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 $p+1$.

27 pages

References in corpus (1)

• Sur la structure des ensembles de Fatou p-adiques