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

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}.