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arxiv: 2607.01501 · v1 · pith:5J4EQ67Onew · submitted 2026-07-01 · 🧮 math.DS

Uniformly Positive Mean Dimension

Tal Barak , Elon Lindenstrauss This is my paper

Pith reviewed 2026-07-03 18:09 UTC · model grok-4.3

classification 🧮 math.DS
keywords hub-and-spoke systemsmean dimensionuniformly positive entropycompletely positive mean dimensionsymbolic dynamicsirrational rotationsdynamical systems
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The pith

A hub-and-spoke construction on symbolic systems transfers uniformly positive entropy to uniformly positive mean dimension and produces examples with completely positive mean dimension but neither uniform property.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper associates to each symbolic system X a hub-and-spoke dynamical system Spoke(X) obtained by replacing each symbol with a one-dimensional spoke attached to a common hub. It proves that a shift-invariant measure of full support on X implies completely positive mean dimension on Spoke(X). It further proves that uniformly positive entropy on X implies uniformly positive mean dimension on Spoke(X). Using symbolic codings of irrational rotations, the authors construct nondegenerate examples of Spoke systems that have completely positive mean dimension yet lack both uniformly positive mean dimension and uniformly positive entropy, where base covers contribute zero but iterated covering numbers are unbounded.

Core claim

To a symbolic system X we associate the hub-and-spoke system Spoke(X) obtained by replacing each symbol by a one-dimensional spoke attached to a common hub. If X admits a shift-invariant measure of full support, then Spoke(X) has completely positive mean dimension. If X has uniformly positive entropy, then Spoke(X) has uniformly positive mean dimension. Using symbolic codings of irrational rotations on tori, we construct hub-and-spoke systems with completely positive mean dimension but without uniformly positive mean dimension or uniformly positive entropy. The examples are nondegenerate: the relevant covers have zero mean dimension and zero entropy, but when refined by iterating under the d

What carries the argument

The hub-and-spoke construction Spoke(X), which replaces each symbol of the symbolic system X with a one-dimensional spoke attached to a common hub.

If this is right

  • A symbolic system with a full-support invariant measure yields a hub-and-spoke system with completely positive mean dimension.
  • Uniformly positive entropy on the base system implies uniformly positive mean dimension on the associated spoke system.
  • There exist hub-and-spoke systems with completely positive mean dimension that lack uniformly positive mean dimension and uniformly positive entropy.
  • In the constructed examples, certain open covers have zero mean dimension and zero entropy while their iterated covering numbers under the dynamics are unbounded.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The spoke construction may allow separation of complete positivity from uniform positivity in mean dimension across a wider class of systems.
  • Similar geometric attachments could relate mean dimension to other invariants in systems that are not originally symbolic.
  • The phenomenon of zero single-cover contribution yet unbounded iterated numbers may appear in other dimension-like invariants under iteration.

Load-bearing premise

The spoke replacement preserves the relevant dynamical properties at the level of fixed open covers when associating the system and when proving the implications from entropy or full-support measures.

What would settle it

A symbolic system with uniformly positive entropy whose Spoke version admits an open cover whose mean dimension contribution is zero would falsify the uniform positivity transfer.

read the original abstract

We study the relation between uniformly positive entropy and uniformly positive mean dimension at the level of fixed open covers. To a symbolic system X, we associate a hub-and-spoke system Spoke(X), obtained by replacing each symbol by a one-dimensional spoke attached to a common hub. We prove that if X admits a shift-invariant measure of full support, then Spoke(X) has completely positive mean dimension. We also prove that if X has uniformly positive entropy, then Spoke(X) has uniformly positive mean dimension. Finally, using symbolic codings of irrational rotations on tori, we construct hub-and-spoke systems with completely positive mean dimension but without uniformly positive mean dimension or uniformly positive entropy. The examples are nondegenerate: the relevant covers have zero mean dimension and zero entropy, but when refined by iterating under the dynamics the corresponding covering numbers are unbounded.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 0 minor

Summary. The paper studies the relation between uniformly positive entropy and uniformly positive mean dimension at the level of fixed open covers. To a symbolic system X, it associates a hub-and-spoke system Spoke(X) obtained by replacing each symbol by a one-dimensional spoke attached to a common hub. It proves that if X admits a shift-invariant measure of full support, then Spoke(X) has completely positive mean dimension. It also proves that if X has uniformly positive entropy, then Spoke(X) has uniformly positive mean dimension. Finally, using symbolic codings of irrational rotations on tori, it constructs hub-and-spoke systems with completely positive mean dimension but without uniformly positive mean dimension or uniformly positive entropy. The examples are nondegenerate: the relevant covers have zero mean dimension and zero entropy, but when refined by iterating under the dynamics the corresponding covering numbers are unbounded.

Significance. If the results hold, the Spoke(X) construction and the separation examples provide a concrete way to distinguish completely positive mean dimension from the uniformly positive variant (and from uniformly positive entropy) while preserving nondegeneracy via unbounded iterated covering numbers. The implications from full-support measures and from uniformly positive entropy are useful for relating these invariants at the fixed-cover level.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript and for highlighting the significance of the Spoke(X) construction and the separation examples. The recommendation is listed as uncertain, but the report contains no major comments or specific criticisms. We therefore have no points to address point-by-point and propose no changes.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes theorems relating uniformly positive entropy to uniformly positive mean dimension via explicit constructions of Spoke(X) systems from symbolic X, along with counterexamples using irrational rotation codings. These are presented as proved implications and constructions at the level of fixed open covers, with no reduction of target quantities (mean dimension, entropy) to fitted parameters, self-definitions, or load-bearing self-citations. The derivation chain relies on direct dynamical arguments and examples rather than renaming or importing uniqueness from prior author work. The central claims remain independent of the inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

This is a pure-mathematics paper in dynamical systems. No numerical free parameters are introduced. The central claims rest on standard background facts about mean dimension and entropy for symbolic systems and on the specific definition of the hub-and-spoke construction.

axioms (2)
  • domain assumption Mean dimension and entropy are well-defined for the open covers of the constructed Spoke(X) system and behave functorially under the hub-and-spoke replacement.
    Invoked when the paper associates Spoke(X) to X and states the implications for completely positive and uniformly positive mean dimension.
  • domain assumption Symbolic codings of irrational rotations on tori exist and can be used to produce systems whose covers have zero entropy and zero mean dimension before iteration.
    Used to construct the separating examples.
invented entities (1)
  • Spoke(X) hub-and-spoke system no independent evidence
    purpose: To transfer entropy and mean-dimension properties from a symbolic system X to a new system while allowing control over uniformity.
    The construction is introduced in the paper to prove the stated implications and to build the counterexamples; no independent evidence outside the paper is supplied.

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