Shadowing, Finite Order Shifts and Ultrametric Spaces

TL;DR

This paper explores the connections between shadowing, shifts of finite order, and ultrametric spaces, developing a comprehensive theory that links these concepts and applies to $p$-adic dynamics.

Contribution

It introduces the theory of shifts of finite order for infinite alphabets and establishes the equivalence of shadowing and finite shadowing in ultrametric spaces.

Findings

Finite shadowing property characterized by inverse limits of shifts of finite order.

Shadowing property holds for maps with Lipschitz constant 1 in ultrametric spaces.

Results applied to $p$-adic integers and rationals.

Abstract

Inspired by a recent novel work of Good and Meddaugh, we establish fundamental connections between shadowing, finite order shifts, and ultrametric complete spaces. We develop a theory of shifts of finite type for infinite alphabets. We call them shifts of finite order. We develop the basic theory of the shadowing property in general metric spaces, exhibiting similarities and differences with the theory in compact spaces. We connect these two theories in the setting of zero-dimensional complete spaces, showing that a uniformly continuous map of an ultrametric complete space has the finite shadowing property if, and only if, it is an inverse limit of a system of shifts of finite order satisfying the Mittag-Leffler Condition. Furthermore, in this context, we show that the shadowing property is equivalent to the finite shadowing property and the fulfillment of the Mittag-Leffler Condition in the inverse limit description of the system. As corollaries, we obtain that a variety of maps in ultrametric spaces have the shadowing property, such as similarities and, more generally, maps which themselves, or their inverses, have Lipschitz constant 1. Finally, we apply our results to the dynamics of $p$-adic integers and $p$-adic rationals.