TThe locus of post-critically finite maps in the moduli space of self-maps of P^n

A degree d>1 self-map f of P^n is called post critically finite (PCF) if its critical hypersurface C_f is pre-periodic for f, that is, if there exist integers r ≥ 0 and k>0 such that f^{r+k}(C_f) is contained in f^{r}(C_f). 

I will discuss the question: what does the locus of PCF maps look like as a subset of the moduli space of degree d maps on P^n? I’ll give a survey of many known results and some conjectures in dimension 1. I’ll then present a result, joint with Joseph Silverman and Patrick Ingram, that suggests that in dimensions two or greater, PCF maps are comparatively scarce in the moduli space of all self-maps.