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

TL;DR

This paper explores shadowing and stability in non-invertible $p$-adic dynamical systems, providing conditions for strong stability properties and introducing new stable $p$-adic examples.

Contribution

It extends stability theory to non-invertible $p$-adic dynamics, offering new criteria and examples in zero-dimensional spaces.

Findings

Identifies conditions for shadowing in right-invertible $p$-adic maps

Establishes stability for left-invertible contraction maps

Provides new examples of stable $p$-adic dynamical systems

Abstract

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.