Relative Sofic Mean Dimension Tuples

• The paper introduces relative sofic mean dimension tuples as local invariants that signal positive global relative sofic mean dimension through every admissible open cover.
• It employs finite open covers and factor images of microstate spaces to localize and measure the complexity of continuous actions by sofic groups.
• The tuple framework mirrors conditional tuple theory, preserving functorial properties under factor maps and providing a precise local detection method for mean dimension positivity.

Relative sofic mean dimension tuples are local invariants attached to a factor map

$\pi:(X,G)\to (Y,G)$

between compact metric $G$-systems, where $G$ is a countable sofic group with a fixed sofic approximation sequence $\Sigma$. They were introduced as part of a localization theory for relative sofic mean dimension: an $n$-tuple in $Y^n$ is declared positive when every admissible open cover around that tuple has positive relative sofic mean dimension. In this way, tuple sets provide a local criterion for the positivity of the global invariant $\operatorname{mdim}_{\Sigma}(Y|X)$, and they play the target-side role parallel to sofic conditional mean dimension tuples on $X^n$ (Li et al., 16 Aug 2025).

1. Foundational setting in sofic mean dimension

The ambient framework comes from sofic mean dimension for continuous actions of countable sofic groups on compact metrizable spaces. In the foundational theory, one fixes a sofic approximation

$$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$

and studies approximately equivariant maps on finite model spaces. For a compatible pseudometric $\rho$ on $G$0, a finite set $G$1, $G$2, and $G$3, the model space is

$G$4

These microstate spaces are the basic objects from which both global and localized mean-dimension invariants are built (Li, 2011).

The original sofic mean-dimension paper defined sofic mean topological dimension $G$5 and sofic metric mean dimension $G$6, established monotonicity for closed invariant subsets and factors, proved the comparison theorem

$G$7

and showed that in the amenable case these quantities recover the Lindenstrauss–Weiss mean-dimension invariants. It also proved that if $G$8, then $G$9 has a largest factor $G$0 satisfying $G$1. However, it did not explicitly define sofic mean dimension tuples, relative sofic mean dimension tuples, or tuple invariants analogous to entropy tuples (Li, 2011).

A later stage of the theory developed conditional sofic mean dimension for factor maps and a sofic version of Tsukamoto’s relative mean dimension. That work introduced fiberwise and conditional model-space invariants, showed independence of dynamically generating pseudometrics, recovered the amenable theory, and established inequalities such as

$G$2

It still did not formulate an explicit tuple theory, but it supplied the conditional and relative model-space formalism on which the tuple localization was later built (Liang, 2024).

2. Definition of relative sofic mean dimension tuples

For a factor map

$G$3

the relative tuple notion is defined on the target space $G$4. A tuple

$G$5

is called a relative sofic mean dimension tuple relevant to $G$6 and $G$7 if for every admissible open cover $G$8 with respect to $G$9, one has

$\Sigma$0

The set of all such $\Sigma$1-tuples is denoted

$\Sigma$2

This definition is the basic local object in the relative theory (Li et al., 16 Aug 2025).

The admissibility condition is expressed in terms of finite open covers localized near the points of the tuple. The paper describes an admissible cover with respect to $\Sigma$3 as a finite open cover $\Sigma$4 satisfying the relevant avoidance condition at each $\Sigma$5, and then gives an equivalent formulation using covers whose complements form neighborhood systems of the points. More precisely, for tuples in $\Sigma$6, the local characterization is: $\Sigma$7 is a relative sofic mean dimension tuple if and only if for every cover

$\Sigma$8

where each $\Sigma$9 is a neighborhood of $n$0 and

$n$1

one has

$n$2

The cover invariant used in the definition is built from the factor images of microstate spaces. For a finite open cover $n$3 of $n$4,

$n$5

and

$n$6

The same theory may also be expressed using the width dimension $n$7 of the factor image

$n$8

Thus, the tuple relation records local positivity of relative sofic mean dimension as seen through the geometry of these model-space factor images.

3. Positivity and local detection

The central theorem of the relative theory is the equivalence between positivity of the global invariant and nonemptiness of the tuple relation. For a factor map $n$9,

$Y^n$0

for some $Y^n$1. In particular, positive relative sofic mean dimension is completely detected by the existence of at least one nontrivial relative tuple (Li et al., 16 Aug 2025).

This theorem is genuinely local: positivity of $Y^n$2 is not merely a global asymptotic statement, but is witnessed by finite tuples and admissible local covers. The extraction mechanism is explicit. If

$Y^n$3

and $Y^n$4 is a finite open cover of $Y^n$5 with

$Y^n$6

then there exist points

$Y^n$7

such that

$Y^n$8

Accordingly, every positive cover yields a tuple witness.

The finite-cover inequalities are part of the mechanism. The paper proves

$Y^n$9

This permits the passage from positivity of a relative mean-dimension cover invariant to the construction of localized tuple data. A plausible implication is that the tuple relation functions as a refined support of positive cover-dimension growth in the factor image of the sofic microstate spaces.

4. Topological structure and functorial behavior

The tuple sets have a closedness property analogous to local relations in entropy theory. For each $\operatorname{mdim}_{\Sigma}(Y|X)$0,

$\operatorname{mdim}_{\Sigma}(Y|X)$1

is closed, and

$\operatorname{mdim}_{\Sigma}(Y|X)$2

Thus, non-diagonal relative tuples form a relation that is closed up to diagonal degeneration (Li et al., 16 Aug 2025).

The theory is also functorial under factor maps. Suppose

$\operatorname{mdim}_{\Sigma}(Y|X)$3

If

$\operatorname{mdim}_{\Sigma}(Y|X)$4

and $\operatorname{mdim}_{\Sigma}(Y|X)$5, then

$\operatorname{mdim}_{\Sigma}(Y|X)$6

Likewise, if

$\operatorname{mdim}_{\Sigma}(Y|X)$7

and $\operatorname{mdim}_{\Sigma}(Y|X)$8, then

$\operatorname{mdim}_{\Sigma}(Y|X)$9

These lifting results show that tuple structure persists through factor maps except when the image collapses onto the diagonal.

The tuple theory is accompanied by a factor-theoretic zero-complexity statement. If $X^n$0 has non-negative relative sofic mean dimension, then $X^n$1 has the maximal relative zero sofic mean dimension factor. The tuple relation therefore sits alongside a canonical decomposition into positive and zero relative mean-dimension parts. This suggests that relative tuples are the local witnesses of the same obstruction that is encoded globally by the maximal relative zero factor.

5. Relation to conditional tuples and positivity notions

The relative and conditional tuple theories are formally parallel, but they live on opposite sides of the factor map. The conditional theory assigns tuples to $X^n$2, whereas the relative theory assigns tuples to $X^n$3. Their positivity tests are different: $X^n$4

Theory	Tuples live in	Positivity tested by
Conditional	$X^n$5	$X^n$6
Relative	$X^n$7	$X^n$8

The conditional tuple set is denoted

$X^n$9

and a tuple $$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$0 belongs to it if every admissible open cover $$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$1 with respect to $$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$2 satisfies

$$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$3

The conditional positivity theorem is the exact analogue of the relative one: $$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$4 for some $$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$5 (Li et al., 16 Aug 2025).

The paper describes the relative theory as the “mirror image” of the conditional theory: the same admissible-cover philosophy, the same tuple-nonemptiness $$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$6 positivity criterion, the same closure and factor-map stability properties, and the same CPCMD/UPMD dichotomy. On the relative side, if the smallest closed $$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$7-invariant equivalence relation containing

$$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$8

is $$\Sigma=\{\sigma_i:G\to \operatorname{Sym}(d_i)\}_{i=1}^\infty,$$9, then $\rho$0 has sofic $\rho$1-CPMD; moreover, if $\rho$2 has sofic $\rho$3-UPMD, then it has sofic $\rho$4-CPMD. On the conditional side, the analogous statement uses the smallest closed $\rho$5-invariant equivalence relation generated by $\rho$6 and the relation $\rho$7.

The tuple localization clarifies the relation between the newer localization theory and earlier work on conditional and relative mean dimension for factor maps. The 2024 conditional sofic mean dimension framework had already established the model-space invariants $\rho$8, $\rho$9, and $G$00, together with amenable recovery and fiberwise comparisons, but without an explicit tuple formalism (Liang, 2024). The tuple theory can therefore be viewed as a local refinement of that factor-map framework.

6. Historical and conceptual position

Relative sofic mean dimension tuples emerged after the global and factorwise invariants were already in place. The 2011 theory of sofic mean dimension introduced the absolute invariants, proved subsystem and factor monotonicity, identified a largest zero sofic mean topological dimension factor, established the comparison

$G$01

and recovered the amenable theory. But it did not define any tuple invariant, either absolute or relative (Li, 2011).

The subsequent factor-map theory of conditional sofic mean dimension broadened the scope to extension complexity and fiberwise relative complexity. It studied a sofic version of Tsukamoto’s relative mean dimension, extended the Shi–Tsukamoto lower bound to the sofic case, and proved comparisons such as

$G$02

Even there, the paper explicitly did not introduce a tuple theory by name, although its fiberwise model-space invariants were the natural precursors of one (Liang, 2024).

Further metric work proved the equivalence between sofic metric mean dimension and sofic $G$03-metric mean dimension and established a product formula, but again did not set up a theory of relative sofic mean dimension tuples. That paper only remarked that Liang had introduced relative sofic metric mean dimension and conditional sofic metric mean dimension earlier, and that Garcia-Ramos and Gutman had introduced sofic mean dimension pairs (Li, 4 Mar 2025).

Against this background, the tuple formalism of 2025 supplies the missing localization layer. It turns positivity of relative sofic mean dimension into a pointwise relation on $G$04, proves that this relation is nonempty exactly when the global invariant is positive, and integrates it with closure, lifting, and complete positivity properties (Li et al., 16 Aug 2025). This suggests that relative sofic mean dimension tuples occupy, within mean-dimension theory, the same conceptual niche that entropy tuples occupy in local entropy theory: they are not the original invariant, but the localized relation that records where positive complexity is forced.