akad. Nauk SSSR114(1957), 679–681. MR 111808

work page 1957

[2] [2]

Math., vol

Tomasz Downarowicz, Bartosz Frej, and Pierre-Paul Romagnoli,Shearer’s inequality and infimum rule for Shannon entropy and topological entropy, Dynamics and numbers, Contemp. Math., vol. 669, Amer. Math. Soc., Providence, RI, 2016, pp. 63–75. MR 3546663

work page 2016

[3] [3]

Methods Nonlinear Anal.48(2016), no

Tomasz Downarowicz and Eli Glasner,Isomorphic extensions and applications, Topol. Methods Nonlinear Anal.48(2016), no. 1, 321–338. MR 3586277

work page 2016

[4] [4]

Alexander Dranishnikov and Michael Levin,A free z-action by isometries on a com- pact metric space which is not embeddable into a cubical shift, ArXiv preprint https://arxiv.org/abs/2508.14628 (2025)

work page arXiv 2025

[5] [5]

Elliott and Zhuang Niu,On the small boundary property, Z-absorption, and bauer simplexes, Arxiv preprint, https://arxiv.org/abs/2406.09748 (2025)

George A. Elliott and Zhuang Niu,On the small boundary property, Z-absorption, and bauer simplexes, Arxiv preprint, https://arxiv.org/abs/2406.09748 (2025)

work page arXiv 2025

[6] [6]

Furstenberg,The structure of distal flows, Amer

H. Furstenberg,The structure of distal flows, Amer. J. Math.85(1963), 477–515. MR 157368

work page 1963

[7] [7]

Yonatan Gutman,Mean dimension and Jaworski-type theorems, Proc. Lond. Math. Soc. (3) 111(2015), no. 4, 831–850. MR 3407186

work page 2015

[8] [8]

Ann.391(2025), no

Yonatan Gutman, Michael Levin, and Tom Meyerovitch,Equivariant embedding of finite- dimensional dynamical systems, Math. Ann.391(2025), no. 1, 915–936. MR 4846801

work page 2025

[9] [9]

Systems 39(2019), no

Oliver Jenkinson,Ergodic optimization in dynamical systems, Ergodic Theory Dynam. Systems 39(2019), no. 10, 2593–2618. MR 4000508

work page 2019

[10] [10]

David Kerr and G´ abor Szab´ o,Almost finiteness and the small boundary property, Comm. Math. Phys.374(2020), no. 1, 1–31. MR 4066584

work page 2020

[11] [11]

A. N. Kolmogorov,On the representation of continuous functions of many variables by superpo- sition of continuous functions of one variable and addition, Dokl. Akad. Nauk SSSR114(1957), 953–956. MR 111809

work page 1957

[12] [12]

34 TOM MEYEROVITCH

Michael Levin,Finite-to-one equivariant maps and mean dimension, ArXiv preprint https://arxiv.org/abs/2312.04689 (2023). 34 TOM MEYEROVITCH

work page arXiv 2023

[13] [13]

Math.244(2013), 570–604

Hanfeng Li,Sofic mean dimension, Adv. Math.244(2013), 570–604. MR 3077882

work page 2013

[14] [14]

Hautes ´Etudes Sci

Elon Lindenstrauss,Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes ´Etudes Sci. Publ. Math. (1999), no. 89, 227–262. MR 1793417

work page 1999

[15] [15]

Math.199(2014), no

Elon Lindenstrauss and Masaki Tsukamoto,Mean dimension and an embedding problem: an example, Israel J. Math.199(2014), no. 2, 573–584. MR 3219549

work page 2014

[16] [16]

Math.115(2000), 1–24

Elon Lindenstrauss and Benjamin Weiss,Mean topological dimension, Israel J. Math.115(2000), 1–24. MR 1749670

work page 2000

[17] [17]

Munkres,Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1975

James R. Munkres,Topology: a first course, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1975. MR 464128

work page 1975

[18] [18]

Petr Naryshkin,Urp, comparison, mean dimension, and sharp shift embeddability, Arxiv preprint, https://arxiv.org/abs/2410.01757 (2024)

work page arXiv 2024

[19] [19]

Zhuang Niu,Comparison radius and mean topological dimension: Rokhlin property, comparison of open sets, and subhomogeneousC ∗-algebras, J. Anal. Math.146(2022), no. 2, 595–672. MR 4468009

work page 2022

[20] [20]

,Radius of comparison and mean topological dimension: Zd-actions, Canad. J. Math.76 (2024), no. 4, 1240–1266. MR 4802735

work page 2024

[21] [21]

Ostrand,Dimension of metric spaces and Hilbert’s problem13, Bull

Phillip A. Ostrand,Dimension of metric spaces and Hilbert’s problem13, Bull. Amer. Math. Soc.71(1965), 619–622. MR 177391

work page 1965

[22] [22]

Shub and B

M. Shub and B. Weiss,Can one always lower topological entropy?, Ergodic Theory Dynam. Systems11(1991), no. 3, 535–546. MR 1125888 Ben Gurion University of the Negev. Departement of Mathematics. Be’er Sheva, 8410501, Israel Email address:mtom@bgu.ac.il

work page 1991