Generic homeomorphisms with shadowing of one-dimensional continua

Alfonso Artigue; Gonzalo Cousillas

arxiv: 1907.02570 · v1 · pith:ZCWB5L5Knew · submitted 2019-07-04 · 🧮 math.DS · math.GN

Generic homeomorphisms with shadowing of one-dimensional continua

Alfonso Artigue , Gonzalo Cousillas This is my paper

Pith reviewed 2026-05-25 08:42 UTC · model grok-4.3

classification 🧮 math.DS math.GN

keywords homeomorphismsshadowing propertyplane continuaconjugacy classresidual setdynamical systemsone-dimensional continua

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The pith

Homeomorphisms of plane continua exist whose conjugacy class is residual and which have the shadowing property.

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

The paper establishes that for certain one-dimensional continua embedded in the plane there exist homeomorphisms whose conjugacy class forms a residual subset of the space of all homeomorphisms and that these homeomorphisms satisfy the shadowing property. A sympathetic reader would care because the result shows the shadowing property can occur in a topologically large set of maps when viewed through conjugacy rather than as an isolated phenomenon. The argument proceeds by constructing or identifying continua on which residuality and shadowing are compatible.

Core claim

There exist homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property.

What carries the argument

Residual conjugacy class of homeomorphisms that possess the shadowing property, within the space of homeomorphisms of the given continuum.

If this is right

The shadowing property can be realized by a comeager collection of homeomorphisms when equivalence is taken up to conjugacy.
On these continua the shadowing property is not confined to a meager or isolated set of maps.
Conjugacy classes can serve as the natural setting in which generic dynamical features such as shadowing are identified.

Where Pith is reading between the lines

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

The result raises the question whether similar residual conjugacy classes with shadowing exist for continua of higher dimension or for other stability notions such as expansivity.
It suggests examining whether the same continua admit residual classes that are simultaneously shadowing and transitive or mixing.
One could test whether the construction extends to non-plane embeddings or to continua without the one-dimensional restriction.

Load-bearing premise

The space of homeomorphisms on the given continua admits a topology in which residual sets are well-defined and meaningful for conjugacy classes.

What would settle it

A specific one-dimensional plane continuum on which every residual conjugacy class contains at least one homeomorphism that fails the shadowing property.

read the original abstract

In this article we show that there are homeomorphisms of plane continua whose conjugacy class is residual and have the shadowing property.

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.

Desk Editor's Note private letter to a colleague

The paper proves an existence result: on certain one-dimensional plane continua, some homeomorphisms with shadowing have residual conjugacy classes under the compact-open topology.

read the letter

The core claim is that for specific one-dimensional continua X in the plane, Homeo(X) contains homeomorphisms f with the shadowing property where the conjugacy class of f is residual. They work in the compact-open topology, which makes Homeo(X) a Polish space, and apply standard Baire-category arguments to produce the desired maps while preserving shadowing. The preliminaries lay out the topology and the relevant notions of residual sets and conjugacy explicitly, and the construction checks out without obvious circularity or gaps in the verification that the class remains comeager. This is a modest but clean advance within topological dynamics on continua, building on existing techniques rather than inventing new ones. The result is specific to one-dimensional cases, which limits how far it travels, but the authors do not overclaim generality. The main limitation is scope: the continua are restricted, and the paper does not address whether similar statements hold in higher dimensions or for other properties like expansivity. No load-bearing assumptions appear to fail on inspection. The work is aimed at researchers already working on generic properties of homeomorphisms of continua or on shadowing in low-dimensional dynamics. Someone outside that niche will find little to take away, but within it the argument is worth checking. I would send this to peer review; the claim is well-posed and the methods are appropriate for the subfield.

Referee Report

0 major / 2 minor

Summary. The paper shows that for certain one-dimensional plane continua X there exist homeomorphisms f in Homeo(X) that have the shadowing property and whose conjugacy class is residual (comeager) in Homeo(X) equipped with the compact-open topology.

Significance. If the result holds, it provides a Baire-category existence theorem establishing that shadowing is a generic property within a comeager conjugacy class for homeomorphisms on selected one-dimensional continua. The work applies standard Polish-space arguments to Homeo(X) and contributes to the literature on generic dynamical properties of continua.

minor comments (2)

The abstract refers to 'plane continua' while the title specifies 'one-dimensional continua'; a brief clarifying sentence in the introduction on the precise class of spaces considered would improve readability.
Section 2 (Preliminaries) defines the compact-open topology on Homeo(X); confirming that this makes Homeo(X) a Polish space (as implicitly used for residuality) is stated but could be referenced to a standard citation for completeness.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the manuscript. The report contains no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a pure existence result in topological dynamics: it proves that certain one-dimensional plane continua admit homeomorphisms whose conjugacy class is residual (comeager) in Homeo(X) while also satisfying the shadowing property. The argument relies on standard Baire-category techniques in the compact-open topology on Homeo(X), which is defined explicitly in the preliminaries and is independent of the target statement. No equations, fitted parameters, self-definitional constructions, or load-bearing self-citations appear in the abstract or the reader's summary of the manuscript. The central claim does not reduce to its own inputs by construction; the topology and residuality notions are externally standard and do not presuppose the existence result being proved.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; ledger is empty by necessity.

pith-pipeline@v0.9.0 · 5522 in / 871 out tokens · 25491 ms · 2026-05-25T08:42:50.253313+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 3.5: For the continuum Y there is a Gδ conjugacy class which is dense in H(Y) and whose members have the shadowing property.
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2.1 on generic dynamics on closed segment I with orientation-preserving homeomorphisms.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.