Shadowing and Stability of Non-Invertible $p$-adic Dynamics

TL;DR

The paper advances the theory of shadowing and stability for non-invertible, zero-dimensional dynamical systems in the $p$-adic setting by establishing concrete sufficient conditions under which right-invertible contractions yield shadowing and strong stability properties on $\mathbb{Z}_{p}$ and $\mathbb{Q}_{p}$. It develops a fixed-point framework on a sequence space to convert pseudo-orbits into true orbits and constructs conjugacies under perturbations to obtain both strong Lipschitz structural stability and topological stability; a covering property of the right inverses is crucial for these results. It also provides a counterexample showing that a single right inverse does not guarantee shadowing, and it extends the stability analysis to left-invertible contractions with bi-Lipschitz or scaling behavior, including affine contractions as canonical examples. The findings yield new stable $p$-adic dynamics, including $(p^{-k},p^{k})$-locally scaling maps, and broaden the zero-dimensional stability landscape by linking non-invertible $p$-adic dynamics with shadowing and Lipschitz robustness, with potential implications for applications in cryptography and physics.

Abstract

The stability theory of compact metric spaces with positive topological dimension is a well-established area in Continuous Dynamics. 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.

Theorems & Definitions (30)

• Theorem 3.1
• Corollary 3.2
• Theorem 3.3
• Definition 3.4: Nowhere Locally Constant maps
• Proposition 3.5
• Theorem 3.6
• Remark 3.7
• Example 3.8
• Lemma 4.1
• proof : Proof of Theorem \ref{['Theor1_Dilations in p-adics']}