Metric realization of mean dimension, the small boundary property at mean dimension zero, and cubical-shift embedding: the open problems of Lindenstrauss–Weiss (2000)

Mean topological dimension, introduced by Gromov and systematically developed by Lindenstrauss and Weiss (Israel Journal of Mathematics 115 (2000) 1–24) for actions of amenable groups, counts the number of real parameters per unit time needed to describe a dynamical system, as topological entropy counts bits per second. That paper proved that metric mean dimension $\mathrm{mdim}_M(X,d,T)$ upper-bounds $\mathrm{mdim}(X,T)$ for every compatible metric $d$, that finite topological entropy forces mean dimension zero, and that the small boundary property implies mean dimension zero; together with its 1999 companion paper it also made mean dimension the known obstruction and partial criterion for embedding dynamical systems into cubical shifts $([0,1]^N)^{\mathbb{Z}}$. Literature status: (1) whether the infimum of metric mean dimension over all compatible metrics equals mean dimension — and whether that infimum is even attained by some compatible metric — is open in general, with the attainment known for systems with the marker property and for systems admitting an aperiodic minimal factor; (2) Lindenstrauss's conjecture that a free system with mean dimension zero has the small boundary property was refuted in 2025 by Dranishnikov and Levin, whose free isometric system has mean dimension zero but neither the marker property nor the small boundary property; (3) the sharp embedding constant for minimal systems is $1/2$ (Gutman–Tsukamoto, 2020), while the general embedding conjecture was refuted by Dranishnikov–Levin, leaving the full characterization of cubical-shift embeddability open (Meyerovitch's 2026 dimension accounts for all currently known obstructions); (4) the subadditive limit foundations over amenable groups have been extended to cancellative amenable semigroups, countable amenable groups, and sofic groups. The open core consists of components (1)–(3), with (2) and (3) cast as characterization questions.

Background

A (topological) dynamical system is a pair $(X,T)$ with $X$ a compact metric space and $T:X\to X$ a homeomorphism; more generally one considers actions of a countable group $G$ by homeomorphisms. Gromov (1999) introduced a dynamical analogue of the Lebesgue covering dimension: the mean topological dimension $\mathrm{mdim}(X,T)$ measures how many independent real parameters per unit time are needed to describe the orbits, just as topological entropy measures the number of bits per second. Formally, for a finite open cover $\mathcal{U}$ of $X$ one forms the iterated join $\mathcal{U}_n=\bigvee_{i=0}^{n-1}T^{-i}\mathcal{U}$ and lets $D(\mathcal{U}_n)$ be its order (the largest number of cover elements with a common point, minus one); then $\mathrm{mdim}(X,T)=\sup_{\mathcal{U}}\lim_{n\to\infty}D(\mathcal{U}_n)/n$. For example, the shift on the infinite product $([0,1]^N)^{\mathbb{Z}}$ has mean dimension $N$, the precise expression of the intuition that this system carries $N$ parameters per unit time.

Lindenstrauss and Weiss (2000) (denoted LW00 below) systematically developed mean dimension for actions of amenable groups. Amenability means the group admits Følner sets — finite sets approximately invariant under translation, as in $\mathbb{Z}$ — which is what makes averaging and limit-existence arguments possible. The paper introduced two companion notions that generate the open problems considered here.

First, metric mean dimension. Given a compatible metric $d$ (a metric inducing the topology of $X$), set $d_n(x,y)=\max_{0\le k<n}d(T^k x,T^k y)$ and let $\mathrm{cover}_n(\varepsilon)$ be the minimal number of $d_n$-balls of radius $\varepsilon$ covering $X$; the metric mean dimension $\mathrm{mdim}_M(X,d,T)$ is the growth rate of $\log \mathrm{cover}_n(\varepsilon)$ in $n$, normalized by $|\log\varepsilon|$ as $\varepsilon\to 0$. LW00 proved that $\mathrm{mdim}(X,T)\le\mathrm{mdim}_M(X,d,T)$ for every compatible $d$, and that finite topological entropy (finite exponential growth rate of orbit-covering numbers) implies $\mathrm{mdim}(X,T)=0$. So metric mean dimension is an a priori strict upper bound on the topological invariant, and the natural question is whether the bound can always be attained by some choice of metric.

Second, the small boundary property (SBP): $X$ has a basis of open sets whose boundaries are null for every $T$-invariant probability measure (sets with this nullness property are called small; the idea goes back to Shub and Weiss (1991) in the context of lowering topological entropy). LW00 proved (their Theorem 5.4, as quoted in the later literature) that SBP implies $\mathrm{mdim}=0$ for actions of amenable groups. The naive converse fails for systems with periodic points — the trivial action on a connected space (for instance, the identity on $[0,1]$) has mean dimension zero but fails the small boundary property — so an honest converse must restrict to aperiodic systems (no periodic points) or free actions (no non-identity group element has a fixed point).

Third, the embedding problem. A system embeds into the cubical shift $([0,1]^N)^{\mathbb{Z}}$ (the homeomorphism shifting coordinates) if it is topologically conjugate to a subsystem; this is the natural coding of the system into $N$ real-valued signals per unit time. A classical theorem of 1974 (stated as Theorem 1.2 in the Gutman–Tsukamoto 2020 paper cited below) says that a finite-dimensional aperiodic system embeds already into $([0,1]^{\mathbb{Z}},\mathrm{shift})$. For infinite-dimensional systems two obstructions are known: mean dimension (an embedded subsystem has $\mathrm{mdim}\le N$) and periodic-point structure, measured by the periodic dimension $\mathrm{perdim}(X,T)=\sup_{n\ge1}\dim P_n/n$ where $P_n$ is the set of points fixed by $T^n$ and $\dim$ is covering dimension. LW00, Proposition 3.5 (as reported in the later literature) constructed a minimal system — a system in which every orbit is dense, hence with no periodic points at all — of mean dimension strictly greater than one, solving negatively the 1980s problem of embedding every minimal system into $[0,1]^{\mathbb{Z}}$. The 1999 companion paper of Lindenstrauss proved the partial converse that a system admitting an aperiodic minimal factor with $\mathrm{mdim}<N/36$ embeds into $([0,1]^N)^{\mathbb{Z}}$, and asked for the largest constant $c$ such that $\mathrm{mdim}<cN$ guarantees embeddability.

The key structural hypothesis that emerged is the marker property: for every $n$ there is an open set $U$ whose forward orbit covers $X$ while $U,TU,\dots,T^nU$ are pairwise disjoint — a continuous analogue of the Rokhlin towers of ergodic theory (a Rokhlin tower is a decomposition of the measure space into disjoint level sets over a base set, the classical tool for approximating a measure-preserving transformation by a periodic one). It implies aperiodicity, and by 2013 the marker property was known to imply everything the program wanted: the existence of a compatible metric realizing $\mathrm{mdim}=\mathrm{mdim}_M$ (Gutman, generalizing the 1999 construction of Lindenstrauss for systems with an aperiodic minimal factor), the equivalence of $\mathrm{mdim}=0$ with SBP and with being an inverse limit of finite-entropy systems, and quantitative embedding theorems. Whether the marker property itself follows from freeness plus small (in particular zero) mean dimension became the central open question of the subject — until it was answered negatively in 2025.

Problem Statement

The problem is the open-question bundle attached to Lindenstrauss–Weiss (2000), stated at its current strength. The common objective is to complete the metric, boundary-theoretic, and embedding-theoretic description of mean dimension for general dynamical systems, together with the combinatorial (subadditivity) foundations of the amenable-group theory. The components and their relationships are as follows; components (1)–(3) form the open core.

(1) Metric realization (open in full generality). LW00 proves $\mathrm{mdim}(X,T)\le\mathrm{mdim}_M(X,d,T)$ for every compatible metric $d$, so $\mathrm{mdim}(X,T)\le\inf_d\mathrm{mdim}_M(X,d,T)$, the infimum taken over all compatible metrics. Is this inequality an equality for every dynamical system $(X,T)$ with $\mathrm{mdim}(X,T)<\infty$? The stronger attainment form — the existence of a compatible metric $d$ with $\mathrm{mdim}_M(X,d,T)=\mathrm{mdim}(X,T)$ — is known for systems with the marker property and for systems admitting an aperiodic minimal factor; both forms are open for general aperiodic systems, including the marker-less systems constructed in 2020–2025.

(2) The small boundary property at mean dimension zero (characterization). LW00 proves SBP implies $\mathrm{mdim}=0$. Determine the structural conditions under which the converse holds. The sharp conjecture — that a free system with $\mathrm{mdim}=0$ has the SBP, attributed to Lindenstrauss and stated as Conjecture 7.5 by Tsukamoto–Tsutaya–Yoshinaga — has been refuted (2025). The remaining question is to characterize which mean-dimension-zero systems have the SBP; the marker property is sufficient for the converse whereas freeness is insufficient.

(3) Cubical-shift embedding (quantitative characterization). Determine exactly which dynamical systems embed into $([0,1]^N)^{\mathbb{Z}}$: (i) sharp constants within structural classes — the constant $1/2$ is sharp for minimal systems and for marker-property systems, and the question is for which further classes it (or any positive constant) is valid; and (ii) a complete set of obstructions for free actions — mean dimension and periodic dimension do not suffice (Dranishnikov–Levin 2025), and the question is whether finiteness of Meyerovitch's dimension $\dim(X,T)$ is necessary and sufficient for embeddability into some cubical shift, or whether yet further obstructions exist; here $\dim(X,T)$ is the dynamical dimension invariant introduced by Meyerovitch (2026), which satisfies $\mathrm{mdim}(X,T)\le\dim(X,T)$, vanishes exactly on systems with the small boundary property, and takes the value $+\infty$ on the Dranishnikov–Levin example, so that it accounts for all currently known obstructions.

(4) Subadditive foundations (technical component). LW00's limit-existence theorem for subadditive set functions over amenable groups underlies the definition of mean dimension for amenable actions. Determine the weakest natural hypotheses (invariance, monotonicity, and the form of the subadditivity inequality) under which normalized Følner limits exist, and the widest classes of groups and semigroups for which they exist; known extensions cover cancellative amenable semigroups (semigroups whose translations are injective), countable amenable groups, and sofic groups (groups admitting almost-action approximations by finite symmetric groups, a class containing all amenable and all residually finite groups), and the open question is whether those hypotheses are optimal and what fails at the boundary.

The question bundle comprises components (1)–(4). Within each component the listed demands are jointly posed — in component (3), for instance, both the sharp constants within structural classes (i) and the complete set of obstructions for free actions (ii).

References

1. E. Lindenstrauss and B. Weiss, Mean topological dimension, Israel Journal of Mathematics 115 (2000) 1–24. DOI 10.1007/BF02810577, https://doi.org/10.1007/BF02810577
2. M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps: I, Mathematical Physics, Analysis and Geometry 2 (1999) 323–415. DOI 10.1023/A:1009841100168, https://doi.org/10.1023/A:1009841100168
3. E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Publications Mathématiques de l'IHÉS 89 (1999) 227–262. DOI 10.1007/BF02698858, https://doi.org/10.1007/BF02698858 (full text also at https://www.numdam.org/item/PMIHES_1999__89__227_0.pdf)
4. M. Shub and B. Weiss, Can one always lower topological entropy?, Ergodic Theory and Dynamical Systems 11 (1991) 535–546. DOI 10.1017/S0143385700006325, https://doi.org/10.1017/S0143385700006325
5. Y. Gutman, Dynamical embedding in cubical shifts & the topological Rokhlin and small boundary properties, arXiv:1301.6072 (2013), https://arxiv.org/abs/1301.6072
6. Y. Gutman, Mean dimension and Jaworski-type theorems, Proceedings of the London Mathematical Society 111 (2015) 831–850. DOI 10.1112/plms/pdv043, https://doi.org/10.1112/plms/pdv043 (arXiv:1208.5248, https://arxiv.org/abs/1208.5248)
7. E. Lindenstrauss and M. Tsukamoto, Mean dimension and an embedding problem: an example, Israel Journal of Mathematics 199 (2014) 573–584. DOI 10.1007/s11856-013-0040-9, https://doi.org/10.1007/s11856-013-0040-9
8. Y. Gutman and M. Tsukamoto, Embedding minimal dynamical systems into Hilbert cubes, Inventiones mathematicae 221 (2020) 113–166. DOI 10.1007/s00222-019-00942-w, https://doi.org/10.1007/s00222-019-00942-w (arXiv:1511.01802, https://arxiv.org/abs/1511.01802)
9. Y. Gutman, Y. Qiao and M. Tsukamoto, Application of signal analysis to the embedding problem of $\mathbb{Z}^k$-actions, Geometric and Functional Analysis 29 (2019) 1440–1502. DOI 10.1007/s00039-019-00499-z, https://doi.org/10.1007/s00039-019-00499-z
10. E. Lindenstrauss and M. Tsukamoto, Double variational principle for mean dimension, Geometric and Functional Analysis 29 (2019) 1048–1109. DOI 10.1007/s00039-019-00501-8, https://doi.org/10.1007/s00039-019-00501-8 (arXiv:1901.05623, https://arxiv.org/abs/1901.05623)
11. Y. Gutman and A. Śpiewak, Around the variational principle for metric mean dimension, Studia Mathematica 261 (2021) 345–360. DOI 10.4064/sm201029-23-2, https://doi.org/10.4064/sm201029-23-2 (arXiv:2010.14772, https://arxiv.org/abs/2010.14772)
12. M. Tsukamoto, M. Tsutaya and M. Yoshinaga, G-index, topological dynamics and the marker property, Israel Journal of Mathematics 251 (2022) 737–764. DOI 10.1007/s11856-022-2433-0, https://doi.org/10.1007/s11856-022-2433-0 (arXiv:2012.15372, https://arxiv.org/abs/2012.15372)
13. R. Shi, Finite mean dimension and marker property, Transactions of the American Mathematical Society 376 (2023) 6123–6139. DOI 10.1090/tran/8840, https://doi.org/10.1090/tran/8840 (arXiv:2102.12197, https://arxiv.org/abs/2102.12197)
14. A. Dranishnikov and M. Levin, A free Z-action by isometries on a compact metric space which is not embeddable into a cubical shift, arXiv:2508.14628 (2025), https://arxiv.org/abs/2508.14628
15. T. Meyerovitch, A new notion of dimension for dynamical systems and shift embeddability, Geometric and Functional Analysis 36 (2026) 947–980. DOI 10.1007/s00039-026-00742-4, https://doi.org/10.1007/s00039-026-00742-4 (arXiv:2601.13161, https://arxiv.org/abs/2601.13161)
16. T. Ceccherini-Silberstein, M. Coornaert and F. Krieger, An analogue of Fekete's lemma for subadditive functions on cancellative amenable semigroups, Journal d'Analyse Mathématique 124 (2014) 59–81. DOI 10.1007/s11854-014-0027-4, https://doi.org/10.1007/s11854-014-0027-4 (arXiv:1209.6179, https://arxiv.org/abs/1209.6179)
17. A. H. Dooley, V. Ya. Golodets and G. Zhang, Sub-additive ergodic theorems for countable amenable groups, Journal of Functional Analysis 267 (2014) 1291–1320. DOI 10.1016/j.jfa.2014.06.008, https://doi.org/10.1016/j.jfa.2014.06.008 (arXiv:1310.1198, https://arxiv.org/abs/1310.1198)
18. T. Downarowicz, D. Huczek and G. Zhang, Tilings of amenable groups, Journal für die reine und angewandte Mathematik 750 (2019) 277–298. DOI 10.1515/crelle-2016-0025, https://doi.org/10.1515/crelle-2016-0025 (arXiv:1502.02413, https://arxiv.org/abs/1502.02413)
19. H. Li, Sofic mean dimension, Advances in Mathematics 244 (2013) 570–604. DOI 10.1016/j.aim.2013.05.005, https://doi.org/10.1016/j.aim.2013.05.005 (arXiv:1105.0140, https://arxiv.org/abs/1105.0140)