Sofic Conditional Mean Dimension Tuples

• Sofic conditional mean dimension tuples are defined as non-diagonal tuples in a single fiber that witness positive conditional mean dimension through every admissible cover.
• They provide a local criterion that equivalently characterizes global positive conditional mean dimension, linking cover invariants with fiber dynamics.
• The theory demonstrates that tuple sets are functorial under factor maps and connect to maximal zero conditional mean dimension factors in dynamical systems.

A sofic conditional mean dimension tuple is a local witness for positive conditional mean-dimensional complexity in a factor map

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

between continuous actions of a countable sofic group on compact metrizable spaces. In the formulation developed in "Sofic conditional mean dimension, relative sofic mean dimension and their localizations" (Li et al., 16 Aug 2025), such tuples are defined through positivity of the cover-level invariant $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$ for every admissible cover separating the tuple. The central result is a localization theorem: $\mathrm{mdim}_{\Sigma}(X|\pi)>0$ holds if and only if there exists a nontrivial sofic conditional mean dimension tuple. The theory places conditional mean dimension alongside entropy tuples and local mean dimension pairs, but in a fiber-relative sofic framework (Li et al., 16 Aug 2025).

1. Dynamical setting and conditional mean dimension

The basic object is a factor map

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

where $G$ is a countable sofic group, $X$ and $Y$ are compact metrizable spaces, $G$ acts continuously on both spaces by homeomorphisms, and $\pi$ is a continuous surjection satisfying

$\pi(gx)=g\pi(x),\qquad \forall g\in G,\ x\in X.$

A fixed sofic approximation sequence

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$0

is assumed throughout, with $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$1. For $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$2, the paper uses

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$3

and for the factor relation,

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$4

Thus $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$5 is the set of tuples lying in a single fiber of $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$6 (Li et al., 16 Aug 2025).

For a continuous pseudometric $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$7 on $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$8, the sofic microstate spaces are

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$9

consisting of all maps $\mathrm{mdim}_{\Sigma}(X|\pi)>0$0 such that

$\mathrm{mdim}_{\Sigma}(X|\pi)>0$1

where $\mathrm{mdim}_{\Sigma}(X|\pi)>0$2, and

$\mathrm{mdim}_{\Sigma}(X|\pi)>0$3

These are the sofic model spaces on which conditional dimension is measured.

The cover-theoretic conditional complexity begins with

$\mathrm{mdim}_{\Sigma}(X|\pi)>0$4

where for a finite open cover $\mathrm{mdim}_{\Sigma}(X|\pi)>0$5,

$\mathrm{mdim}_{\Sigma}(X|\pi)>0$6

For microstate spaces one defines

$\mathrm{mdim}_{\Sigma}(X|\pi)>0$7

and then

$\mathrm{mdim}_{\Sigma}(X|\pi)>0$8

Successively infimizing over $\mathrm{mdim}_{\Sigma}(X|\pi)>0$9 and finite $\pi:(X,G)\to (Y,G),$0, and then taking the supremum over finite open covers $\pi:(X,G)\to (Y,G),$1, yields

$\pi:(X,G)\to (Y,G),$2

The paper proves independence of the compatible metric $\pi:(X,G)\to (Y,G),$3, so one writes $\pi:(X,G)\to (Y,G),$4. It also proves that this cover-based definition agrees with Liang’s embedding formulation: $\pi:(X,G)\to (Y,G),$5 Among the basic properties are monotonicity under refinement, subadditivity under joins, and the bound

$\pi:(X,G)\to (Y,G),$6

(Li et al., 16 Aug 2025).

2. Admissible covers and the definition of tuples

The local theory begins from the notion of an admissible cover. Let $\pi:(X,G)\to (Y,G),$7, and let $\pi:(X,G)\to (Y,G),$8 be a finite open cover of $\pi:(X,G)\to (Y,G),$9. Then $G$0 is admissible with respect to $G$1 if for every $G$2,

$G$3

Equivalently, no single member of the cover has closure containing the whole tuple (Li et al., 16 Aug 2025).

With this notion fixed, the paper defines a tuple $G$4 to be a sofic conditional mean dimension tuple relevant to $G$5 and $G$6 if, for every admissible open cover $G$7 with respect to $G$8,

$G$9

For $X$0, the set of all such tuples is denoted

$X$1

This definition is explicitly local: it does not ask merely for positive global conditional mean dimension, but for positivity to persist under every admissible local attempt to separate the coordinates of the tuple.

The paper gives a more concrete neighborhood criterion. A tuple $X$2 is a sofic conditional mean dimension tuple if and only if for every open cover of the form

$X$3

where each $X$4 is a neighborhood of $X$5 and

$X$6

one has

$X$7

This reformulation reduces admissible-cover testing to $X$8-set covers whose complements are small pairwise separated neighborhoods around the coordinates (Li et al., 16 Aug 2025).

The paper describes this notion as the mean dimension analogue of entropy tuples. A tuple $X$9 is a sofic conditional mean dimension tuple if every local attempt to separate these points by a cover still detects positive conditional mean dimension.

3. Fiber localization and structural constraints

A fundamental feature of the theory is that conditional mean dimension tuples are necessarily fiberwise. The paper proves

$Y$0

Hence such a tuple must be non-diagonal and must lie in a single fiber of $Y$1. In particular, local positive conditional mean dimension is concentrated along fibers (Li et al., 16 Aug 2025).

The mechanism behind this inclusion is also explicit. If the points of a tuple do not lie in one fiber, then their images in $Y$2 can be separated by an open cover of $Y$3. Pulling that cover back to $Y$4 produces an admissible cover $Y$5 with

$Y$6

and therefore

$Y$7

Such a tuple cannot satisfy the defining positivity condition. This argument shows that the fiber condition is not merely formal; it is forced by the conditional cover dimension itself.

The tuple sets also have a closedness property up to collision on the diagonal: $Y$8 Equivalently,

$Y$9

Thus tuple sets persist under limits unless distinct coordinates merge.

The theory is functorial under factor maps. If $G$0 is a factor of $G$1 via $G$2, then

$G$3

imply

$G$4

Accordingly, conditional mean dimension tuples descend to factors unless they collapse to the diagonal.

These properties place the tuple sets inside the geometry of the fiber relation $G$5. A plausible implication is that tuple sets encode which parts of a fiber cannot be collapsed without destroying positive conditional mean-dimensional complexity (Li et al., 16 Aug 2025).

4. The localization theorem

The central theorem is the characterization of positive conditional mean dimension by nonemptiness of a tuple set: $G$6 This is Theorem 4.8 in the paper and is the main localization principle for sofic conditional mean dimension (Li et al., 16 Aug 2025).

The implication from tuple existence to positive global invariant is direct. If

$G$7

one chooses $G$8 so that

$G$9

defines

$\pi$0

and obtains an admissible open cover $\pi$1. By definition,

$\pi$2

hence

$\pi$3

The converse uses two localization steps. First, starting from

$\pi$4

one chooses a finite open cover $\pi$5 with

$\pi$6

By refinement and decomposition, Proposition 4.4 constructs an admissible open cover

$\pi$7

with respect to some tuple $\pi$8, still satisfying

$\pi$9

The mechanism uses removal of redundant cover elements, decomposition into binary covers, subadditivity,

$\pi(gx)=g\pi(x),\qquad \forall g\in G,\ x\in X.$0

and extraction of one positive piece.

Second, Proposition 4.5 begins with a positive cover

$\pi(gx)=g\pi(x),\qquad \forall g\in G,\ x\in X.$1

and constructs points

$\pi(gx)=g\pi(x),\qquad \forall g\in G,\ x\in X.$2

such that

$\pi(gx)=g\pi(x),\qquad \forall g\in G,\ x\in X.$3

The argument iteratively shrinks the closed complements $\pi(gx)=g\pi(x),\qquad \forall g\in G,\ x\in X.$4 while preserving positivity, producing nested nonempty compact sets with diameters tending to zero. The limiting points then satisfy the tuple condition by monotonicity.

This theorem converts the global invariant $\pi(gx)=g\pi(x),\qquad \forall g\in G,\ x\in X.$5 into a local geometric object. The paper states that this is the localization principle it seeks: a global positivity statement is equivalent to a local tuple witness.

5. Zero factors, equivalence relations, and complete positivity

Before the tuple theory, the paper develops the maximal zero conditional mean dimension factor. A factor map $\pi(gx)=g\pi(x),\qquad \forall g\in G,\ x\in X.$6 is a zero sofic conditional mean dimension extension if

$\pi(gx)=g\pi(x),\qquad \forall g\in G,\ x\in X.$7

The paper proves that if

$\pi(gx)=g\pi(x),\qquad \forall g\in G,\ x\in X.$8

then there exists a maximal zero sofic conditional mean dimension factor

$\pi(gx)=g\pi(x),\qquad \forall g\in G,\ x\in X.$9

of $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$00. This means that $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$01 is a factor of $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$02, that

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$03

and that every zero sofic conditional mean dimension factor of $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$04 factors through $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$05. The construction uses the intersection of the equivalence relations corresponding to all zero conditional mean dimension intermediate factors. The associated relation is denoted

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$06

the relative zero sofic conditional mean dimension relation, with

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$07

(Li et al., 16 Aug 2025).

The tuple sets interact directly with this relation. Theorem 4.14 states that if $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$08 is the relative zero sofic conditional mean dimension relation, then the smallest closed $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$09-invariant equivalence relation containing

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$10

is contained in $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$11. Moreover, if the smallest closed $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$12-invariant equivalence relation containing that pair set is $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$13, then $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$14 has completely positive sofic conditional mean dimension.

This identifies conditional mean dimension pairs as generators of the residual fiber structure that survives after collapsing all zero-conditional-mean-dimension behavior. In the language of the paper, tuple sets detect how much of the fiber relation remains after collapsing all zero-conditional-mean-dimension structure.

A plausible implication is that the pair set

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$15

plays for conditional mean dimension the role that local positive relations play in completely positive entropy theory. The paper does not present this as an analogy alone; it incorporates the pair set into the construction of the maximal zero factor itself (Li et al., 16 Aug 2025).

6. Relative tuples and surrounding theories

The paper develops a parallel localization theory for relative sofic mean dimension tuples in $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$16. A tuple $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$17 is a relative sofic mean dimension tuple if for every admissible open cover $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$18 with respect to $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$19,

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$20

The set of such tuples is denoted

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$21

The exact analogue of the conditional localization theorem holds: $\mathrm{mdim}_{\Sigma}(\alpha|\pi)$22 The paper also proves that conditional tuples project to relative tuples: if

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$23

then

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$24

More generally,

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$25

This places conditional tuple sets inside a hierarchy linking absolute, conditional, relative, and factor-level mean dimension (Li et al., 16 Aug 2025).

As surrounding context, "Conditional Mean Dimension" develops conditional mean dimension for amenable group actions and does not discuss tuple notions explicitly (Liang, 2020). "Conditional sofic mean dimension" develops conditional and fiberwise relative sofic invariants for factor maps of sofic actions, including the fiberwise quantity

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$26

but likewise does not define tuple objects (Liang, 2024). "Local mean dimension theory for sofic group actions" introduces mean dimension pairs, uniform positive mean dimension, and completely positive mean dimension, again without conditional tuples (García-Ramos et al., 2024). Against this background, the tuple theory of (Li et al., 16 Aug 2025) supplies the missing localization principle for conditional and relative sofic mean dimension.

Within that progression, sofic conditional mean dimension tuples are the local objects that witness positivity of

$\mathrm{mdim}_{\Sigma}(\alpha|\pi)$27

They are non-diagonal tuples in a single fiber, stable under limits up to the diagonal, functorial under factors, and linked to the maximal zero conditional factor. Their role is therefore both local and structural: they localize positive conditional mean dimension and organize the geometry of zero-dimensional collapse (Li et al., 16 Aug 2025).