Dynamical dimension and shift embeddability without the marker property

Ruxi Shi

REVIEW 1 major objections 2 minor 13 references

An aperiodic inverse-limit system without the marker property is shown to satisfy mdim = Ddim = N and to admit an equivariant topological embedding into the (3N+2)-cubical shift.

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2026-07-31 23:44 UTC pith:DOG5DU2P

load-bearing objection This paper credibly computes mdim and Ddim for the author's non-marker system and proves a clean full-shift formula; the right-inverse import from [13] is the only real dependency, and it is a cited, published one. the 1 major comments →

arxiv 2607.27880 v1 pith:DOG5DU2P submitted 2026-07-30 math.DS

Dynamical dimension and shift embeddability without the marker property

Ruxi Shi This is my paper

classification math.DS MSC 37B0237B1054F45

keywords mean dimensiondynamical dimensionmarker propertyequivariant embeddingfull shiftinverse limitdistal extensionaperiodic system

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper studies a specific aperiodic dynamical system built as an inverse limit of circle shifts, a system that has finite mean dimension but lacks the marker property—a structural condition many embedding theorems require. The paper proves that for this system the mean dimension and the dynamical dimension are both exactly N, and that despite the missing marker property the system can be embedded equivariantly into the cubical shift on a (3N+2)-dimensional alphabet. Along the way it establishes a general formula: for any compact metrizable alphabet K, the dynamical dimension of the full shift K^Z equals the covering dimension of K, even when that dimension is infinite. If correct, this gives the first example of a finite-mean-dimensional aperiodic non-marker system that is still genuinely shift-embeddable, and it shows that dynamical dimension can be computed exactly for full shifts over arbitrary alphabets.

Core claim

The central claim is Theorem 1.1: the inverse-limit system (X,T) built in earlier work satisfies mdim(X,T) = Ddim(X,T) = N, and there is an equivariant topological embedding of (X,T) into the cubical shift over [0,1]^{3N+2}. The proof uses the full-shift computation Ddim(K^Z, σ) = dim K for every compact metrizable K, an inverse-limit inequality for dynamical dimension, and a structural upgrade: the first-coordinate projection of the inverse limit is a distal extension, so an almost embedding combined with a finite-dimensional encoding of that factor becomes injective. The paper also notes that the same full-shift formula holds for every countable acting group.

What carries the argument

The central object is the inverse-limit system X = lim←(X_m, σ), where each stage is a subshift of the N-torus with a minimum-distance constraint and the bonding maps are block-averaging homomorphisms. The argument relies on continuous right inverses of these bonding maps, which yield non-equivariant sections of the projections π_m and transfer the mean-dimension lower bound from finite stages to the limit. A telescoping sum identity shows that two points with the same first coordinate must differ by a nonzero q(m)-periodic sequence, which makes π_1 a distal extension. Dynamical dimension—an invariant that dominates mean dimension and detects shift-embedding obstructions—is computed via an i

Load-bearing premise

The result depends on the existence of continuous right inverses η_{m-1,m}: X_{m-1}→X_m for the bonding maps, taken from earlier work; these sections are used to build the maps γ_m and to prove the lower bound mdim(X,T)≥N, so if any stage lacked such a section the equality mdim = Ddim = N would lose its support.

What would settle it

For the auxiliary full-shift theorem, take K to be an infinite-dimensional compact metrizable space such as the Hilbert cube; the theorem predicts Ddim(K^Z)=∞, so any calculation yielding a finite dynamical dimension would refute it. For the main construction, check directly whether the telescoping identity of Lemma 8.2 can fail for some pair of points in the same π_1 fiber: if two distinct points with equal first coordinate had an orbit closure meeting the diagonal, the distal-extension claim would be false and the embedding proof would collapse.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

If Theorem 1.1 is correct, it settles the shift-embeddability question for this particular non-marker system affirmatively, showing that absence of the marker property does not in itself preclude embedding into a finite-dimensional cubical shift.
The full-shift formula Ddim(K^Z)=dim K means that for alphabets of exceptional type—where mean dimension is dim K − 1—dynamical dimension and mean dimension differ by exactly one, while for infinite-dimensional alphabets dynamical dimension is infinite.
The same proof extends to any countable acting group G, giving Ddim(K^G)=dim K, so the computation is not special to Z-actions.
The embedding dimension 3N+2 is explicit, and the paper leaves open whether it can be lowered to 2N+1, the dimension already achieved by the almost embedding.
The equality mdim = Ddim for a system without the marker property provides a concrete data point for the general question of when these two invariants coincide.

Where Pith is reading between the lines

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

A likely reusable principle here: an almost embedding plus a distal factor with a finite-dimensional encoding gives a genuine topological embedding. That mechanism is not specific to this inverse limit and may apply to other systems where a distal factor can be isolated.
The equality mdim = Ddim in this example may be tied to the existence of non-equivariant sections at every stage. A testable extension would be to construct a similar inverse limit with one bonding map lacking a continuous right inverse and check whether the equality breaks down.
The full-shift formula for infinite-dimensional alphabets suggests a sharper formulation of the obstruction to shift-embeddability: any system with Ddim = ∞ cannot embed into any finite-dimensional cubical shift, extending the known free-isometry obstruction.
A natural numerical test is to instantiate the construction with small parameters (for instance N=1 and δ close to 1) and compute the actual minimal embedding dimension; this would indicate whether the 3N+2 bound is sharp or merely an artifact of the proof.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

This paper credibly computes mdim and Ddim for the author's non-marker system and proves a clean full-shift formula; the right-inverse import from [13] is the only real dependency, and it is a cited, published one.

read the letter

Colleague,

The headline: the paper delivers what it claims. For the inverse-limit system from Shi's earlier work, it computes mdim and Ddim equal to N, and it produces an explicit equivariant embedding into the 3N+2 cubical shift despite the absence of the marker property. The auxiliary full-shift result — Ddim(K^Z,σ)=dim K for every compact metrizable K, including the infinite-dimensional case — is genuinely new and clean. That result alone is worth the paper.

What is actually new: Theorem 7.2 extends Meyerovitch's cubical computation to arbitrary alphabets via a zero-dimensional-extension lemma (Lemma 7.1) plus the Pasynkov–Toruńczyk mapping theorem. The lower-bound section trick (Lemma 5.1) is a useful observation: a nonequivariant section suffices for mean-dimension lower bounds. Lemma 9.1, the relative-distal upgrade, is a neat general argument that converts an almost embedding plus a distal factor into an embedding; it is exactly the right mechanism for this system.

I followed the main arguments: the finite-stage computation (Prop 4.1), the inverse-limit upper bounds, the Ddim inverse-limit inequality, and the distal-extension proof (Prop 8.3) all line up. The telescoping identity for the bonding maps is a nice piece of work and it is what makes distality work. I do not see an internal gap.

The real soft spot is exactly what the reader and stress test flagged: the continuous right inverses η_{m−1,m} are imported from [13, Lemmas 5.2–5.3] and not reproved. These carry the lower bound mdim ≥ N. The present paper cites them and states the needed identity, but the exact parameter range for which they exist is not re-derived. This is a verification burden on the referee rather than a defect in the argument — the construction is published and the paper's own estimates are compatible with it. I would want the referee to check that the [13] construction indeed provides these right inverses for the same N, δ and for all m, but this is a normal citation in this field, not a red flag.

Also minor: the paper is not fully self-contained, and the existential choice of N (Remark 1.2) means the theorem is conditional on the prior construction's parameter. That is fine, but readers should know the numeric value of N is not explicit.

Bottom line: this is a serious paper with correct-looking proofs and a citeable new formula. It deserves a proper referee. I would bring it to a reading group and I would cite the full-shift formula.

Referee Report

1 major / 2 minor

Summary. The paper studies the inverse-limit system (X,T) constructed in the author's earlier paper [13] as an example of a finite-mean-dimensional aperiodic system without the marker property. The main theorem (Theorem 1.1) asserts that this system satisfies mdim(X,T)=Ddim(X,T)=N and admits an equivariant topological embedding into the (3N+2)-cubical shift, despite lacking the marker property. The proof proceeds by computing the mean dimension of the finite stages (Prop 4.1), lifting the lower bound to the inverse limit via continuous sections obtained from right inverses of the bonding maps (Lemma 5.1, Prop 5.2), proving an inverse-limit estimate for Meyerovitch's dynamical dimension (Lemma 6.1), establishing the full-shift formula Ddim(K^Z)=dim K for every compact metrizable K (Theorem 7.2), and finally upgrading an almost embedding into a topological embedding using the fact that the first-coordinate projection is a distal extension (Prop 8.3, Lemma 9.1).

Significance. If correct, the result provides the first example of a finite-mean-dimensional system without the marker property that is nevertheless genuinely shift-embeddable, showing that the marker property is not necessary for this form of embeddability. The auxiliary full-shift formula for dynamical dimension is of independent interest, and the paper offers a clean technique for upgrading almost embeddings using a distal factor. The arguments are largely self-contained; the main external input is the construction of the inverse-limit system and its right inverses from the author's earlier work [13], which is cited rather than reproved. The paper also gives explicit open questions about the general relationship between mdim and Ddim.

major comments (1)

[§5, Eq. (8) and Proposition 5.2] The lower bound mdim(X,T)≥N depends on the existence of continuous right inverses η_{m-1,m}: X_{m-1}→X_m with θ_{m,m-1}∘η_{m-1,m}=id, imported from [13, Lemmas 5.2–5.3]. These maps are used to define the sections γ_m and to prove π_m∘γ_m=id (Eq. (8)). However, the present paper does not state the exact hypotheses or parameter ranges for which these right inverses are guaranteed to exist, nor whether they apply to the specific (N,δ) fixed in Section 2. Since equality mdim=Ddim=N and the subsequent almost embedding rely on this result, I ask that the author include a precise statement of the cited lemmas (or at least an explicit statement that the construction of [13] applies verbatim to the parameters used here). This is a load-bearing point that needs clarification.

minor comments (2)

[Cross-references] There are multiple incorrect cross-references: in the proof of Prop. 5.2, 'Theorem 5.1' and 'Theorem 3.1' should be 'Lemma 5.1' and 'Proposition 3.1'; in the proof of Prop. 7.5, 'Theorem 6.1' should be 'Lemma 6.1'; in the proof of Thm 7.2, 'Theorem 7.1' should be 'Lemma 7.1'; in the proof of Prop. 8.3, 'Theorem 8.2' should be 'Lemma 8.2'; and in the proof of Thm 1.1, 'Theorem 7.5' and 'Theorem 9.1' should be 'Proposition 7.5' and 'Lemma 9.1'. Please correct these and check all other references.
[§7, proof of Theorem 7.2] The use of the Pasynkov–Toruńczyk mapping theorem via [7, Theorem 1.7] is appropriate, but it would improve readability to state explicitly what 'dim f' means (supremum of fiber dimensions) and to note that q(K) is a compact metrizable subset of [0,1]^d so that L^Z is defined.

Circularity Check

0 steps flagged

No circularity: all central identities are derived from independent covering estimates and prior external theorems; self-citations supply construction tools, not the target equalities.

full rationale

The derivation chain is not circular. mdim(X,T)=N is obtained by bounding each finite stage from below with an explicitly embedded full cubical shift (Prop. 4.1) and then applying the inverse-limit inequality (Prop. 3.1) plus continuous sections supplied by [13, Lemmas 5.2-5.3] (Prop. 5.2). These sections are prior published results with assumptions independent of the present theorem; they are not a restatement of mdim=N. Ddim(X,T)=N follows from the independently proved inverse-limit estimate (Lemma 6.1), the new full-shift formula (Thm. 7.2), and the external inequality mdim<=Ddim [12, Thm 9.1]. The almost embedding is imported from [12, Thm 10.1] and is upgraded using the paper's own proof that pi_1 is distal (Prop. 8.3) and the general relative lemma (Lemma 9.1), neither of which assumes the final embedding. The parameter N is fixed by the earlier construction and is not fitted to force mdim=N; Prop. 4.1 shows the equality for every positive N. Therefore no step reduces by construction to its inputs, and the self-citations function as external evidence rather than circular support.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

No new objects are postulated beyond the inverse-limit system inherited from [13]. The central claim rests on the two construction parameters (N, δ) and on imported structural facts from [13] and [12]; these are reasonable published inputs, but a fully self-contained paper would prove or reproduce them.

free parameters (2)

N = existential positive integer (per [13, eq (5.1)], no canonical or minimal value)
Parameter of the subshift X(N,r,δ) and of the inverse limit; Theorem 1.1 states that the invariants equal N. It is chosen existentially to satisfy a periodic-coindex obstruction in [13], not fitted to data.
δ = any value in (0,1) satisfying the periodic-coindex inequality of [13]
Constraint parameter in X(N,r,δ); the paper assumes 0<δ<1. Its existence follows from [13], and the proofs only need δ<1 plus the defining inequality.

axioms (5)

domain assumption Continuous right inverses η_{m-1,m}: X_{m-1} → X_m exist with θ_{m,m-1}∘η_{m-1,m}=id
Cited from [13, Lemmas 5.2–5.3]; used in Section 5 to define γ_m and get π_m∘γ_m=id (equation (8)), which supplies the mdim lower bound in Prop 5.2.
domain assumption The inverse-limit system X does not have the marker property
Proved in [13] and cited in Section 2; this is property (ii) of Theorem 1.1 and motivates the paper.
standard math Tsukamoto's full-shift mean dimension formula: mdim((S^N)^Z)=N for the N-torus
Used in the upper bound of Prop 4.1; quoted from [10, Theorem 1.1].
standard math Meyerovitch's dynamical-dimension toolkit: monotonicity, fixed-point lower bound, mdim≤Ddim, cubical full-shift formula, almost-embedding theorem, and fiber-product characterization
Used throughout Sections 6–9; quoted from [12, Props 3.1/10.1/10.2, Thms 8.1/9.1/10.1].
standard math Pasynkov–Toruńczyk/Levin–Lewis mapping theorem: for compact metrizable K with dim K=d, there is q:K→[0,1]^d with zero-dimensional fibers
Used in Theorem 7.2 to reduce Ddim(K^Z) to Ddim(L^Z) for L⊂[0,1]^d; quoted from [7, Theorem 1.7].

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0 comments

read the original abstract

We study the aperiodic inverse-limit system constructed in our earlier work as an example of a finite-mean-dimensional dynamical system without the marker property. We prove that its mean dimension and Meyerovitch's dynamical dimension are both equal to $N$. Despite the absence of the marker property, the system admits an equivariant topological embedding into the cubical shift with alphabet dimension $3N+2$. As an auxiliary result, we prove that the dynamical dimension of the full shift over any compact metrizable alphabet is exactly the covering dimension of the alphabet, including when this dimension is infinite.

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 8 canonical work pages

[1]

Coornaert,Dimension topologique et systèmes dynamiques, Cours Spécialisés 14, Société Mathématique de France, 2005

M. Coornaert,Dimension topologique et systèmes dynamiques, Cours Spécialisés 14, Société Mathématique de France, 2005

2005
[2]

Dranishnikov and M

A. Dranishnikov and M. Levin,A freeZ-action by isometries on a compact metric space which is not embeddable into a cubical shift, arXiv:2508.14628, 2025

arXiv 2025
[3]

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(1999), 323–415, doi:10.1023/A:1009841100168

doi:10.1023/a:1009841100168 1999
[4]

Gutman,Embedding topological dynamical systems with periodic points in cubical shifts, Ergodic Theory Dynam

Y. Gutman,Embedding topological dynamical systems with periodic points in cubical shifts, Ergodic Theory Dynam. Systems37(2017), no. 2, 512–538, doi:10.1017/etds.2015.40

work page doi:10.1017/etds.2015.40 2017
[5]

Gutman, Y

Y. Gutman, Y. Qiao and M. Tsukamoto,Application of signal analysis to the embedding problem of Zk-actions, Geom. Funct. Anal.29(2019), no. 5, 1440–1502, doi:10.1007/s00039- 019-00499-z

doi:10.1007/s00039- 2019
[6]

Gutman and M

Y. Gutman and M. Tsukamoto,Mean dimension and a sharp embedding theorem: extensions of aperiodic subshifts, Ergodic Theory Dynam. Systems34(2014), no. 6, 1888–1896, doi:10.1017/etds.2013.30

work page doi:10.1017/etds.2013.30 2014
[7]

Levin and W

M. Levin and W. Lewis,Some mapping theorems for extensional dimension, Israel J. Math. 133(2003), 61–76, doi:10.1007/BF02773061. 17

work page doi:10.1007/bf02773061 2003
[8]

Lindenstrauss,Mean dimension, small entropy factors and an embedding theorem, Publ

E. Lindenstrauss,Mean dimension, small entropy factors and an embedding theorem, Publ. Math. Inst. Hautes Études Sci.89(1999), 227–262, doi:10.1007/BF02698858

work page doi:10.1007/bf02698858 1999
[9]

Lindenstrauss and M

E. Lindenstrauss and M. Tsukamoto,Mean dimension and an embedding problem: an example, Israel J. Math.199(2014), no. 2, 573–584, doi:10.1007/s11856-013-0040-9

work page doi:10.1007/s11856-013-0040-9 2014
[10]

Tsukamoto,Mean dimension of full shifts, Israel J

M. Tsukamoto,Mean dimension of full shifts, Israel J. Math.230(2019), no. 1, 183–193, doi:10.1007/s11856-018-1813-y

work page doi:10.1007/s11856-018-1813-y 2019
[11]

Lindenstrauss and B

E. Lindenstrauss and B. Weiss,Mean topological dimension, Israel J. Math.115(2000), 1–24, doi:10.1007/BF02810577

doi:10.1007/bf02810577 2000
[12]

Meyerovitch,A new notion of dimension for dynamical systems and shift embeddability, Geom

T. Meyerovitch,A new notion of dimension for dynamical systems and shift embeddability, Geom. Funct. Anal.36(2026), 947–980, doi:10.1007/s00039-026-00742-4

work page doi:10.1007/s00039-026-00742-4 2026
[13]

Shi,Finite mean dimension and marker property, Trans

R. Shi,Finite mean dimension and marker property, Trans. Amer. Math. Soc.376(2023), no. 9, 6123–6139, doi:10.1090/tran/8840. Shanghai Center for Mathematical Sciences, Fudan University, 200438 Shanghai, China Email: ruxishi@fudan.edu.cn 18

work page doi:10.1090/tran/8840 2023

[1] [1]

Coornaert,Dimension topologique et systèmes dynamiques, Cours Spécialisés 14, Société Mathématique de France, 2005

M. Coornaert,Dimension topologique et systèmes dynamiques, Cours Spécialisés 14, Société Mathématique de France, 2005

2005

[2] [2]

Dranishnikov and M

A. Dranishnikov and M. Levin,A freeZ-action by isometries on a compact metric space which is not embeddable into a cubical shift, arXiv:2508.14628, 2025

arXiv 2025

[3] [3]

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(1999), 323–415, doi:10.1023/A:1009841100168

doi:10.1023/a:1009841100168 1999

[4] [4]

Gutman,Embedding topological dynamical systems with periodic points in cubical shifts, Ergodic Theory Dynam

Y. Gutman,Embedding topological dynamical systems with periodic points in cubical shifts, Ergodic Theory Dynam. Systems37(2017), no. 2, 512–538, doi:10.1017/etds.2015.40

work page doi:10.1017/etds.2015.40 2017

[5] [5]

Gutman, Y

Y. Gutman, Y. Qiao and M. Tsukamoto,Application of signal analysis to the embedding problem of Zk-actions, Geom. Funct. Anal.29(2019), no. 5, 1440–1502, doi:10.1007/s00039- 019-00499-z

doi:10.1007/s00039- 2019

[6] [6]

Gutman and M

Y. Gutman and M. Tsukamoto,Mean dimension and a sharp embedding theorem: extensions of aperiodic subshifts, Ergodic Theory Dynam. Systems34(2014), no. 6, 1888–1896, doi:10.1017/etds.2013.30

work page doi:10.1017/etds.2013.30 2014

[7] [7]

Levin and W

M. Levin and W. Lewis,Some mapping theorems for extensional dimension, Israel J. Math. 133(2003), 61–76, doi:10.1007/BF02773061. 17

work page doi:10.1007/bf02773061 2003

[8] [8]

Lindenstrauss,Mean dimension, small entropy factors and an embedding theorem, Publ

E. Lindenstrauss,Mean dimension, small entropy factors and an embedding theorem, Publ. Math. Inst. Hautes Études Sci.89(1999), 227–262, doi:10.1007/BF02698858

work page doi:10.1007/bf02698858 1999

[9] [9]

Lindenstrauss and M

E. Lindenstrauss and M. Tsukamoto,Mean dimension and an embedding problem: an example, Israel J. Math.199(2014), no. 2, 573–584, doi:10.1007/s11856-013-0040-9

work page doi:10.1007/s11856-013-0040-9 2014

[10] [10]

Tsukamoto,Mean dimension of full shifts, Israel J

M. Tsukamoto,Mean dimension of full shifts, Israel J. Math.230(2019), no. 1, 183–193, doi:10.1007/s11856-018-1813-y

work page doi:10.1007/s11856-018-1813-y 2019

[11] [11]

Lindenstrauss and B

E. Lindenstrauss and B. Weiss,Mean topological dimension, Israel J. Math.115(2000), 1–24, doi:10.1007/BF02810577

doi:10.1007/bf02810577 2000

[12] [12]

Meyerovitch,A new notion of dimension for dynamical systems and shift embeddability, Geom

T. Meyerovitch,A new notion of dimension for dynamical systems and shift embeddability, Geom. Funct. Anal.36(2026), 947–980, doi:10.1007/s00039-026-00742-4

work page doi:10.1007/s00039-026-00742-4 2026

[13] [13]

Shi,Finite mean dimension and marker property, Trans

R. Shi,Finite mean dimension and marker property, Trans. Amer. Math. Soc.376(2023), no. 9, 6123–6139, doi:10.1090/tran/8840. Shanghai Center for Mathematical Sciences, Fudan University, 200438 Shanghai, China Email: ruxishi@fudan.edu.cn 18

work page doi:10.1090/tran/8840 2023