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.
Shadowing property and transitivity of a set-valued map and its inverse limit
School of Computer Science and Technology, Dongguan University of Technology（东莞理工学院计算机科学与技术学院）；School of Data Science, Zhuhai College of Science and Technology（珠海科技学院数据科学学院）
