Shift embeddability of topological dynamical systems: completing the Lindenstrauss embedding and factor-separation program beyond the marker-property class

Lindenstrauss's 1999 embedding theorem states that an extension of a minimal $\mathbb{Z}$-action with mean dimension below $\dim(K)/36$ embeds equivariantly into the shift on $K^{\mathbb{Z}}$, and the same paper asks for the optimal value of this constant and for extensions of the theorem and of the companion small-entropy-factor theory to all $\mathbb{Z}$-actions and to more general acting groups, with periodic points flagged as the essential obstruction. The constant question is settled: the optimal threshold is mean dimension strictly below $d/2$ for minimal systems (Gutman-Tsukamoto, Invent. Math. 2020, explicitly solving the posed problem), for extensions of aperiodic subshifts, and for marker-property $\mathbb{Z}^k$-actions, while $1/2$ is a ceiling for every infinite amenable group (Jin-Park-Qiao). The unrestricted extension is refuted: Dranishnikov and Levin (2025) constructed a free $\mathbb{Z}$-action by isometries - mean dimension and topological entropy zero - that does not embed into any cubical shift, destroying the conjecture that mean dimension and periodic-orbit dimension are the only obstructions. The open problem is the surviving core in the form the current literature gives it: decide whether the dynamical dimension introduced by Meyerovitch (GAFA 2026), which accounts for all known obstructions, gives necessary and sufficient conditions for equivariant embedding into $([0,1]^d)^{\mathbb{Z}}$ (in particular whether $\dim<d/2$, or even finiteness of $\dim$, suffices); determine the sharp embedding threshold for aperiodic systems without the marker property (between the known $3N+2$ and $d/2$ scales); extend the sharp embedding theorem beyond the uniform-Rokhlin-property and almost-finite classes to general amenable group actions; and extend the small-entropy-factor and maximal zero-mean-dimension-factor theory from minimal and marker-property systems to general systems and groups.

Background

A topological dynamical system $(X,T)$ is a homeomorphism $T$ of a compact metrizable space $X$; more generally one considers actions of a discrete group $G$ on $X$ by homeomorphisms. A factor of $(X,T)$ is a system $(Y,S)$ together with a continuous surjection $\pi:X\to Y$ satisfying $\pi\circ T=S\circ\pi$, and $(X,T)$ is then an extension of $(Y,S)$. The $d$-dimensional cubical shift is the system $\left(([0,1]^d)^{\mathbb{Z}},\sigma\right)$, where $\sigma$ shifts a bi-infinite sequence one step, $(\sigma x)_n=x_{n+1}$; a system is shift embeddable if for some finite $d$ it admits an equivariant topological embedding into the $d$-shift, that is, a homeomorphism of $X$ onto a closed shift-invariant subset that intertwines $T$ with $\sigma$. Equivalently, finitely many continuous observables $f:X\to[0,1]^d$ separate every pair of points once sampled along their orbits, $x\mapsto(f(T^nx))_{n\in\mathbb{Z}}$ - the finite-sampling-rate problem. For real flows, the classical Beboutov theorem, proved by Kakutani, embeds a flow into the shift on the space $C(\mathbb{R},\mathbb{R})$ of continuous real functions exactly when its fixed-point set embeds in $\mathbb{R}$; for $\mathbb{Z}$-actions a 1974 theorem of Jaworski, as recorded in the primary source of this problem, states that every aperiodic system on a finite-dimensional compact metric space embeds into the one-dimensional cubical shift, and the finite-dimensionality is essential.

For infinite-dimensional spaces the governing invariant is mean dimension, introduced by Gromov and developed by Lindenstrauss and Weiss: for an open cover $\mathcal{U}$ of $X$ let $\bigvee_{i=0}^{n-1}T^{-i}\mathcal{U}$ be the join (common refinement) of its first $n$ iterates and let $D(\cdot)$ be the dimension of a cover (the order of its nerve minus one); then

$$\operatorname{mdim}(X,T)=\sup_{\mathcal{U}}\lim_{n\to\infty}\frac{1}{n}D\left(\bigvee_{i=0}^{n-1}T^{-i}\mathcal{U}\right),$$

a number in $[0,\infty]$ counting the average number of real parameters per time step needed to describe orbits. It is meaningful precisely for systems of infinite topological dimension and infinite entropy: $\operatorname{mdim}\left(([0,1]^d)^{\mathbb{Z}}\right)=d$, embeddability into the $d$-cubical shift forces $\operatorname{mdim}\le d$, finite topological entropy forces $\operatorname{mdim}=0$, and Lindenstrauss and Weiss constructed minimal systems of arbitrary prescribed mean dimension, so the necessary condition is far from sufficient in general.

The primary source proved the first substantive converse. Its Theorem 5.1: if $(X,T)$ is an extension of an (infinite, hence aperiodic) minimal system and $K$ is a compact convex set with non-empty interior, then $\operatorname{mdim}(X)<\dim K/36$ implies that $(X,T)$ embeds into $K^{\mathbb{Z}}$; taking $K=[0,1]^d$ gives the constant $c\ge 1/36$ referred to below. The paper simultaneously develops a factor-separation theory for the same class: two points can be distinguished by finite-entropy factors exactly when the mean dimension vanishes, a maximal zero-mean-dimension factor exists, and the small boundary property - the existence of a basis of open sets whose boundaries have zero orbit capacity, i.e. cannot be detected by orbits at positive density - implies zero mean dimension. Two questions were posed: (i) extend these results to more general $\mathbb{Z}$-actions and to more general groups such as $\mathbb{Z}^2$ and amenable groups (discrete groups admitting a finitely additive, translation-invariant probability measure; every abelian group is amenable), the obstruction being 'not purely technical' - periodic points, since a non-zero-dimensional set of periodic points precludes the small boundary property and genuinely obstructs embedding; and (ii) determine the largest constant $c$ such that $\operatorname{mdim}(X)<cd$ implies embeddability into $([0,1]^d)^{\mathbb{Z}}$.

The later literature reshaped these questions. Gutman isolated the marker property - for every $n$ there is an open $U\subset X$ with $\bigcup_kT^kU=X$ and $U,TU,\dots,T^nU$ pairwise disjoint - as the true hypothesis behind the embedding theorem; it holds for aperiodic subshifts, aperiodic finite-dimensional systems, and extensions of aperiodic minimal systems. He also introduced the periodic dimension, the growth rate of the covering dimension of the fixed-point sets $\operatorname{Fix}(T^n)$, which quantifies the periodic-point obstruction. Within the marker-property class the constant question is closed at the optimal value $1/2$: minimal systems and extensions of aperiodic subshifts of mean dimension $<d/2$ embed into $([0,1]^d)^{\mathbb{Z}}$ and the constant cannot be improved (Gutman-Tsukamoto; Lindenstrauss-Tsukamoto), and the same optimal $1/2$ holds for $\mathbb{Z}^k$-actions with the marker property (Gutman-Qiao-Tsukamoto, via signal analysis) and for actions of an arbitrary amenable group with the uniform Rokhlin property and comparison (Naryshkin) - the uniform Rokhlin property asking that the action decompose, up to an arbitrarily small error, into finitely many disjoint Rokhlin towers, families of sets $gU$ indexed by the elements $g$ of a finite subset of the group, and comparison asking that continuous functions on such towers be comparable in an order-theoretic sense - as well as for extensions of almost finite actions of amenable groups, those admitting such tower decompositions by clopen sets (Lanckriet-Szabo), while Jin-Park-Qiao constructed minimal actions of every infinite amenable group with mean dimension exactly $d/2$ that do not embed into $([0,1]^d)^G$. Outside this class the situation changed abruptly: Tsukamoto-Tsutaya-Yoshinaga and Shi produced free systems without the marker property (with arbitrarily small positive mean dimension in Shi's case), and Dranishnikov-Levin constructed a free $\mathbb{Z}$-action by isometries - mean dimension and entropy zero - that does not embed into any cubical shift, refuting the belief that mean dimension and periodic-orbit dimensions are the only obstructions. Meyerovitch then introduced a new dimension $\dim(X,T)$ for actions of any countable group, defined variationally from invariant measures and open covers, which dominates mean dimension, takes the value $d$ on the $d$-shift, vanishes exactly on systems with the small boundary property, coincides with mean dimension wherever finite mean dimension is known to suffice, and for distal systems (systems where distinct points stay at positive distance under iteration) characterizes shift embeddability by finiteness; the Dranishnikov-Levin example has $\dim=\infty$. Whether $\dim(X,T)<d/2$, or even finiteness of $\dim$, suffices for genuine embeddability in general, what the sharp threshold is for markerless systems, and how the factor-separation theory extends beyond the minimal and marker classes, constitute the surviving open core stated below.

Problem Statement

The problem is the surviving open core of the program posed in the source paper, in the form the current literature gives it. Its single scientific objective is a complete embedding-and-factor theory for topological dynamical systems: sharp, checkable dimension-type conditions that are necessary and sufficient for equivariant embedding into cubical shifts, together with the corresponding factor-separation theory, valid for general $\mathbb{Z}$-actions and actions of amenable groups, with the optimal-constant question now understood to be $1/2$ in the classes where it is settled. Four linked, source-rooted questions constitute the core.

(1) Characterization problem for $\mathbb{Z}$-actions. Decide whether finiteness of the dynamical dimension $\dim(X,T)$ of Meyerovitch - the variational invariant defined from invariant measures and open covers that dominates mean dimension, vanishes exactly on systems with the small boundary property, and takes the value $d$ on the shift $([0,1]^d)^{\mathbb{Z}}$ - is sufficient for shift embeddability; in quantitative form, decide whether $\dim(X,T)<d/2$ implies that $(X,T)$ embeds into $([0,1]^d)^{\mathbb{Z}}$. Sufficiency is known for distal systems and for all classes where finite mean dimension suffices (there dim coincides with mean dimension), and $\dim(X,T)<d/2$ is known to give almost embeddability - a continuous equivariant map that is injective on the points generic for every invariant measure, hence a measure-theoretic isomorphism for each invariant measure, though not necessarily injective on all of $X$ (Meyerovitch, GAFA 2026); the gap to genuine embeddability for arbitrary systems is open. Equivalently: prove that $\dim$ accounts for all obstructions to shift embeddability, or exhibit a further obstruction and refine the invariant accordingly.

(2) Sharp threshold without the marker property. For aperiodic $\mathbb{Z}$-actions of finite dynamical dimension that lack the marker property, determine the sharp alphabet dimension needed for embedding as a function of the dimension-type invariant - in particular, whether the $1/2$ threshold extends beyond marker-property systems. The current bounds for Shi's inverse-limit example (mean dimension and dynamical dimension $N$, embedding into the $(3N+2)$-cubical shift) leave a genuine gap in both directions.

(3) General amenable groups. Extend the sharp embedding theorem beyond the uniform-Rokhlin-property, almost-finite and marker classes: for which actions of a countably infinite amenable group $G$ does $\operatorname{mdim}(X,G)<d/2$ imply equivariant embeddability into $([0,1]^d)^G$? The Jin-Park-Qiao examples (minimal actions with mean dimension exactly $d/2$ that do not embed) show the constant $1/2$ cannot be improved for any infinite amenable group, so the question is exactly the class of actions covered.

(4) Factor-separation side. Extend from extensions of minimal systems and marker-property systems to arbitrary $\mathbb{Z}$-actions and amenable group actions the structure theory of the source paper: existence of the maximal zero-mean-dimension factor, separation of points by finite-entropy (arbitrarily low entropy) factors, and the relation between vanishing mean dimension and the small boundary property. The Dranishnikov-Levin example (mean dimension zero, no small boundary property) shows the naive extension fails, so the correct formulation must be found.

The four parts (1)-(4) are scientifically linked: they together determine the exact obstruction landscape for finite-sampling-rate models of dynamics. The four parts overlap rather than being independent: (2) is (1) restricted to aperiodic actions without the marker property, so a complete answer to (1) settles (2).

References

1. Elon Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Publications Mathematiques de l'IHES 89 (1999), 227-262. DOI 10.1007/BF02698858, https://doi.org/10.1007/bf02698858 (open-access PDF: https://pmihes.centre-mersenne.org/item/10.1007/BF02698858.pdf)
2. Misha 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. Elon Lindenstrauss and Benjamin Weiss, Mean topological dimension, Israel Journal of Mathematics 115 (2000), 1-24. DOI 10.1007/BF02810577, https://doi.org/10.1007/BF02810577
4. Elon Lindenstrauss, Lowering topological entropy, Journal d'Analyse Mathematique 67 (1995), 231-267. DOI 10.1007/BF02787792, https://doi.org/10.1007/BF02787792
5. S. Kakutani, A proof of Beboutov's theorem, Journal of Differential Equations 4 (1968), 194-201. DOI 10.1016/0022-0396(68)90036-3, https://doi.org/10.1016/0022-0396(68)90036-3
6. Yonatan Gutman, Embedding Z^k-actions in cubical shifts and Z^k-symbolic extensions, Ergodic Theory and Dynamical Systems 31 (2011), 383-403. DOI 10.1017/S0143385709001096, https://doi.org/10.1017/S0143385709001096
7. Yonatan Gutman, Elon Lindenstrauss and Masaki Tsukamoto, Mean dimension of Z^k-actions, Geometric and Functional Analysis 26 (2016), 778-817. DOI 10.1007/s00039-016-0372-9, https://doi.org/10.1007/s00039-016-0372-9
8. Yonatan Gutman and Masaki Tsukamoto, Mean dimension and a sharp embedding theorem: extensions of aperiodic subshifts, Ergodic Theory and Dynamical Systems 34 (2014), 1888-1896. DOI 10.1017/etds.2013.30, https://doi.org/10.1017/etds.2013.30 (preprint arXiv:1207.3906, https://arxiv.org/abs/1207.3906)
9. Elon Lindenstrauss and Masaki 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
10. Yonatan 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 (preprint arXiv:1208.5248, https://arxiv.org/abs/1208.5248)
11. Yonatan Gutman, Embedding topological dynamical systems with periodic points in cubical shifts, Ergodic Theory and Dynamical Systems 37 (2017), 512-538. DOI 10.1017/etds.2015.40, https://doi.org/10.1017/etds.2015.40 (preprint arXiv:1301.6072, https://arxiv.org/abs/1301.6072)
12. Yonatan Gutman and Masaki 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 (preprint arXiv:1511.01802, https://arxiv.org/abs/1511.01802)
13. Yonatan Gutman, Yixiao Qiao and Masaki Tsukamoto, Application of signal analysis to the embedding problem of 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 (preprint arXiv:1709.00125, https://arxiv.org/abs/1709.00125)
14. Yonatan Gutman and Lei Jin, Mean dimension and an embedding theorem for real flows, Fundamenta Mathematicae 251 (2020), 161-181. DOI 10.4064/fm597-2-2020, https://doi.org/10.4064/fm597-2-2020 (preprint arXiv:1806.01897, https://arxiv.org/abs/1806.01897)
15. Ruxi Shi, Embedding theorems for discrete dynamical systems and topological flows, preprint (2020). arXiv:2011.05121, https://arxiv.org/abs/2011.05121
16. Dawid Huczek, Zero-dimensional extensions of amenable group actions, preprint (2015). arXiv:1503.02827, https://arxiv.org/abs/1503.02827
17. Lei Jin, Kyewon Koh Park and Yixiao Qiao, Mean dimension and a non-embeddable example for amenable group actions, Fundamenta Mathematicae 257 (2022), 19-38. DOI 10.4064/fm921-1-2021, https://doi.org/10.4064/fm921-1-2021 (preprint arXiv:2101.01458, https://arxiv.org/abs/2101.01458)
18. Masaki Tsukamoto, Mitsunobu Tsutaya and Masahiko 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 (preprint arXiv:2012.15372, https://arxiv.org/abs/2012.15372)
19. Ruxi 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 (preprint arXiv:2102.12197, https://arxiv.org/abs/2102.12197)
20. Emiel Lanckriet and Gabor Szabo, On embeddings of extensions of almost finite actions into cubical shifts, Colloquium Mathematicum 174 (2023), 229-240. DOI 10.4064/cm9106-10-2023, https://doi.org/10.4064/cm9106-10-2023 (preprint arXiv:2202.10198, https://arxiv.org/abs/2202.10198)
21. Yonatan Gutman, Michael Levin and Tom Meyerovitch, Equivariant embedding of finite-dimensional dynamical systems, Mathematische Annalen 391 (2025), 915-936. DOI 10.1007/s00208-024-02911-y, https://doi.org/10.1007/s00208-024-02911-y
22. Petr Naryshkin, URP, comparison, mean dimension, and sharp shift embeddability, preprint (2024). arXiv:2410.01757, https://arxiv.org/abs/2410.01757
23. Ruxi Shi, Lowering mean topological dimension, preprint (2024). arXiv:2411.10749, https://arxiv.org/abs/2411.10749
24. Alexander Dranishnikov and Michael Levin, A free Z-action by isometries on a compact metric space which is not embeddable into a cubical shift, preprint (2025). arXiv:2508.14628, https://arxiv.org/abs/2508.14628
25. Tom 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 (preprint arXiv:2601.13161, https://arxiv.org/abs/2601.13161)
26. Ruxi Shi, Dynamical dimension and shift embeddability without the marker property, preprint (2026). arXiv:2607.27880, https://arxiv.org/abs/2607.27880