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arxiv: 2408.04779 · v3 · submitted 2024-08-08 · 🧮 math.NT · math.DS

Shadowing and Stability of Non-Invertible p-adic Dynamics

D.A. Caprio , F. Lenarduzzi , A. Messaoudi , I. Tsokanos This is my paper

Pith reviewed 2026-05-23 21:55 UTC · model grok-4.3

classification 🧮 math.NT math.DS
keywords p-adic dynamicsshadowing propertytopological stabilitynon-invertible mapscontractionsZ_pQ_pright-invertible maps
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The pith

Non-invertible p-adic maps that are right-invertible through contractions or left-invertible contractions exhibit strong shadowing and topological stability.

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

The paper establishes sufficient conditions for strong shadowing and stability in non-invertible dynamics on the p-adic integers and numbers. It targets two families: maps right-invertible via contractions and left-invertible contractions. A sympathetic reader cares because these conditions extend the known Walters-type connection between shadowing and stability from invertible systems on positive-dimensional spaces to the zero-dimensional non-invertible setting. The work supplies concrete new examples of stable p-adic systems that prior invertible-focused results left unaddressed.

Core claim

The main result provides sufficient conditions under which the following families of maps exhibit strong shadowing and stability properties: 1) p-adic dynamical systems that are right-invertible through contractions, and 2) left-invertible contractions. Consequently, new examples of stable p-adic dynamics are presented.

What carries the argument

Right-invertibility through contractions and left-invertibility of contractions on Z_p or Q_p, which supply the structural control needed to prove shadowing and stability for these non-invertible maps.

If this is right

  • Such maps possess the strong shadowing property.
  • Such maps are topologically stable.
  • New families of stable non-invertible p-adic dynamical systems become available for study.

Where Pith is reading between the lines

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

  • The same invertibility-through-contraction conditions could be tested for stability in other zero-dimensional compact metric spaces.
  • Iterates or finite compositions of these maps may inherit the shadowing and stability properties.
  • The conditions might yield explicit constructions useful for arithmetic dynamics on p-adic fields.

Load-bearing premise

The maps under study are right-invertible through contractions or left-invertible contractions when acting on the p-adic integers or numbers.

What would settle it

A concrete counterexample consisting of a map on Z_p that satisfies right-invertibility through a contraction yet lacks the strong shadowing property would disprove the sufficient conditions.

read the original abstract

The stability theory of compact metric spaces with positive topological dimension is a well-established area in Dynamical Systems. A central result, attributed to Walters, connects the concepts of topological stability and the shadowing property in invertible dynamics. In contrast, zero-dimensional stability theory is a developing field, with an analogue of Walters' theorem for Cantor spaces being fully established only in 2019 by Kawaguchi. In this paper, we investigate the shadowing and stability properties of non-invertible dynamics in zero-dimensional spaces, focusing on the $p$-adic integers $\mathbb{Z}_{p} $ and the $p$-adic numbers $\mathbb{Q}_{p}$, where $p \geq 2$ is a prime number. The main result provides sufficient conditions under which the following families of maps exhibit strong shadowing and stability properties: 1) $p$-adic dynamical systems that are right-invertible through contractions, and 2) left-invertible contractions. Consequently, new examples of stable $p$-adic dynamics are presented.

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 paper extends stability theory from invertible dynamics on positive-dimensional spaces (Walters) and Cantor spaces (Kawaguchi 2019) to non-invertible maps on the zero-dimensional spaces Z_p and Q_p. It asserts sufficient conditions under which two families—right-invertible maps realized through contractions, and left-invertible contractions—possess strong shadowing and topological stability, and it supplies new concrete examples of stable p-adic dynamical systems.

Significance. If the stated sufficient conditions are correctly proved, the work supplies the first explicit non-invertible analogues of the Walters–Kawaguchi theorems inside the p-adics. The provision of new families and examples is a concrete advance for the still-developing zero-dimensional stability theory; the ultrametric setting and the explicit invertibility hypotheses make the extension falsifiable and potentially useful for further p-adic dynamics.

minor comments (3)
  1. §1 (Introduction): the precise statements of the two sufficient conditions are only alluded to in the abstract; they should be displayed as numbered theorems immediately after the statement of Kawaguchi’s result so that the reader can see exactly which hypotheses are added.
  2. The paper should include a short table or list that records, for each new example, which of the two sufficient conditions it satisfies and which shadowing/stability conclusion follows.
  3. Notation: the distinction between “right-invertible through contractions” and “left-invertible contractions” is used repeatedly; a single displayed definition or diagram in §2 would remove any ambiguity for readers unfamiliar with the p-adic literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work extending stability theory to non-invertible maps on Z_p and Q_p. The recommendation of minor revision is noted. No specific major comments were listed in the report, so we have no points to address point-by-point at this stage and will incorporate any minor editorial suggestions in the revision.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim consists of sufficient conditions for strong shadowing and stability in two families of non-invertible p-adic maps (right-invertible contractions and left-invertible contractions). These conditions are asserted to follow from the ultrametric structure on Z_p and Q_p together with the cited external results of Walters (invertible case) and Kawaguchi (Cantor-space analogue). No equations, parameter fits, or self-citations appear in the provided abstract or description that reduce the stated conditions to the inputs by construction; the argument instead extends prior theorems to explicitly named new families and supplies concrete examples. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on two standard theorems from the literature and introduces no new free parameters or invented entities.

axioms (2)
  • standard math Walters' theorem connecting topological stability and shadowing for invertible dynamics on compact metric spaces
    Cited in abstract as the central result for the invertible case.
  • standard math Kawaguchi's 2019 analogue of Walters' theorem for Cantor spaces
    Cited in abstract as the established result for zero-dimensional invertible dynamics.

pith-pipeline@v0.9.0 · 5723 in / 1188 out tokens · 33553 ms · 2026-05-23T21:55:04.952609+00:00 · methodology

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Reference graph

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