Shadowing, finite order shifts and ultrametric spaces

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