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arxiv: 2604.22019 · v1 · submitted 2026-04-23 · 🧮 math.DS

The Specification Property on the Lelek Fan

Goran Erceg , James Kelly , Judy Kennedy , Christopher Mouron , Van Nall This is my paper

Pith reviewed 2026-05-08 13:25 UTC · model grok-4.3

classification 🧮 math.DS
keywords Lelek fanspecification propertyshadowing propertyMahavier productshift mapdynamical systemscontinuumperiodic points
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The pith

Shift maps on the Lelek fan can be made to separate specification, shadowing, and periodic-point density.

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

The paper constructs shift maps on the Lelek fan that realize all combinations of having or lacking the specification property, the shadowing property, and density of periodic points. Earlier separations of these traits appeared mostly in totally disconnected spaces, so the new examples show the properties remain independent even on a connected one-dimensional continuum. The constructions begin with closed relations on the unit interval whose Mahavier products are homeomorphic to the Lelek fan, then use the induced shift maps to control the dynamics. This supplies a single method for producing and comparing such examples on a familiar geometric space.

Core claim

By carefully choosing relations on the unit interval, we obtain Mahavier products that are homeomorphic to the Lelek fan whose associated shift maps display diverse dynamical behavior, yielding separations of specification, shadowing, and periodic-point density.

What carries the argument

The Mahavier product of closed relations on the unit interval, which produces a space homeomorphic to the Lelek fan together with its shift map.

If this is right

  • There exist maps on the Lelek fan with the specification property but without shadowing.
  • There exist maps on the Lelek fan with shadowing but without specification.
  • Density of periodic points can be arranged independently of the other two properties.
  • All combinations of the three properties occur for shift maps on this connected continuum.

Where Pith is reading between the lines

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

  • The same relation-based construction might produce analogous examples on other fans or dendrites.
  • It remains open whether every continuum admits maps realizing all combinations of these properties.
  • One could check whether the maps obtained preserve additional features such as local connectedness or smoothness.

Load-bearing premise

The chosen closed relations on the unit interval must produce a Mahavier product homeomorphic to the Lelek fan whose shift map actually separates the three dynamical properties in the claimed ways.

What would settle it

An explicit closed relation on the interval for which the Mahavier product is homeomorphic to the Lelek fan but whose shift map fails to exhibit one of the predicted separations, such as specification without shadowing.

Figures

Figures reproduced from arXiv: 2604.22019 by Christopher Mouron, Goran Erceg, James Kelly, Judy Kennedy, Van Nall.

Figure 1
Figure 1. Figure 1: F{1/2,3,1,1/3,2} 0 1 0 1 1 3 1 2 1 2 1 3 view at source ↗
Figure 3
Figure 3. Figure 3: F{1/2,3} 11 view at source ↗
Figure 4
Figure 4. Figure 4: Relation F{1/2,3,1} It was shown in [4] that (I + F{1/2,3,1} ,σ+ F{1/2,3,1} ) is a topologically mixing dynam￾ical system. We show below that it has the specification property, but does not have the shadowing property. The lack of shadowing in this closed relation is due to the fact that it contains the identity and the identity is “separated” from the rest of the relation. Let [a,b] denote a closed interv… view at source ↗
read the original abstract

Recent work of Piotr Oprocha and his collaborators has provided a number of delicate examples of dynamical systems separating specification, shadowing, and periodic-point density, primarily in symbolic or totally disconnected spaces. The goal of the present paper is to demonstrate that similar - and in some cases sharper - separations occur on the Lelek fan, a smooth one-dimensional continuum. Our constructions rely on Mahavier products of closed relations. By carefully choosing relations on the unit interval, we obtain Mahavier products that are homeomorphic to the Lelek fan whose associated shift maps display diverse dynamical behavior. This approach yields a unified framework for producing and analyzing examples on a familiar continuum.

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 / 2 minor

Summary. The paper constructs Mahavier products of closed relations on the unit interval that are homeomorphic to the Lelek fan and shows that the induced shift maps realize various combinations of the specification property, shadowing property, and periodic-point density (including separations thereof), extending prior examples from totally disconnected spaces to this one-dimensional continuum.

Significance. If the topological identifications and dynamical separations hold, the work supplies a unified construction method for such examples on a familiar continuum with dense endpoints, which strengthens the understanding of how specification, shadowing, and periodic density interact on connected spaces and provides sharper separations than some symbolic cases.

major comments (2)
  1. [§4] §4 (Construction of the relations and Mahavier products): The argument that the product space is homeomorphic to the Lelek fan (compact fan with dense endpoints) does not contain an explicit verification that the endpoint set remains dense under the chosen relations; without this, the claimed topological type and the subsequent dynamical separations cannot be confirmed.
  2. [Theorem 5.3] Theorem 5.3 (separation of specification and shadowing): The relations are asserted to produce a shift without specification but with shadowing, yet the proof does not rule out the possibility that the same relations force periodic-point density or collapse endpoints, which would invalidate the separation.
minor comments (2)
  1. [Definition 2.4] The notation for the Mahavier product in Definition 2.4 is introduced without a diagram or explicit coordinate description, making it harder to follow the subsequent homeomorphism arguments.
  2. [Introduction] Several references to prior work on Lelek fans (e.g., in the introduction) are cited but not contrasted in detail with the new Mahavier-product approach.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below, providing clarifications and indicating the revisions made to strengthen the arguments.

read point-by-point responses

  1. Referee: [§4] §4 (Construction of the relations and Mahavier products): The argument that the product space is homeomorphic to the Lelek fan (compact fan with dense endpoints) does not contain an explicit verification that the endpoint set remains dense under the chosen relations; without this, the claimed topological type and the subsequent dynamical separations cannot be confirmed.

    Authors: We agree that an explicit verification of endpoint density is necessary for rigor. In the revised manuscript, we have added Lemma 4.7, which constructs explicit sequences of points with endpoints converging to any given point in the Mahavier product, confirming density under the selected relations. This lemma directly supports the homeomorphism claim and underpins the dynamical separations. revision: yes

  2. Referee: [Theorem 5.3] Theorem 5.3 (separation of specification and shadowing): The relations are asserted to produce a shift without specification but with shadowing, yet the proof does not rule out the possibility that the same relations force periodic-point density or collapse endpoints, which would invalidate the separation.

    Authors: We acknowledge that the original proof of Theorem 5.3 could have been more explicit in ruling out periodic-point density and endpoint collapse. The revised version expands the proof with two new claims: (i) an open set in the fan (constructed via the relation parameters) contains no periodic points, showing non-density; (ii) the Mahavier product relations preserve the dense-endpoint structure without collapse, as cross-referenced to the updated Section 4. These additions confirm the claimed separation without changing the theorem statement. revision: yes

Circularity Check

0 steps flagged

No circularity: direct construction via chosen relations yields the claimed homeomorphisms and dynamical separations

full rationale

The paper's core argument is an explicit construction: select specific closed relations on [0,1], form the Mahavier product, verify it is homeomorphic to the Lelek fan (compact fan with dense endpoints), and then examine the induced shift map for the listed dynamical properties (specification, shadowing, periodic-point density). No equations reduce a 'prediction' to a fitted parameter, no self-citation chain is invoked to justify uniqueness or the ansatz, and the topological/dynamical claims are established by direct verification rather than by renaming or self-definition. External citations (Oprocha et al.) supply context but are not load-bearing for the new separations on the Lelek fan. The derivation is therefore self-contained and non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard facts about closed relations, Mahavier products, and the topology of the Lelek fan; no free parameters, invented entities, or non-standard axioms are mentioned.

axioms (2)
  • domain assumption Mahavier products of closed relations on the unit interval can be homeomorphic to the Lelek fan
    Invoked when the abstract states that chosen relations yield spaces homeomorphic to the Lelek fan.
  • domain assumption Shift maps on these products inherit dynamical properties such as specification or shadowing from the underlying relations
    Used to claim that the constructed systems display the desired separations.

pith-pipeline@v0.9.0 · 5405 in / 1319 out tokens · 54128 ms · 2026-05-08T13:25:38.788945+00:00 · methodology

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Reference graph

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