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

Abstract

Given a family $\{ f_λ \}_{λ\in Λ}$ of polynomial maps of degree $d$ where $Λ$ is the set of parameters, a polynomial map $f_{λ_0}$ is called {\it $J$-stable in $Λ$} if there exists a neighborhood of $λ_0$ in $Λ$ such that for any element $λ$ in the neighborhood, there exists a conjugacy between the dynamics on the Julia sets of $f_λ$ and $f_{λ_0}$. The aim of this paper is to show that a polynomial map $f_{λ_0}$ 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 $f_{λ_0}$ is {\it immediately expanding}.