Universal metric selection and the exchange of supremum and scale limit in the variational principle for metric mean dimension

For a topological dynamical system, Lindenstrauss and Tsukamoto proved that the metric mean dimension is given by a variational principle: it is the scale limit of the suprema, over invariant measures, of rate distortion functions (under a mild growth condition on the metric, and unconditionally for a modified $L^\infty$ version). Three linked completion questions are open for systems without the marker property. (A) The metric-attainment conjecture, open since 1999: does every topological dynamical system admit a compatible metric whose metric mean dimension (lower or upper) equals its topological mean dimension? (B) The double variational principle: does mean dimension equal the minimax value of the rate distortion dimension, minimized over compatible metrics and maximized over invariant measures, for every system? (C) The exchange problem: for which metrics does a single invariant measure — rather than a scale-dependent family of measures — capture the full metric mean dimension through its rate distortion dimension? All three are settled affirmatively for systems with the marker property (e.g. free minimal systems); the exchange is known to fail in general for the rate distortion functional, convexity of the rate distortion dimension under ergodic decomposition supplies the known obstruction, and recently introduced alternative per-measure functionals (the Katok and Shapira $\epsilon$-entropies, i.e. per-measure scale-dependent orbit complexities, and the mean quantization dimension, built from optimal quantization of invariant measures) do admit the exchange in full generality. The problem asks for a resolution without marker-type hypotheses, or a counterexample, together with a characterization of the exchange.

Background

A topological dynamical system (TDS) is a pair $(X,T)$ where $X$ is a compact metrizable space and $T:X\to X$ is a homeomorphism. Its topological entropy $h_{\mathrm{top}}(T)$ measures orbit complexity, and the classical variational principle identifies it with the supremum of the Kolmogorov–Sinai entropies $h_\mu(T)$ over the set $\mathscr{M}^T(X)$ of $T$-invariant Borel probability measures. Many natural systems — for instance the shift $\sigma$ on $[0,1]^{\mathbb{Z}}$ — have infinite topological entropy, and entropy no longer distinguishes them. To quantify such systems, Gromov introduced the (topological) mean dimension $\mathrm{mdim}(X,T)$, developed by Lindenstrauss and Weiss: roughly, the number of real parameters per iterate needed to describe a typical orbit. It is defined via the topological dimension of the nerves of joins of open covers under iteration (the nerve of a cover being the simplicial complex whose vertices are the elements of the cover and whose simplices are its finite intersecting subfamilies); for example $\mathrm{mdim}([0,1]^{\mathbb{Z}},\sigma)=1$ although the entropy is infinite.

To make mean dimension computable, Lindenstrauss and Weiss introduced the metric mean dimension, a dynamical analogue of the Minkowski (box) dimension of fractal geometry. Given a compatible metric $d$ on $X$, define the Bowen metric $d_n(x,y)=\max_{0\le k<n}d(T^kx,T^ky)$ and let $\#(X,d_n,\varepsilon)$ be the least cardinality of a cover of $X$ by sets of $d_n$-diameter below $\varepsilon$. The entropy at scale $\varepsilon$ is $S(X,T,d,\varepsilon)=\lim_{n\to\infty}\frac{1}{n}\log\#(X,d_n,\varepsilon)$ (the limit exists by subadditivity), and $h_{\mathrm{top}}(T)=\lim_{\varepsilon\to0}S(X,T,d,\varepsilon)$. The upper and lower metric mean dimensions are $\overline{\mathrm{mdim}}_M(X,T,d)=\limsup_{\varepsilon\to0}\frac{S(X,T,d,\varepsilon)}{|\log\varepsilon|},\qquad \underline{\mathrm{mdim}}_M(X,T,d)=\liminf_{\varepsilon\to0}\frac{S(X,T,d,\varepsilon)}{|\log\varepsilon|}.$ For every compatible $d$ one has $\mathrm{mdim}(X,T)\le\underline{\mathrm{mdim}}_M(X,T,d)\le\overline{\mathrm{mdim}}_M(X,T,d)$, and metric mean dimension is the main tool for bounding mean dimension from above, with applications to dynamical systems from geometric analysis (spaces of Brody curves, i.e. holomorphic maps with bounded derivative, and Yang–Mills gauge fields) and to embedding dynamical systems into shift spaces.

On the information-theoretic side, rate distortion theory studies lossy data compression of stochastic processes. For $\mu\in\mathscr{M}^T(X)$, the rate distortion function $R(d,\mu,\varepsilon)$ is the infimum of the mutual information rate over all communication channels that reproduce the stationary process $\{T^nx\}_{x\sim\mu}$ within average distortion $\varepsilon$ with respect to $d$; it measures how many nats (natural-logarithm units of information) per iterate are needed to reproduce the process at resolution $\varepsilon$. Kawabata and Dembo introduced the rate distortion dimension of a process as the polynomial growth rate of $R$ in $1/\varepsilon$; here one writes $\overline{\mathrm{rdim}}(X,T,d,\mu)=\limsup_{\varepsilon\to0}R(d,\mu,\varepsilon)/|\log\varepsilon|$ and the liminf version $\underline{\mathrm{rdim}}$.

Lindenstrauss and Tsukamoto connected these two worlds: if $d$ has tame growth of covering numbers (for every $\theta>0$, $\varepsilon^{\theta}\log r_1(T,d,\varepsilon,X)\to0$ as $\varepsilon\to0$, where $r_1$ is a one-step covering number — a mild condition satisfied, e.g., by product-type metrics on subshifts over finite-dimensional alphabets), then $\overline{\mathrm{mdim}}_M(X,T,d)=\limsup_{\varepsilon\to0}\sup_{\mu\in\mathscr{M}^T(X)}\frac{R(d,\mu,\varepsilon)}{|\log\varepsilon|}$ together with the analogous liminf statement; an unconditional version holds for a modified $L^\infty$ rate distortion function (the variant in which the average-distortion reproduction constraint is replaced by an almost-sure one), and without any condition on $d$ the principle fails. In the published version of that paper (Section VIII), the authors also showed that the order of the scale limit and the supremum cannot be exchanged in general: for the alphabet $A=\{1,1/2,1/3,\dots\}\cup\{0\}$, the subshift $(A^{\mathbb{Z}},\sigma)$ with the metric $d(x,y)=\sum_{n\in\mathbb{Z}}2^{-|n|}|x_n-y_n|$ has $\overline{\mathrm{mdim}}_M=1/2$ while $\overline{\mathrm{rdim}}(X,\sigma,d,\mu)=0$ for every invariant $\mu$; the complexity at scale $\varepsilon$ is captured only by a family of maximizing measures that drift with $\varepsilon$.

The same authors' companion "double variational principle" paper restricted the study to systems with the marker property: $(X,T)$ has the marker property if for every $N$ there is an open set $U\subset X$ with $X=\bigcup_{n\in\mathbb{Z}}T^{-n}U$ and $U\cap T^{-n}U=\emptyset$ for $1\le n\le N$. Free minimal systems, their extensions, and aperiodic finite-dimensional systems have this property. For marker-property systems they proved the minimax identity $\mathrm{mdim}(X,T)=\min_{d\in\mathscr{D}(X)}\sup_{\mu\in\mathscr{M}^T(X)}\overline{\mathrm{rdim}}(X,T,d,\mu)$ (where $\mathscr{D}(X)$ is the set of compatible metrics, the minimum is attained, and the analogous identity holds for the lower rate distortion dimension), and, en route, that such a system admits a metric $d$ with $\overline{\mathrm{mdim}}_M(X,T,d)=\mathrm{mdim}(X,T)$ — the upper version of a statement open since Lindenstrauss's 1999 paper, which had proved the lower version for marker systems. For that metric, all the relevant quantities — $\mathrm{mdim}$, both metric mean dimensions, and both suprema of rate distortion dimensions — coincide. The inequality $\mathrm{mdim}(X,T)\le\inf_{d}\sup_{\mu}\overline{\mathrm{rdim}}$ holds without the marker property. Ruxi Shi later constructed systems of arbitrary prescribed finite mean dimension without the marker property, so the hypothesis cannot be removed by proving that all systems carry markers.

Subsequent work refined the measure-theoretic side without removing the marker hypothesis: Gutman and Śpiewak showed that ergodic measures suffice in the tame-growth principle and gave an unconditional variational principle via a dynamical Rényi information dimension (a per-measure scale exponent built from Rényi entropy rates of the process observed at resolution $\varepsilon$); Shi expressed metric mean dimension through the Shapira, Katok, and Brin–Katok local entropies (per-measure, scale-dependent analogues of the Kolmogorov–Sinai entropy); Yang–Chen–Zhou and Chen–Yang–Zhou unified these into a family of per-measure $\epsilon$-entropies — functionals measuring the orbit complexity of a single invariant measure at scale $\epsilon$ — whose scale limits all coincide with the rate distortion dimension; Tsukamoto proved that the upper rate distortion dimension is convex under ergodic decomposition whenever $\overline{\mathrm{mdim}}_M<\infty$, and constructed a system whose ergodic invariant measures all have rate distortion dimension $0$ while some non-ergodic invariant measure has $+\infty$; and Carvalho and Pessil showed that for the Katok and Shapira $\epsilon$-entropies, and for a new mean quantization dimension (a per-measure scale exponent built from the optimal quantization error of the stationary process), the exchange of limit and supremum does hold in full generality, with a maximizing measure. What remains untouched is the exchange for the rate distortion function itself, and the marker-free case of the metric-attainment conjecture.

Problem Statement

Let $(X,T)$ be a topological dynamical system ($X$ compact metrizable, $T:X\to X$ a homeomorphism). Let $\mathscr{D}(X)$ denote the set of metrics on $X$ compatible with the topology, $\mathscr{M}^T(X)$ the set of $T$-invariant Borel probability measures, $\mathrm{mdim}(X,T)$ the topological mean dimension, $\overline{\mathrm{mdim}}_M$ and $\underline{\mathrm{mdim}}_M$ the upper and lower metric mean dimensions, and $\overline{\mathrm{rdim}}(X,T,d,\mu)$ and $\underline{\mathrm{rdim}}(X,T,d,\mu)$ the upper and lower rate distortion dimensions of $\mu$, all as defined in the Background. The problem is to settle the following three linked questions for arbitrary systems, without hypotheses such as the marker property, minimality, aperiodicity, or finite dimensionality.

(A) Metric attainment (universality of metric selection; Lindenstrauss 1999 conjecture, restated in the source paper; Lindenstrauss–Tsukamoto 2019, Problem 1.6). Prove, or disprove by an explicit example, that for every $(X,T)$ there exists $d\in\mathscr{D}(X)$ with $\mathrm{mdim}(X,T)=\underline{\mathrm{mdim}}_M(X,T,d),$ and independently settle the same question for the upper version $\overline{\mathrm{mdim}}_M$. Both versions are known for systems with the marker property (the lower version since 1999, the upper since 2019); both are open in general.

(B) Double variational principle (Lindenstrauss–Tsukamoto 2019, Problem 1.5). Prove, or disprove, that for every $(X,T)$ $\mathrm{mdim}(X,T)=\min_{d\in\mathscr{D}(X)}\sup_{\mu\in\mathscr{M}^T(X)}\overline{\mathrm{rdim}}(X,T,d,\mu)$ with the minimum attained by some $d$, together with the analogous statement with $\underline{\mathrm{rdim}}$. The inequality $\mathrm{mdim}\le\inf_d\sup_\mu\overline{\mathrm{rdim}}$ is known unconditionally; the remaining content is the reverse inequality and attainment, which are known under the marker property.

(C) Exchange of supremum and scale limit (Lindenstrauss–Tsukamoto 2019, Problem 1.3; posed independently by Velozo–Velozo 2017). Determine necessary and sufficient conditions on the triple $(X,T,d)$ under which $\overline{\mathrm{mdim}}_M(X,T,d)=\sup_{\mu\in\mathscr{M}^T(X)}\overline{\mathrm{rdim}}(X,T,d,\mu),$ together with the corresponding liminf/lower statements and the variants obtained by restricting the supremum to ergodic measures.

These three questions share one scientific objective: to determine whether topological mean dimension, metric mean dimension, and the measure-theoretic rate distortion dimension can always be reconciled by a single choice of compatible metric and invariant measure — completing the measure-theoretic variational program for mean dimension. The three questions are posed jointly for arbitrary systems; either direction counts as a resolution of (A) or (B).

References

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