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.

受最近Good和Meddaugh的新颖作品的启发，我们建立了阴影，有限阶移位和超完整空间之间的基本联系。我们开发了一种用于无限字母的有限类型转换的理论。我们称它们为有限阶移位。我们发展了一般度量空间中遮蔽属性的基本理论，与紧空间中的阴影属性具有相同之处和不同之处。我们在零维完整空间的设置中将这两个理论联系在一起，这表明，当且仅当它是有限阶移位系统的逆极限时，超度量完整空间的一致连续映射才具有有限的阴影属性。满足Mittag-Leffler条件。此外，在这种情况下，我们证明，在系统的逆极限描述中，遮蔽属性与有限遮蔽属性以及Mittag-Leffler条件的满足是等效的。作为推论，我们获得了超度量空间中的各种图具有阴影属性，例如相似性，更一般地说，其本身或它们的逆具有Lipschitz常数1的图。最后，我们将结果应用于的动力学。p -adic整数和p -adic有理数。