2023/11/04 by Michael McQuillan, McQuillan, Michael
Mathematics · #Algebraic Geometry (math.AG) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #History and Theory of Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2311.02518
openalex publication_date 2023/11/04 · openalex created_date 2023/11/08 · openalex updated_date 2026/07/28
We show how to extend Epstein's algebraic transversality principles for rational maps f of \mathbb P1\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.