Shadowing, Finite Order Shifts and Ultrametric Spaces

Abstract

Inspired by recent novel work of Good and Meddaugh, we establish fundamental connections between shadowing, finite order shifts and ultrametric Polish 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 Polish spaces, exhibiting similarities and differences with the theory in compact spaces. We connect these two theories in the setting of zero dimensional Polish spaces, showing that a surjective uniformly continuous map of an ultrametric Polish space has the shadowing property if, and only if, it is the inverse limit of shifts of finite order. As corollaries we obtain that a variety of maps in ultrametric Polish 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. 1

Some notes on the classification of shift spaces: Shifts of Finite Type; Sofic Shifts; and Finitely Defined Shifts

“…While the above definition coincides with the traditional one when considering finite-alphabet shifts, it allows to consider them as subset of infinite-alphabet shifts. On the other hand, for infinite-alphabet shift spaces, this definition includes several shift spaces over infinite alphabets that behave similar to the classical SFTs over finite alphabets (e.g., countable topological Markov chains [10,17] and shift spaces with the shadowing property [5,9]).…”

“…A SFT may also be referred as a finite-step shift, a finite-order shift, a shift with finite memory or shift of bounded type. Such names allude the finiteness of number of coordinates used to define the forbidden words (see, for instance, [5,15,16]). Note that since lattices N or Z are totally ordered, and so the forbidden words can always be thought to be defined on consecutive coordinates starting at coordinate zero, in the context of SFTs on these lattices the "step", "memory" or "order" of a SFT can be taken as single numbers m ∈ N related to the maximum quantity of consecutive coordinates needed to write any forbidden word.…”