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Abstract <p> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-adic integers <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb{Z}_{p} \)</EquationSource> </InlineEquation> and the <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-adic numbers <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathbb{Q}_{p}\)</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(p \geq 2\)</EquationSource> </InlineEquation> is a prime number. The main result provides sufficient conditions under which the following families of maps exhibit strong shadowing and stability properties: 1) <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-adic dynamical systems that are right-invertible through contractions, and 2) left-invertible contractions. Consequently, new examples of stable <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(p\)</EquationSource> </InlineEquation>-adic dynamics are presented. </p>

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Shadowing and Stability of Non-Invertible \(p\)-Adic Dynamics

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • Danilo A. Caprio,
  • Fernando Lenarduzzi,
  • Ali Messaoudi,
  • Ioannis Tsokanos

摘要

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.