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arxiv: 2604.06368 · v2 · submitted 2026-04-07 · 🧮 math.DS · math.OA

Paterson compactifications, inverse limits and shadowing for Deaconu-Renault systems

Daniel Gon\c{c}alves , Danilo Royer , Felipe Augusto Tasca This is my paper

Pith reviewed 2026-05-10 18:16 UTC · model grok-4.3

classification 🧮 math.DS math.OA
keywords Deaconu-Renault systemsPaterson compactificationshadowinginverse limitsultrametriclocal homeomorphismszero-dimensional spacessymbolic dynamics
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The pith

Shadowing in inverse-limit Deaconu-Renault systems is equivalent to shadowing in their compactified bases under uniform contraction on inverse branches.

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

The paper develops a metric and inverse-limit framework for Deaconu-Renault systems coming from local homeomorphisms on locally compact zero-dimensional spaces. It starts from Paterson-type compactifications of infinite product spaces, equips them with an explicit ultrametric, and builds an inverse-limit space that includes both infinite backward orbits and finite configurations that arise as their limits. For systems whose spaces admit tame defining sequences and that satisfy a separation property via uniformly contracting inverse branches, it proves that the inverse-limit extension has shadowing if and only if the compactified base system does. This supplies a concrete transfer mechanism that extends classical shadowing descriptions from compact zero-dimensional dynamics into the noncompact setting.

Core claim

Under the separation property expressed through uniformly contracting inverse branches, and for spaces admitting tame defining sequences, shadowing for the inverse-limit Deaconu-Renault system is equivalent to shadowing for the compactified base system. This produces a noncompact inverse-limit shadowing theory for a broad class of partially defined local-homeomorphism dynamics.

What carries the argument

The Paterson-type compactification of infinite product spaces, together with the associated inverse-limit space that incorporates finite configurations arising as limits of backward orbits, equipped with an explicit compatible ultrametric whose generalized cylinder topology is described explicitly.

If this is right

  • Shadowing admits a topological characterization directly in terms of the defining partitions of the ultrametric on the compactification.
  • The framework applies to one-sided shifts over infinite alphabets and to path spaces of graphs and higher-rank graphs.
  • The inverse-limit construction yields a canonical shift system that extends the original partially defined dynamics.
  • Shadowing questions for noncompact Deaconu-Renault systems can be reduced to the corresponding questions on their compactifications.

Where Pith is reading between the lines

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

  • The equivalence may allow verification of shadowing in infinite-alphabet symbolic systems by checking only the compact base.
  • Similar transfer arguments could apply to other noncompact systems whose inverse branches contract uniformly with respect to a suitable ultrametric.
  • Because the construction is tied to groupoid models, the shadowing equivalence might carry over to corresponding statements for the associated groupoids.

Load-bearing premise

The spaces admit tame defining sequences and the Deaconu-Renault system satisfies a separation property with uniformly contracting inverse branches.

What would settle it

A counterexample Deaconu-Renault system with tame defining sequences and uniformly contracting inverse branches in which the inverse-limit system has shadowing while the compactified base does not, or vice versa.

read the original abstract

We develop a new metric and inverse-limit framework for Deaconu-Renault systems arising from local homeomorphisms between open subsets of locally compact zero-dimensional spaces. Our starting point is the Paterson-type compactification of infinite product spaces, which underlies several symbolic and groupoid models, including one-sided shifts over infinite alphabets and path spaces of graphs and higher-rank graphs. We construct an explicit compatible ultrametric on this compactification and give a concrete description of its generalized cylinder topology and convergence. Within this framework, we introduce an inverse-limit-type space naturally associated to a Deaconu-Renault system. In contrast with the classical compact theory, the correct inverse-limit object must incorporate not only infinite backward orbits but also finite configurations arising as limits of such orbits. This produces a canonical shift system extending the original dynamics. We then study shadowing in the ultrametric setting. For spaces admitting tame defining sequences, we characterize shadowing in terms of the defining partitions, extending the topological description of shadowing from compact zero-dimensional dynamics to the locally compact setting relevant for Deaconu-Renault systems. As a main application, we prove a transfer theorem showing, under a separation property expressed through uniformly contracting inverse branches, that shadowing for the inverse-limit Deaconu-Renault system is equivalent to shadowing for the compactified base system. This provides a noncompact inverse-limit shadowing theory for a broad class of partially defined local-homeomorphism dynamics.

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 / 3 minor

Summary. The paper develops a metric and inverse-limit framework for Deaconu-Renault systems arising from local homeomorphisms on locally compact zero-dimensional spaces. It begins with a Paterson-type compactification of infinite product spaces, equips it with an explicit compatible ultrametric, and describes the associated cylinder topology. An inverse-limit-type space is constructed that incorporates both infinite backward orbits and finite configurations as limits thereof, yielding a canonical shift extension of the original dynamics. For spaces with tame defining sequences, shadowing is characterized in terms of the defining partitions. The main result is a transfer theorem: under a separation property expressed via uniformly contracting inverse branches, shadowing for the inverse-limit Deaconu-Renault system is equivalent to shadowing for the compactified base system.

Significance. If the constructions and transfer theorem are verified, the work supplies a noncompact inverse-limit shadowing theory for a broad class of partially defined local-homeomorphism dynamics. The explicit ultrametric on the Paterson compactification, the incorporation of finite configurations into the inverse-limit object, and the partition-based characterization of shadowing extend classical compact zero-dimensional results in a concrete way. These tools are likely to be useful for symbolic dynamics over infinite alphabets, path spaces of graphs, and groupoid models.

major comments (2)
  1. [Transfer theorem and separation property] The transfer theorem (stated in the abstract and presumably proved in the final section) equates shadowing on the inverse-limit system to shadowing on the compactified base only after assuming both tame defining sequences and the separation property with uniformly contracting inverse branches. The manuscript should clarify whether the separation property is preserved under the inverse-limit construction or must be checked separately on the base; if the latter, an explicit non-trivial example verifying the property would strengthen the applicability claim.
  2. [Inverse-limit space construction] In the construction of the inverse-limit-type space, the incorporation of finite configurations as limits of backward orbits is central to extending the dynamics beyond the classical compact case. The proof that the resulting shift system is well-defined and that its shadowing property transfers correctly should explicitly address potential issues with the ultrametric compatibility at these finite points.
minor comments (3)
  1. [Abstract] The abstract refers to 'tame defining sequences' without a forward reference; a brief definition or pointer to the relevant section would aid readers unfamiliar with the term.
  2. [Ultrametric and cylinder topology] Notation for the ultrametric and the generalized cylinder sets should be introduced consistently; ensure that the same symbols are used in the topology description and in the shadowing characterization.
  3. [Inverse-limit framework] A short comparison table or diagram contrasting the new inverse-limit object with the classical compact inverse limit would clarify the role of the finite configurations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive suggestions. We address the two major comments below, indicating the revisions that will be incorporated into the revised manuscript.

read point-by-point responses

  1. Referee: [Transfer theorem and separation property] The transfer theorem (stated in the abstract and presumably proved in the final section) equates shadowing on the inverse-limit system to shadowing on the compactified base only after assuming both tame defining sequences and the separation property with uniformly contracting inverse branches. The manuscript should clarify whether the separation property is preserved under the inverse-limit construction or must be checked separately on the base; if the latter, an explicit non-trivial example verifying the property would strengthen the applicability claim.

    Authors: We agree that the distinction requires explicit clarification. The separation property (uniform contraction of inverse branches together with the separation condition) is imposed on the original Deaconu-Renault system and its compactified base; it is not automatically inherited by the inverse-limit space and must be verified on the base. We will revise the statement of the transfer theorem (Theorem 5.3) and the surrounding discussion to make this explicit. In addition, we will insert a short subsection containing a non-trivial example: the one-sided shift on the infinite alphabet {0,1,2,…} equipped with the ultrametric d(x,y)=2^{-min{n:x_n≠y_n}} and the natural contracting inverse branches; this system satisfies the separation property on the base and therefore transfers shadowing to its inverse-limit extension. revision: yes

  2. Referee: [Inverse-limit space construction] In the construction of the inverse-limit-type space, the incorporation of finite configurations as limits of backward orbits is central to extending the dynamics beyond the classical compact case. The proof that the resulting shift system is well-defined and that its shadowing property transfers correctly should explicitly address potential issues with the ultrametric compatibility at these finite points.

    Authors: We acknowledge that the ultrametric compatibility at the newly added finite points deserves a dedicated paragraph. The finite configurations arise as limits in the Paterson compactification, and the ultrametric is defined so that it extends continuously to these points (see the explicit formula in Section 3). The shift map remains continuous at these points by construction, and the shadowing transfer argument uses only the uniform contraction and the cylinder topology, both of which remain valid at finite points. Nevertheless, we will add a new remark immediately after the definition of the inverse-limit space (Definition 4.2) that explicitly verifies the ultrametric inequalities and the continuity of the shift at finite configurations, thereby removing any ambiguity. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper constructs a Paterson-type compactification with an explicit ultrametric, defines an associated inverse-limit space that incorporates finite orbit limits, characterizes shadowing via tame defining sequences and partitions (extending the compact zero-dimensional case), and proves a conditional equivalence for shadowing transfer under a separation property with uniformly contracting inverse branches. All steps are presented as direct topological constructions and proofs rather than reductions to fitted inputs, self-definitions, or load-bearing self-citations. No equations or claims reduce by construction to prior inputs within the paper; the framework relies on standard external references (e.g., Paterson compactification) without smuggling ansatzes or renaming known results as new derivations. The central equivalence is a proved theorem, not an identity by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

Central claims rest on standard assumptions from topology and dynamics plus newly defined objects; no data-fitting parameters.

axioms (2)
  • domain assumption Paterson compactification admits a compatible ultrametric with explicit cylinder topology
    Invoked to ground the metric framework and convergence description.
  • domain assumption Existence of tame defining sequences for the spaces under consideration
    Required for the topological characterization of shadowing.
invented entities (2)
  • Inverse-limit-type space incorporating finite configurations no independent evidence
    purpose: Canonical extension of the dynamics that includes limits of backward orbits which are finite sequences
    Newly constructed object central to the framework
  • Explicit ultrametric on the Paterson compactification no independent evidence
    purpose: Concrete metric inducing the generalized cylinder topology
    Constructed in the paper to make the compactification usable for dynamics

pith-pipeline@v0.9.0 · 5564 in / 1490 out tokens · 67992 ms · 2026-05-10T18:16:44.891946+00:00 · methodology

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Reference graph

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13 extracted references · 13 canonical work pages

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