Relative Mean Dimension in Dynamical Systems

• Relative Mean Dimension is a dynamical invariant that measures the average topological complexity of fibers in factor maps, generalizing mean dimension and topological entropy.
• It employs both metric and covering formulations, using Følner sequences for amenable groups and sofic approximations for non-amenable actions.
• The theory bridges variational principles and structure theorems, enabling sharper analysis of embedding properties and dimensional counterexamples in complex dynamical systems.

Relative mean dimension is a dynamical invariant that quantifies the average information-theoretic or topological complexity contributed by the fibers of a factor map between dynamical systems, generalizing the notion of mean dimension to the relative or conditional setting. Specifically, relative mean dimension measures the asymptotic width (in the sense of embedding dimension or covering numbers) of fibers, normalized by group orbit size, and provides a nontrivial extension of both topological entropy and mean dimension for both amenable and sofic group actions.

1. Definitions and Foundational Concepts

For a countable discrete amenable group $G$, compact metrizable $G$-systems $(X, G)$ and $(Y, G)$, and a continuous surjective factor map $\pi: X \to Y$, the relative mean dimension $\mdim(G, X \mid Y)$ is defined as follows. Fix a compatible metric $d$ on $X$ and a Følner sequence $(F_n)$ in $G$. For $\varepsilon > 0$, let $\mathrm{Widim}_\varepsilon(A, d_F)$ denote the minimal integer $k$ such that there is an $\varepsilon$-embedding $A \to P$ into a $k$-dimensional simplicial complex $P$ (the map's fibers have diameter $< \varepsilon$ for the $F$-orbit pseudometric). The relative mean dimension is then given by

$\mdim(G, X \mid Y) = \lim_{\varepsilon \to 0} \limsup_{n \to \infty} \frac{1}{|F_n|} \sup_{y \in Y} \mathrm{Widim}_\varepsilon(\pi^{-1}(y), d_{F_n}).$

This definition is robust under changing metric and Følner sequence (Liu et al., 22 Nov 2025).

For sofic groups, analogous definitions use sofic approximations and model spaces to define both relative and conditional mean dimensions, maintaining consistency with the classical amenable setting (Li et al., 16 Aug 2025, Liang, 2024).

2. Relations between Relative Mean Dimension, Topological Entropy, and Factors

Relative mean dimension serves as a bridge between relative topological entropy and mean dimension, providing a hierarchy of invariants for factor maps. The interplay is highlighted by recent structural results. For any factor map $\pi: (X, G) \to (Y, G)$ of compact metrizable amenable $G$-systems, with induced factor map $$\tilde{\pi}: (\mathcal{P}(X), G) \to (\mathcal{P}(Y), G)$$, the following holds:

• $h_{\mathrm{top}}(G, X \mid Y) = 0$ if and only if $$h_{\mathrm{top}}(G, \mathcal{P}(X) \mid \mathcal{P}(Y)) = 0$$.
• $h_{\mathrm{top}}(G, X \mid Y) > 0$ if and only if $\mdim(G, \mathcal{P}(X) \mid \mathcal{P}(Y)) = +\infty$.

The proof strategies employ combinatorial independence notions and block structures in the group action; independence density in the factor system lifts to the induced system on measures, leading to divergence of mean dimension when relative entropy is positive (Liu et al., 22 Nov 2025).

3. Methodologies: Metric and Topological Formulations

Relative mean dimension admits equivalent metric and covering formulations. The metric approach considers the complexity of fibers under the $F$-orbit pseudometric $d_F$, measuring the fiberwise embedding width into simplicial complexes at increasingly fine scales. The topological approach uses open covers, with a Lebesgue number condition adapted to the fibers, leading to a definition via covering and separation entropy (Liu et al., 22 Nov 2025, Shi, 2024).

For sofic groups, one uses sofic approximation sequences $(\sigma_i : G \to \operatorname{Sym}(d_i))$, considers approximate model spaces $\mathrm{Map}(\rho, F, \delta, \sigma_i)$, and analyzes both $\varepsilon$-separated subsets (metric) and open cover joins (topological). The metric and topological invariants coincide for amenable groups via Følner reductions (Li et al., 16 Aug 2025, Liang, 2024).

4. Variational Principles and Measure-Theoretic Extensions

Relative metric mean dimension may be extended to incorporate continuous potentials $\varphi \in C(X, \mathbb{R})$, producing a notion of relative mean metric dimension with potential (Wu, 2021). This admits four parallel variational principles:

• In terms of metric entropy of partitions.
• Using Shapira's covering entropy.
• Katok's entropy via Bowen balls.
• Local Brin–Katok entropy.

Explicitly, the upper relative mean metric dimension with potential is given by

$$\overline{\mathrm{mdim}_M}(X, d, T, \varphi \mid Y) = \limsup_{\varepsilon \to 0} \frac{P(d, T, \varphi, \varepsilon \mid Y)}{|\log \varepsilon|},$$

where $P(\cdot)$ involves supremizing over $(n, \varepsilon)$-separated sets in fibers. All four principles express the invariant as an extremal (usually maximal) sum of conditional entropy and potential averaged over invariant Borel probability measures, providing a direct bridge to ergodic-theoretic methods and extending classical variational principles to the relative/dynamical and fibered setting (Wu, 2021).

5. Structure Theorems, Invariants, and Examples

Structure theorems establish the existence of maximal intermediate factors with zero (relative or conditional) mean dimension. For a factor $\pi: (X, G) \to (Y, G)$:

• There is a unique largest factor $X^\Sigma_\pi \to Y$ with $\mdim_\Sigma(X^\Sigma_\pi \mid Y) = 0$ (Li et al., 16 Aug 2025).
• Relative mean dimension is monotone under subsystem restriction and behaves functorially under isomorphisms.
• Product formulas hold for amenable $G$:

$\mdim(X \mid \pi) = \frac{1}{n}\mdim(X^n \mid \pi^n), \qquad \mdim(Y \mid X) = \frac{1}{n}\mdim(Y^n \mid X^n).$

• In the case of projections from a product system $X = Y \times Z$, $\mdim(X \mid Y) = \mdim(Z)$ (Liang, 2024).

For systems with the marker property and positive mean dimension, there always exist factor maps to arbitrary small mean dimensional systems with zero relative mean dimension, providing sharp counterexamples to Hurewicz-type inequalities and enabling inverse-limit decompositions of positive-mean-dimension systems via zero-relative-dimension factors (Shi, 2024).

6. Generalizations: Sofic Invariants and Tuples

The theory extends to sofic groups through sofic approximations and model spaces, yielding sofic conditional and relative mean dimension invariants that generalize amenable cases (Li et al., 16 Aug 2025, Liang, 2024). Characterizations involve:

• Sofic conditional mean dimension tuples: $n$-tuples in $X^n$ for which every admissible open cover yields positive sofic conditional mean dimension.
• Relative mean dimension tuples: $n$-tuples in $Y^n$ for which every open cover yields positive relative sofic mean dimension.

Positivity of these tuple sets characterizes positivity of relative (resp. conditional) mean dimension. Maximal zero-mean-dimension factors correspond to the largest intermediate factors which trivialize the invariants, aligning with classical analogues in entropy theory (Li et al., 16 Aug 2025).

7. Applications, Consequences, and Open Problems

Relative mean dimension allows fine quantification of the complexity in dynamical systems beyond what is captured by entropy. It enables sharp dichotomies for induced systems on spaces of measures: positive topological entropy at the base level implies infinite mean dimension for induced factors, while zero entropy propagates to zero mean dimension (Liu et al., 22 Nov 2025). This underlies strict drop phenomena in mean dimension for certain factor maps and has ramifications for embedding theorems, the construction of low-dimensional models, and the study of maximal zero-dimension factors. Applications also include variational inequalities linking mean dimension to local entropy and metric complexity, and illustrate the robustness and flexibility of the relative mean dimension framework for modern dynamical systems. The identification of sharp variational principles, relations to box dimension, and the study of “very well-partitionable” spaces delineate key directions for ongoing research (Wu, 2021).