Sur la structure des ensembles de Fatou p-adiques
  A classification of the periodic components of the Fatou set of $p$-adic rational maps. Each such periodic component is either an immediate attracting basin or an open affinoid, where the dynamics is quasi-periodic (the $p$-adic analogues of Siegel discs and Herman rings.)   The main tool is a fixed point theorem for connected subsets of Berkovich's projective line (or p-adic hyperbolic space). 
Juan Rivera-Letelier