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

A canonical decomposition of postcritically finite rational maps and their maximal expanding quotients