Wandering Fatou components on p-adic polynomial dynamics

Abstract

We will study perturbations of the polynomials $P_\lambda$, of the form $$Q_\lambda= P_\lambda +Q$$ in the space of centered monic polynomials, where $P_\lambda$ is the polynomial family defined by $$P_{\lambda}(z)=\frac{\lambda}{p}z^p+(1-\frac{\lambda}{p}) z ^{p+1}$$ with $\lambda \in \Lambda= \{z: |z-1| <1\}$, studied by Benedetto, who showed that for a dense set of parameters, the polynomials $P_\lambda$ have a wandering disc contained in the filled Julia set. We will show an analogous result for the family $Q_\lambda$, obtaining the following consequence: The polynomials $P_\lambda$ 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 $p+1$.