  We establish a principle that we call the Fatou-Shishikura injection for
Newton maps of polynomials: there is a dynamically natural injection from the
set of non-repelling periodic orbits of any Newton map to the set of its
critical orbits. This injection obviously implies the classical
Fatou-Shishikura inequality, but it is stronger in the sense that every
non-repelling periodic orbit has its own critical orbit.
  Moreover, for every Newton map we associate a forward invariant graph (a
puzzle) which provides a dynamically defined partition of the Riemann sphere
into closed topological disks (puzzle pieces). This puzzle construction is for
rational Newton maps what Yoccoz puzzles are for polynomials: it provides the
foundation for all kinds of rigidity results of Newton maps beyond our
Fatou-Shishikura injection. Moreover, it gives necessary structure for a
classification of the postcritically finite maps in the spirit of Thurston
theory.
