  We provide a natural canonical decomposition of postcritically finite
rational maps with non-empty Fatou sets based on the topological structure of
their Julia sets. The building blocks of this decomposition are maps where all
Fatou components are Jordan disks with disjoint closures (Sierpi\'{n}ski maps),
as well as those where any two Fatou components can be connected through a
countable chain of Fatou components with common boundary points (crochet or
Newton-like maps). We provide several alternative characterizations for our
decomposition, as well as an algorithm for its effective computation. We also
show that postcritically finite rational maps have dynamically natural
quotients in which all crochet maps are collapsed to points, while all
Sierpi\'{n}ski maps become small spheres; the quotient is a maximal expanding
cactoid. The constructions work in the more general setup of B\"{o}ttcher
expanding maps, which are metric models of postcritically finite rational maps.

Dynamical Systems