影子与非可逆 $p$-adic 动力学的稳定性
Shadowing and Stability of Non-Invertible $p$-adic Dynamics
AI总结：
本文研究零维空间中非可逆动力学的影子与稳定性性质，聚焦于 $p$-adic 整数和 $p$-adic 数，探讨右可逆收缩和左可逆收缩的动力学系统稳定性条件。
摘要：
紧致度量空间的稳定性理论是动力系统中已建立的领域。Walters 的核心结果将拓扑稳定性与影子性质联系起来，适用于可逆动力学。相比之下，零维稳定性理论仍处于发展初期，Kawaguchi 在2019年才为Cantor空间建立了Walters定理的类比。本文研究零维空间中非可逆动力学的影子与稳定性性质，聚焦于 $p$-adic 整数 $\mathbb{Z}_{p} $ 和 $p$-adic 数 $\mathbb{Q}_{p}$，其中 $p \geq 2$ 是素数。主要结果提供了以下映射族在强影子和稳定性性质下的充分条件：1) 通过收缩右可逆的 $p$-adic 动力学系统；2) 左可逆收缩。因此，新的稳定 $p$-adic 动力学示例被提出。
英文摘要：
The stability theory of compact metric spaces with positive topological dimension is a well-established area in Dynamical Systems. A central result, attributed to Walters, connects the concepts of topological stability and the shadowing property in invertible dynamics. In contrast, zero-dimensional stability theory is a developing field, with an analogue of Walters' theorem for Cantor spaces being fully established only in 2019 by Kawaguchi. In this paper, we investigate the shadowing and stability properties of non-invertible dynamics in zero-dimensional spaces, focusing on the $p$-adic integers $\mathbb{Z}_{p} $ and the $p$-adic numbers $\mathbb{Q}_{p}$, where $p \geq 2$ is a prime number. The main result provides sufficient conditions under which the following families of maps exhibit strong shadowing and stability properties: 1) $p$-adic dynamical systems that are right-invertible through contractions, and 2) left-invertible contractions. Consequently, new examples of stable $p$-adic dynamics are presented.
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