Wandering Fatou components on p-adic polynomial dynamics

Abstract

We will study perturbations of the polynomials $P_λ$, of the form $$Q_λ= P_λ+Q$$ in the space of centered monic polynomials, where $P_λ$ is the polynomial family defined by $$P_λ(z)=\fracλ{p}z^p+(1-\fracλ{p}) z ^{p+1}$$ with $λ\in Λ= \{z: |z-1| <1\}$, studied by Benedetto, who showed that for a dense set of parameters, the polynomials $P_λ$ have a wandering disc contained in the filled Julia set. We will show an analogous result for the family $Q_λ$, obtaining the following consequence: The polynomials $P_λ$ belong to $\bar{\mathrm{E}}_{p+1}$ where $\mathrm{E}_{p+1}$ denotes the set of polynomials that have a wandering disc in the filled Julia set, in the space of centered monic polynomials of degree $p+1$.