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arxiv: 2604.15818 · v1 · submitted 2026-04-17 · 🧮 math.DS

Amorphic complexity and entropy for symbolic model sets

Jamal Drewlo This is my paper

Pith reviewed 2026-05-10 08:05 UTC · model grok-4.3

classification 🧮 math.DS
keywords amorphic complexitytopological entropymodel setssymbolic subshiftsWeyl pseudometricmaximal equicontinuous factorcut-and-project schemes
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The pith

Subshifts generated by model sets have a continuous Weyl pseudometric that enables constructions with independent behaviors in entropy, amorphic complexity, and the maximal equicontinuous factor.

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

The paper proves that the Weyl pseudometric remains continuous when restricted to subshifts arising from model sets. This continuity is then applied to build several families of subshifts that realize different combinations of values for topological entropy, amorphic complexity, and the structure of their maximal equicontinuous factor. A sympathetic reader would care because the constructions demonstrate that these three dynamical quantities are not forced to vary together even inside geometrically regular symbolic systems. The work supplies explicit examples in which entropy and amorphic complexity can be adjusted separately while the equicontinuous factor stays fixed or changes in a controlled manner.

Core claim

The central claim is that the Weyl pseudometric is continuous on subshifts generated by model sets. This fact is used to construct multiple subshifts that exhibit different behavior with respect to entropy, amorphic complexity, and their maximal equicontinuous factor.

What carries the argument

The continuity of the Weyl pseudometric on model set subshifts, which serves as the technical device that permits the construction of examples separating entropy from amorphic complexity relative to the maximal equicontinuous factor.

If this is right

  • Subshifts exist in which positive entropy occurs together with zero amorphic complexity and a prescribed maximal equicontinuous factor.
  • Subshifts exist in which zero entropy occurs together with positive amorphic complexity and a prescribed maximal equicontinuous factor.
  • The maximal equicontinuous factor can be held fixed while entropy and amorphic complexity are varied independently across different constructions.
  • These examples show that entropy and amorphic complexity are not rigidly determined by the maximal equicontinuous factor inside the class of model set subshifts.

Where Pith is reading between the lines

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

  • The same continuity technique may be useful for constructing examples that separate other dynamical invariants in symbolic systems with long-range order.
  • Model set subshifts could serve as a test bed for studying how geometric regularity from cut-and-project schemes constrains or liberates complexity measures.
  • The constructions might be adapted to produce symbolic systems whose orbit closures have prescribed properties in both topological and measure-theoretic senses.

Load-bearing premise

The subshifts must be generated by model sets via a cut-and-project scheme that preserves continuity of the Weyl pseudometric.

What would settle it

A concrete model set subshift on which the Weyl pseudometric fails to be continuous, or an attempted construction in which entropy, amorphic complexity, and the maximal equicontinuous factor cannot be made to vary independently.

Figures

Figures reproduced from arXiv: 2604.15818 by Jamal Drewlo.

Figure 1
Figure 1. Figure 1: The construction of W after 2 steps. The green area describes U1 ∪ U2 and the red area is V1 ∪ V2. The fact that qn+1 = 2qn−1 readily implies #M (l) n = qn+1/2 so that ν(U (l) n ) = ν(V (l) n ) = ν([q1 − 1, . . . , qn−1 − 1, l])/2. An easy induction yields H \ Sn j=1(Uj ∪ Vj ) = [q1 − 1, . . . , qn − 1]. We now define (similarly to the proof of Proposition 5.5) U = S n∈N Un, V = S n∈N Vn and W = U, so that… view at source ↗
read the original abstract

We show a continuity result for the Weyl pseudometric on subshifts which are generated by model sets. This fact is then used for multiple constructions of subshifts that exhibit different behavior regarding entropy, amorphic complexity and their maximal equicontinuous factor.

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

2 major / 1 minor

Summary. The manuscript establishes a continuity result for the Weyl pseudometric on subshifts generated by model sets via cut-and-project schemes. This result is then applied to construct multiple families of subshifts realizing distinct combinations of topological entropy, amorphic complexity, and the structure of their maximal equicontinuous factor.

Significance. If the continuity theorem holds under clearly stated hypotheses on the model sets and the constructions are shown to satisfy those hypotheses, the work supplies concrete symbolic examples that separate entropy, amorphic complexity, and equicontinuous factors in the setting of aperiodic order. Such examples would be useful for clarifying the relationships among these invariants beyond the classical entropy theory.

major comments (2)
  1. [§3] §3 (continuity theorem): The statement of the continuity result for the Weyl pseudometric must include the precise hypotheses on the cut-and-project data (compactness and regularity of the window, properties of the lattice, etc.) that are required for the proof.
  2. [§5] §5 (constructions): Each constructed subshift must be accompanied by an explicit check that its underlying model set satisfies the hypotheses of the continuity theorem; without this verification the claims that the examples exhibit different combinations of entropy, amorphic complexity, and maximal equicontinuous factor rest on an unverified transfer of the continuity property.
minor comments (1)
  1. The abstract is terse and does not indicate the hypotheses under which the continuity result holds; a single sentence summarizing the required conditions on the model sets would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (continuity theorem): The statement of the continuity result for the Weyl pseudometric must include the precise hypotheses on the cut-and-project data (compactness and regularity of the window, properties of the lattice, etc.) that are required for the proof.

    Authors: We agree that the theorem statement should make the hypotheses explicit. The proof in Section 3 relies on compactness and regularity of the window together with standard discreteness and cocompactness properties of the lattice. In the revised manuscript we have updated the statement of the continuity theorem to list these conditions verbatim, so that the result is stated under precisely the hypotheses used in the argument. revision: yes

  2. Referee: [§5] §5 (constructions): Each constructed subshift must be accompanied by an explicit check that its underlying model set satisfies the hypotheses of the continuity theorem; without this verification the claims that the examples exhibit different combinations of entropy, amorphic complexity, and maximal equicontinuous factor rest on an unverified transfer of the continuity property.

    Authors: We accept the point. For each family of subshifts constructed in Section 5 we have added a short verification paragraph confirming that the corresponding model sets meet the hypotheses now stated in the continuity theorem (regularity of the window, lattice properties, etc.). These checks ensure the continuity result applies directly and thereby justify the claimed distinctions among entropy, amorphic complexity, and maximal equicontinuous factors. revision: yes

Circularity Check

0 steps flagged

No circularity: continuity theorem followed by independent constructions

full rationale

The paper states a continuity result for the Weyl pseudometric on subshifts generated by model sets and then applies this result to construct examples exhibiting distinct combinations of entropy, amorphic complexity, and maximal equicontinuous factor. No derivation step reduces by definition to its own inputs, no fitted parameters are relabeled as predictions, and no load-bearing claims rest on self-citations or imported uniqueness theorems. The abstract and described structure indicate a standard theorem-plus-applications chain with independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are identifiable.

pith-pipeline@v0.9.0 · 5311 in / 1145 out tokens · 32951 ms · 2026-05-10T08:05:37.448347+00:00 · methodology

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Reference graph

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