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arxiv: 2604.11980 · v1 · submitted 2026-04-13 · 🧮 math.DS

Mean dimension of general iterated function systems

Welington Cordeiro , Maria Jos\'e Pacifico , Xuan Zhang This is my paper

Pith reviewed 2026-05-10 15:24 UTC · model grok-4.3

classification 🧮 math.DS
keywords mean dimensionmetric mean dimensiongeneralized iterated function systemssmall boundary propertygluing orbit propertytopological entropydynamical systems
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The pith

In generalized iterated function systems, mean dimension is bounded above by the lower and upper metric mean dimensions and equals zero under the small boundary property.

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

The paper introduces mean dimension and metric mean dimension as invariants for generalized iterated function systems. It proves that mean dimension never exceeds the lower or upper metric mean dimension. Systems satisfying the small boundary property are shown to have vanishing mean dimension. The work also defines a gluing orbit property and proves that, together with transitivity and non-rigidity, this property forces positive topological entropy. These relations give concrete ways to bound or eliminate certain complexity measures in such systems.

Core claim

In generalized iterated function systems, the mean dimension is bounded above by the lower metric mean dimension and the upper metric mean dimension. Generalized IFS possessing the small boundary property have zero mean dimension. The gluing orbit property, under suitable transitivity and non-rigidity assumptions, guarantees positive topological entropy.

What carries the argument

The small boundary property and gluing orbit property defined on generalized iterated function systems, which control mean dimension and connect it to topological entropy.

Load-bearing premise

The notions of generalized iterated function systems, small boundary property, and gluing orbit property are defined so that the stated bounds and entropy conclusions apply directly to the dynamics they describe.

What would settle it

Constructing or observing a concrete generalized iterated function system that satisfies the small boundary property yet has positive mean dimension would disprove the zero mean dimension claim.

read the original abstract

In this paper, we introduce and investigate the notions of Mean Dimension and Metric Mean Dimension for generalized iterated function systems (IFS). We establish basic properties of these invariants and prove that Mean Dimension is always bounded above by the lower Metric Mean Dimension and the upper Metric Mean Dimension in this setting. We further show that generalized iterated function systems with the Small Boundary Property have zero Mean Dimension. Finally, we introduce a Gluing Orbit Property for generalized iterated function systems and prove that, under suitable transitivity and non-rigidity assumptions, it guarantees positive topological entropy.

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 / 3 minor

Summary. The manuscript introduces mean dimension and metric mean dimension for generalized iterated function systems (gIFS). It proves that mean dimension is bounded above by both the lower and upper metric mean dimensions. It further establishes that gIFS satisfying the small boundary property have zero mean dimension. Finally, it defines a gluing orbit property for gIFS and shows that, under suitable transitivity and non-rigidity assumptions, this property implies positive topological entropy.

Significance. If the results hold, the work extends classical mean-dimension inequalities and zero-mean-dimension theorems under the small-boundary property to the setting of generalized IFS, while also linking a gluing-orbit condition to positive entropy. These extensions supply new invariants and criteria for analyzing dimension and entropy in a broader class of iterated systems, building directly on standard dynamical-systems tools without introducing free parameters or circular definitions.

minor comments (3)
  1. In the introduction, the notation distinguishing lower and upper metric mean dimension from ordinary mean dimension should be introduced with a brief comparison table or explicit inequalities to aid readers unfamiliar with the generalized IFS setting.
  2. Section 3 (definitions of gIFS and small boundary property): the statement that the small boundary property implies zero mean dimension would benefit from an explicit reference to the classical result being generalized, even if the proof is self-contained.
  3. The gluing orbit property is defined in Section 5; a short remark comparing it to the classical gluing orbit property for single maps would clarify the extension.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the positive assessment, including the recommendation to accept. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper defines Mean Dimension and Metric Mean Dimension for generalized IFS, then derives the inequality Mean Dimension ≤ lower/upper Metric Mean Dimension directly from those definitions and standard dynamical properties. It further shows zero Mean Dimension under the Small Boundary Property and positive entropy under the Gluing Orbit Property plus transitivity/non-rigidity, all following from the newly fixed definitions without any fitted parameters, self-referential equations, or load-bearing self-citations that reduce the claims to their inputs by construction. The derivation chain is self-contained against the introduced notions and classical facts.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard definitions from dynamical systems and ergodic theory (e.g., topological entropy, transitivity) plus newly introduced notions whose consistency is assumed. No free parameters or invented physical entities appear; the work is definitional and proof-based.

axioms (2)
  • standard math Standard properties of topological entropy and transitivity in dynamical systems hold for generalized IFS.
    Invoked implicitly when linking the gluing orbit property to positive entropy under transitivity and non-rigidity.
  • domain assumption The small boundary property is a well-defined and applicable condition for generalized IFS.
    Used to conclude zero mean dimension; its precise formulation is not detailed in the abstract.

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Reference graph

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