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arxiv: 2605.17473 · v1 · pith:UCL4CFONnew · submitted 2026-05-17 · 🧮 math.DS

Metric mean dimension of factor maps

Rui Yang This is my paper

Pith reviewed 2026-05-19 22:32 UTC · model grok-4.3

classification 🧮 math.DS
keywords metric mean dimensionfactor mapsweighted topological entropyrelative conditional entropyvariational principlerandom dynamical systemsAbramov-Rokhlin formula
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The pith

Factor maps with infinite weighted topological entropy are characterized using three types of weighted metric mean dimensions that relate to those of the factor and extension systems.

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

This paper develops metric mean dimension tools for factor maps in dynamical systems where entropy is infinite. Three weighted versions are defined to quantify the maps, compared against the metric mean dimensions of the connected systems, and supported by variational principles. A relative conditional metric mean dimension is introduced for cases of infinite relative topological conditional entropy and shown to match the relative metric mean dimension. In random dynamical systems the natural projection is treated as a one-Lipschitz map, and a random average metric mean dimension yields a topological Abramov-Rokhlin formula that relates the dimensions of the driving system, skew product, and associated non-autonomous systems.

Core claim

For factor maps with infinite weighted topological entropy, three types of weighted metric mean dimensions characterize the maps and satisfy comparisons with the factor and extension systems together with variational principles; relative conditional metric mean dimension coincides with relative metric mean dimension when relative topological conditional entropy is infinite; in random systems a random average metric mean dimension establishes a topological Abramov-Rokhlin formula for the projection, producing an inequality among the metric mean dimensions of the driving system, skew product, and inherent non-autonomous systems.

What carries the argument

weighted metric mean dimension (three types) and relative conditional metric mean dimension for factor maps with infinite entropy

If this is right

  • The weighted metric mean dimensions provide consistent comparisons between the complexity of a factor map and the complexities of its factor and extension systems.
  • Variational principles hold, allowing the weighted dimensions to be expressed via invariant measures.
  • The coincidence of relative conditional and relative metric mean dimensions unifies two approaches to measuring relative complexity under infinite conditional entropy.
  • The topological Abramov-Rokhlin formula yields concrete inequalities relating metric mean dimensions across driving systems, skew products, and non-autonomous systems in the random setting.

Where Pith is reading between the lines

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

  • These notions could be applied to classify factor maps in concrete infinite-entropy systems such as certain subshifts or geodesic flows on non-compact manifolds.
  • Similar weighted or relative constructions might extend other dimension-like invariants, such as upper box dimension, to factor maps.
  • The random average version could be tested on explicit random dynamical systems to check whether the derived inequalities become equalities under additional mixing assumptions.

Load-bearing premise

The underlying dynamical systems possess infinite weighted topological entropy or infinite relative topological conditional entropy, so the new dimensions are required and can be meaningfully compared without reducing to finite-entropy cases.

What would settle it

A concrete factor map with infinite weighted topological entropy for which at least one of the three weighted metric mean dimensions fails to satisfy the stated comparison inequalities with the factor and extension systems, or a map with infinite relative topological conditional entropy where the relative conditional metric mean dimension differs from the relative metric mean dimension.

read the original abstract

Metric mean dimension is a metric-depedent quantity to characterize the topological complexity of systems with infinite topological entropy. In this paper, we investigate metric mean dimension of factor maps. (1) We introduce three types of weighted metric mean dimensions to characterize factor maps with infinite weighted topological entropy, and compare them with the metric mean dimensions of the factor system and the extension system. Furthermore, we establish variational principles for weighted metric mean dimension. (2) We introduce relative conditional metric mean dimension for factor maps with infinite relative topological conditional entropy, and prove that it coincides with relative metric mean dimension. (3) In the context of random dynamical systems, the natural projection from the skew product to its driving system is a one-Lipschitz map. We introduce random average metric mean dimension and use it to establish a topological Abramov-Rokhlin formula for the certain one-Lipschitz map. As an application, we obtain an inequality that links the metric mean dimensions of the driving system, the skew product system, and the inherent non-autonomous dynamical systems from the random dynamical system.

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 / 4 minor

Summary. The paper investigates metric mean dimension for factor maps in dynamical systems. It introduces three types of weighted metric mean dimensions to characterize factor maps with infinite weighted topological entropy, compares them with the metric mean dimensions of the factor and extension systems, and establishes variational principles. It defines relative conditional metric mean dimension for factor maps with infinite relative topological conditional entropy and proves that it coincides with relative metric mean dimension. In the setting of random dynamical systems, it introduces random average metric mean dimension and uses it to establish a topological Abramov-Rokhlin formula for the one-Lipschitz projection from the skew product to the driving system, yielding an inequality relating the metric mean dimensions of the driving system, skew product, and associated non-autonomous systems.

Significance. If the results hold, this work meaningfully extends metric mean dimension theory to factor maps and random dynamical systems in the infinite-entropy regime. The variational principles connect the new weighted dimensions to measure-theoretic quantities, while the coincidence result for the relative conditional case simplifies analysis. The topological Abramov-Rokhlin formula provides a concrete relation for one-Lipschitz maps in random systems and leads to a useful inequality across driving, skew-product, and non-autonomous components. These contributions strengthen the toolkit for studying complexity in systems where standard entropy is infinite.

minor comments (4)
  1. Abstract: 'metric-depedent' is a typo and should be 'metric-dependent'.
  2. Introduction: The motivation section would benefit from a brief concrete example of a factor map exhibiting infinite weighted topological entropy to illustrate why the weighted extensions are needed.
  3. Section on random dynamical systems (likely §6 or §7): In the statement and proof of the topological Abramov-Rokhlin formula, explicitly note whether the one-Lipschitz property of the projection is used only for the inequality or is essential to the dimension equality itself.
  4. Notation throughout: The distinction between the three weighted metric mean dimensions could be made clearer by adding a short comparison table or diagram summarizing their definitions and relations to the factor/extension dimensions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and for highlighting its significance in extending metric mean dimension theory to factor maps with infinite entropy and to random dynamical systems. We appreciate the recommendation for minor revision and will incorporate any editorial improvements in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained extensions of standard definitions

full rationale

The paper defines three weighted metric mean dimensions, relative conditional metric mean dimension, and random average metric mean dimension as direct extensions of metric mean dimension using weighted/conditional covers and limits in the infinite-entropy regime. It then proves comparisons to factor/extension systems, variational principles, coincidence with relative metric mean dimension, and a topological Abramov-Rokhlin formula via the one-Lipschitz projection. All steps rely on explicit constructions and limits rather than reducing by the paper's own equations to fitted inputs, self-citations, or ansatzes. No load-bearing claim is equivalent to its inputs by construction, and the work remains internally consistent without hidden reductions to finite-entropy cases.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review relies only on the abstract; the paper appears to rest on standard definitions of metric mean dimension and topological entropy from prior dynamical systems literature, with new weighted and relative variants introduced as definitions rather than derived from axioms.

axioms (1)
  • standard math Standard properties of metric spaces and continuous maps on compact spaces hold for the dynamical systems under study.
    Invoked implicitly when defining metric mean dimension and factor maps.

pith-pipeline@v0.9.0 · 5706 in / 1393 out tokens · 44713 ms · 2026-05-19T22:32:57.775553+00:00 · methodology

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