For a topological dynamical system with the marker property, a topological Rokhlin-type covering condition, Lindenstrauss and Tsukamoto (GAFA 2019) proved the double variational principle: the mean dimension equals the minimum over compatible metrics of the supremum over invariant measures of the rate-distortion dimension, with the minimum attained at a compatible metric whose upper metric mean dimension equals the mean dimension. The marker property enters their proof only through this metric construction, and they pose its removal as the paper's central open problem: prove that every dynamical system admits a compatible metric realizing its mean dimension as upper (hence also lower) metric mean dimension, so that the double variational principle holds for all systems — a statement already conjectured by Gutman–Lindenstrauss–Tsukamoto (GAFA 2016) for actions of all countable amenable groups. The objective is to determine whether every dynamical system admits a compatible metric realizing its mean dimension as upper (hence also lower) metric mean dimension — equivalently, whether the double variational principle holds without any marker-type hypothesis.

Background

A topological dynamical system is a pair (X,T)(\mathcal{X},T) with X\mathcal{X} a compact metrizable space and T:X→XT:\mathcal{X}\to\mathcal{X} a homeomorphism; D(X)\mathscr{D}(\mathcal{X}) denotes the compatible metrics on X\mathcal{X} and MT(X)\mathscr{M}^T(\mathcal{X}) the TT-invariant Borel probability measures. Topological entropy, the classical complexity invariant, is infinite for many infinite-dimensional systems, and Gromov introduced mean dimension in 1999 as an invariant that remains informative there. It is defined without reference to a metric: for a finite open cover U\mathcal{U} of X\mathcal{X} one forms the join ⋁i=0n−1T−iU\bigvee_{i=0}^{n-1}T^{-i}\mathcal{U} over an orbit segment and lets mdim⁡(X,T)\operatorname{mdim}(\mathcal{X},T) be the supremum over U\mathcal{U} of the growth rate of the topological (covering) dimension of these joins, lim⁡n1ndim⁡(⋁i<nT−iU)\lim_n\frac{1}{n}\dim(\bigvee_{i<n}T^{-i}\mathcal{U}).

A computable metric counterpart was introduced by Lindenstrauss and Weiss: for d∈D(X)d\in\mathscr{D}(\mathcal{X}) let dN(x,y)=max⁡0≤k<Nd(Tkx,Tky)d_N(x,y)=\max_{0\le k<N}d(T^kx,T^ky) and let #(X,dN,ε)\#(\mathcal{X},d_N,\varepsilon) be the minimal size of an ε\varepsilon-spanning set; with S(X,T,d,ε)=lim⁡N1Nlog⁡#(X,dN,ε)S(\mathcal{X},T,d,\varepsilon)=\lim_N\frac{1}{N}\log\#(\mathcal{X},d_N,\varepsilon), the upper and lower metric mean dimensions are

mdim⁡‾M(X,T,d)=lim sup⁡ε→0S(X,T,d,ε)log⁡(1/ε),mdim⁡‾M(X,T,d)=lim inf⁡ε→0S(X,T,d,ε)log⁡(1/ε).\overline{\operatorname{mdim}}_{\mathrm{M}}(\mathcal{X},T,d)=\limsup_{\varepsilon\to0}\frac{S(\mathcal{X},T,d,\varepsilon)}{\log(1/\varepsilon)},\qquad\underline{\operatorname{mdim}}_{\mathrm{M}}(\mathcal{X},T,d)=\liminf_{\varepsilon\to0}\frac{S(\mathcal{X},T,d,\varepsilon)}{\log(1/\varepsilon)}.

They proved the sandwich mdim⁡(X,T)≤mdim⁡‾M(X,T,d)≤mdim⁡‾M(X,T,d)\operatorname{mdim}(\mathcal{X},T)\le\underline{\operatorname{mdim}}_{\mathrm{M}}(\mathcal{X},T,d)\le\overline{\operatorname{mdim}}_{\mathrm{M}}(\mathcal{X},T,d) for every compatible dd, so metric mean dimension bounds mean dimension from above; whether the bound can always be made tight by a suitable choice of dd is the subject of the present problem.

The measure-theoretic side comes from Shannon rate-distortion theory, playing the role that Kolmogorov–Sinai entropy (the measure-theoretic entropy of an invariant measure) plays for topological entropy in the classical variational principle (which identifies topological entropy with the supremum of Kolmogorov–Sinai entropies over invariant measures). For (X,T,d,μ)(\mathcal{X},T,d,\mu) with μ∈MT(X)\mu\in\mathscr{M}^T(\mathcal{X}), the L∞L^{\infty} rate-distortion function Rμ(ε)R_\mu(\varepsilon) is the minimal asymptotic coding rate required to reproduce the process within distortion ε\varepsilon measured in dd; the rate-distortion dimension is rdim⁡‾(X,T,d,μ)=lim sup⁡ε→0Rμ(ε)/log⁡(1/ε)\overline{\operatorname{rdim}}(\mathcal{X},T,d,\mu)=\limsup_{\varepsilon\to0}R_\mu(\varepsilon)/\log(1/\varepsilon), with the lower variant rdim⁡‾\underline{\operatorname{rdim}} using lim inf⁡\liminf. Lindenstrauss and Tsukamoto first proved a fixed-metric variational principle expressing mdim⁡M(X,T,d)\operatorname{mdim}_{\mathrm{M}}(\mathcal{X},T,d) through suprema of rate-distortion quantities over MT(X)\mathscr{M}^T(\mathcal{X}) (IEEE Trans. Inform. Theory 2018), with refinements by Gutman-Śpiewak (Studia Math. 2021).

The marker property is a topological analogue of Rokhlin towers (the classical structure theorem decomposing an aperiodic measure-preserving transformation into disjoint orbit segments of a prescribed common length): (X,T)(\mathcal{X},T) has it if for every N>0N>0 there is an open set U⊂XU\subset\mathcal{X} with X=⋃n∈ZTnU\mathcal{X}=\bigcup_{n\in\mathbb{Z}}T^nU and U∩T−nU=∅U\cap T^{-n}U=\emptyset for 1≤n≤N1\le n\le N. Free minimal systems (those in which every orbit is dense) and their extensions have it, and every finite-dimensional aperiodic system (one having no periodic points) has it; however, it fails for some aperiodic infinite-dimensional systems, and Shi (Trans. AMS 2023) showed that for every positive number there exists an aperiodic system with exactly that mean dimension and no marker property, so the hypothesis is a genuine restriction precisely in the infinite-dimensional regime where mean dimension is interesting.

Within this framework the metric-existence problem has a long history. Lindenstrauss (Publ. Math. IHÉS 1999, Theorem 4.3) proved, under the marker property, that some compatible dd satisfies mdim⁡‾M(X,T,d)=mdim⁡(X,T)\underline{\operatorname{mdim}}_{\mathrm{M}}(\mathcal{X},T,d)=\operatorname{mdim}(\mathcal{X},T), and Gutman–Lindenstrauss–Tsukamoto (GAFA 2016) extended the construction to Zk\mathbb{Z}^k-actions and explicitly conjectured the statement without markers, "indeed not just for Zk\mathbb{Z}^k systems but for any countable amenable group". The upper-variant statement resisted attack for about twenty years until Lindenstrauss–Tsukamoto (GAFA 2019) proved the double variational principle: if (X,T)(\mathcal{X},T) has the marker property, then

mdim⁡(X,T)=min⁡d∈D(X)sup⁡μ∈MT(X)rdim⁡‾(X,T,d,μ)=min⁡d∈D(X)sup⁡μ∈MT(X)rdim⁡‾(X,T,d,μ),\operatorname{mdim}(\mathcal{X},T)=\min_{d\in\mathscr{D}(\mathcal{X})}\sup_{\mu\in\mathscr{M}^T(\mathcal{X})}\overline{\operatorname{rdim}}(\mathcal{X},T,d,\mu)=\min_{d\in\mathscr{D}(\mathcal{X})}\sup_{\mu\in\mathscr{M}^T(\mathcal{X})}\underline{\operatorname{rdim}}(\mathcal{X},T,d,\mu),

where the minimum is attained at a "nice" metric for which mdim⁡(X,T)=mdim⁡‾M(X,T,d)\operatorname{mdim}(\mathcal{X},T)=\overline{\operatorname{mdim}}_{\mathrm{M}}(\mathcal{X},T,d) (hence, by the sandwich, also the lower metric mean dimension and the mean Hausdorff dimension, a dynamical analogue of Hausdorff dimension, all coincide). The proof has three steps, and the marker property is used only in Step 3, the metric construction via dynamical tiling (a decomposition of X\mathcal{X} into tiles carved out by the iterates of a marker set) and an embedding into a Banach space; Steps 1 and 2 are marker-free and yield the one-sided inequality mdim⁡(X,T)≤inf⁡dsup⁡μrdim⁡‾(X,T,d,μ)\operatorname{mdim}(\mathcal{X},T)\le\inf_{d}\sup_{\mu}\underline{\operatorname{rdim}}(\mathcal{X},T,d,\mu) valid for every system (Remark 1.2(3), where the authors conjecture the marker hypothesis is unnecessary). The paper's Section 1.4 formulates the removal of the marker assumption as "the most important open problem".

Problem Statement

Let (X,T)(\mathcal{X},T) be a dynamical system (X\mathcal{X} compact metrizable, TT a homeomorphism), D(X)\mathscr{D}(\mathcal{X}) its compatible metrics, and MT(X)\mathscr{M}^T(\mathcal{X}) its TT-invariant Borel probability measures. Prove, without any marker-type hypothesis (Problems 1.5 and 1.6 of Lindenstrauss–Tsukamoto, GAFA 2019), that there exists d∈D(X)d\in\mathscr{D}(\mathcal{X}) with

mdim⁡(X,T)=mdim⁡‾M(X,T,d),\operatorname{mdim}(\mathcal{X},T)=\overline{\operatorname{mdim}}_{\mathrm{M}}(\mathcal{X},T,d),

so that, by the Lindenstrauss–Weiss sandwich, the same metric also satisfies mdim⁡‾M(X,T,d)=mdim⁡(X,T)\underline{\operatorname{mdim}}_{\mathrm{M}}(\mathcal{X},T,d)=\operatorname{mdim}(\mathcal{X},T); equivalently, prove the double variational principle

mdim⁡(X,T)=min⁡d∈D(X)sup⁡μ∈MT(X)rdim⁡‾(X,T,d,μ)=min⁡d∈D(X)sup⁡μ∈MT(X)rdim⁡‾(X,T,d,μ)\operatorname{mdim}(\mathcal{X},T)=\min_{d\in\mathscr{D}(\mathcal{X})}\sup_{\mu\in\mathscr{M}^T(\mathcal{X})}\overline{\operatorname{rdim}}(\mathcal{X},T,d,\mu)=\min_{d\in\mathscr{D}(\mathcal{X})}\sup_{\mu\in\mathscr{M}^T(\mathcal{X})}\underline{\operatorname{rdim}}(\mathcal{X},T,d,\mu)

for every dynamical system (the source shows Problem 1.5 reduces to Problem 1.6 because Steps 1–2 of its proof are marker-free). The source separately emphasizes the companion question for lower metric mean dimension — find dd with mdim⁡‾M(X,T,d)=mdim⁡(X,T)\underline{\operatorname{mdim}}_{\mathrm{M}}(\mathcal{X},T,d)=\operatorname{mdim}(\mathcal{X},T) for arbitrary systems — which is a strictly weaker statement (implied by the upper variant through the sandwich inequality, but not conversely), is open in the same generality, and is a legitimate partial target. A refutation is equally admissible: exhibit an explicit dynamical system for which no compatible metric realizes the equality — whether because inf⁡d∈D(X)mdim⁡‾M(X,T,d)>mdim⁡(X,T)\inf_{d\in\mathscr{D}(\mathcal{X})}\overline{\operatorname{mdim}}_{\mathrm{M}}(\mathcal{X},T,d)>\operatorname{mdim}(\mathcal{X},T) or because the infimum is not attained — which would disprove the conjecture of Lindenstrauss–Tsukamoto (Remark 1.2(3)) and that of Gutman–Lindenstrauss–Tsukamoto. The natural generalization — the same metric-existence statement for actions of any countable amenable group, conjectured by Gutman–Lindenstrauss–Tsukamoto — belongs to the problem's full scope; a proof for Z\mathbb{Z}-actions alone already resolves the source question.

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