Shadowing and Lipschitz Shadowing in Symbolic Dynamics: Finite vs. Infinite Alphabets

TL;DR

This paper investigates the shadowing and Lipschitz shadowing properties in symbolic dynamics over infinite alphabets, revealing a fundamental difference between the classical product-topology and the OTW compactification models, especially regarding metric dependence.

Contribution

It demonstrates that in the OTW model, shadowing is always present, but Lipschitz shadowing can vary with the choice of compatible ultrametrics, highlighting a clear distinction from the product-topology setting.

Findings

Shadowing coincides with Lipschitz shadowing in the product-topology model for prefix ultrametrics.

The OTW full shift has the shadowing property for all OTW metrics.

Lipschitz shadowing can depend on the specific OTW ultrametric chosen, even within the same uniform equivalence class.

Abstract

We point out a basic dichotomy between the shadowing and Lipschitz shadowing properties for one-sided shift spaces in two infinite-alphabet frameworks: the classical product-topology model $X\subseteq A^{\mathbb{N}}$ and the compact Ott--Tomforde--Willis (OTW) model obtained by adjoining finite words. In the product-topology setting, for the natural class of prefix ultrametrics, shadowing and Lipschitz shadowing coincide. However, since $A^{\mathbb{N}}$ is non-compact when $A$ is countably infinite, it remains unclear whether Lipschitz shadowing is stable under arbitrary uniformly equivalent changes of compatible metric in the product-topology model. In contrast, for OTW shift spaces the topology admits a canonical family of compatible ultrametrics indexed by enumerations of finite words, and these metrics are all uniformly equivalent. Using the Deaconu--Renault viewpoint and known shadowing results for local homeomorphisms on zero-dimensional compact spaces, we show that the OTW full shift has the shadowing property for every OTW metric. Nevertheless, Lipschitz shadowing can depend on the chosen OTW metric even within this fixed uniform equivalence class: we construct two uniformly equivalent OTW ultrametrics on the full shift for which Lipschitz shadowing holds in one case and fails in the other. Thus the OTW compactification provides a compact infinite-alphabet setting where the metric dependence of Lipschitz shadowing can be resolved explicitly, in sharp contrast with what is currently known for the product-topology model.