Dynamics Seminar

The critical locus of a self-map f on a variety is the set of all points where f fails to be locally invertible. This locus is important in dynamics because in some sense it governs the global behaviour of f under iteration. While for rational maps of the Riemann sphere the critical locus is a finite set of points, in higher dimensions it is generally a hypersurface. Whether or not this hypersurface is irreducible is an important technical point in the study of post-critically finite maps. We show that critical irreducibility is an open condition: "most" degree-d endomorphisms of P^n, defined over an arbitrary field, have irreducible critical locus (n, d > 1). This extends a theorem of Ingram--Ramadas--Silverman and is completely new in positive characteristic. Our construction is elementary, using only basic facts about polynomial rings. All this is joint work with Max Weinreich.