bibkey: goncalvessilva2026shadowing
authors: Daniel Goncalves and Sofia Meneghel Silva
year: 2026
title: 'Shadowing and Lipschitz Shadowing in Symbolic Dynamics: Finite vs. Infinite Alphabets'
doi: null
url: https://arxiv.org/abs/2603.03200v1
claim: The Section 3 open question asks whether uniform equivalence preserves Lipschitz shadowing for compatible product metrics on a countably infinite alphabet.
strata_touched: []
license: citation-only
triage: anchor

Uniform metric changes on the countably infinite product shift

The literal open question on printed/PDF p.6 quantifies over arbitrary metrics uniformly equivalent to the standard product metric on a product subshift of a countably infinite discrete alphabet. It is not restricted to the globally symbol-independent prefix metrics of Definition 3.3.

Definitions 2.2 and 2.3 use strict pseudo-orbit and tracing inequalities. Tracing includes time zero and every subsequent natural time. Lipschitz shadowing chooses positive L and a positive cutoff before every control parameter in the positive interval ending at that cutoff and every infinite pseudo-orbit. Definition 2.4 requires both identity maps to be uniformly continuous.

The head-dependent metric and its counterexample are developed in D5/S1/Dynamics/Shadowing/ProductMetricCounterexample. They are new deductions addressing the source question, not results attributed to the paper. The constructor and infinite diagonal-splice suppliers are in HeadWeightedMetric and MetricTracing in the same directory. Mathlib’s PiNat supplies the first difference, cylinders and the actual standard product metric; the existing UniformTracing module supplies the noncompact uniform shadowing contract.

Definition 2.1’s preliminary Deaconu–Renault terminology assumes local compactness. The full product of a countably infinite discrete alphabet lacks that property. The formal target is the unrestricted literal Section 3 question, and makes no local compactness claim. A locally compact variant is outside these declarations.

The checked source is v1. The bounded later-literature search has not located a later answer; it does not certify global uniqueness or publication priority.

Verified locator

Versioned primary source: https://arxiv.org/abs/2603.03200v1. PDF: https://arxiv.org/pdf/2603.03200v1.

Definitions 2.1–2.4 are on printed/PDF p.3. Definition 3.3 is on p.5; Corollary 3.5, Remark 3.6 and the literal product-metric open question are on printed/PDF p.6. The supplied primary PDF has SHA-256 9e30b597eee45fb3cb1fb1e1612a9042c13ef13ef2fa6f6b23bf08d933c01494.

Quantitative boundary supplied by the repository

HeadWeightedLipschitz gives the exact equivalence between Lipschitz shadowing for the raw head-weight metric and a single C>0 satisfying weight(s,n) <= C * weight(t,n-1) for all heads s,t and n>=1. The weights are positive, nonincreasing in depth and normalized to one at depth zero. No uniform lower bound, common vanishing upper envelope, continuity assumption or product uniformity is added to the raw tracing predicate.

Sufficiency uses the infinite diagonal-mismatch telescope. For necessity, proposed positive controls L,c give C=max(2L,max(2/c,1)). A violating ratio has C*weight(t,n-1)<1, so its doubled edge error is already below c. The existing single-defect infinite pseudo-orbit then excludes every full tracer at the proposed Lipschitz radius. This replaces the earlier finite-depth cutoff argument and removes its UniformBounds hypothesis.

When UniformBounds are available, the product-uniform metric has exactly the same distance, so the theorem supplies the original product-compatible classification as an exact application. The constant weight family is also covered: it gives the discrete metric and has no uniformly vanishing upper envelope. This is repository-derived, not attributed to the source and not another count of the already proved unrestricted metric counterexample. The separately scoped locally compact problem and arbitrary compatible-metric classification remain outside this statement.

The unversioned primary arXiv page was read back on 2026-10-21: submission history still lists only v1 (3 March 2026), and its abstract still describes arbitrary uniform-metric product-model stability as unclear. This primary version check does not certify that no other publication has answered it.