  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).
