$J$-Stability of immediately expanding polynomial maps in $p$-adic dynamics

TL;DR

This paper investigates the stability of polynomial maps over the $p$-adic complex numbers, showing that immediately expanding maps are $J$-stable within their family, which has implications for understanding their dynamical behavior.

Contribution

It establishes that immediately expanding polynomial maps over $ extbf{C}_p$ are $J$-stable in their parameter families, extending stability results to the $p$-adic setting.

Findings

Immediately expanding maps are $J$-stable in $p$-adic dynamics.

Provides conditions for stability of polynomial maps over $ extbf{C}_p$.

Enhances understanding of $p$-adic Julia sets and their stability.

Abstract

Given a family $\{ f_{\lambda} \}_{\lambda \in \Lambda}$ of polynomial maps of degree $d$ where $\Lambda$ is the set of parameters, a polynomial map $f_{\lambda_0}$ is called {\it $J$-stable in $\Lambda$} if there exists a neighborhood of $\lambda_0$ in $\Lambda$ such that for any element $\lambda$ in the neighborhood, there exists a conjugacy between the dynamics on the Julia sets of $f_\lambda$ and $f_{\lambda_0}$. The aim of this paper is to show that a polynomial map $f_{\lambda_0}$ over the field $\mathbb{C}_p$ of $p$-adic complex numbers is $J$-stable in the family of polynomial maps over $\mathbb{C}_p$ if $f_{\lambda_0}$ is {\it immediately expanding}.