Thurston Vanishing

We show how to extend Epstein's algebraic transversality principles for rational maps $f$ of ${\mathbb P}^1_{\mathbb C}$ to infinite forward invariant subsets of the Fatou set. The key, at least conceptually, to doing this is to have a topos of $f$ invariant sheaves, and Grothendieck's six operations on the same in which Epstein's theory naturally takes place. Thus the resulting count of non-repelling invariant cycles is strictly better than the minimum of Epstein and Shishikura. En passant (in functorially applying the Epstein/Thurston methodology at parabolic fixed points) we calculate the dualising sheaf of a real blow up which is a remarkably algebraic object of independent interest with the capacity to enormously simplify the theory of resurgent functions and Stokes' phenomenon.