Post-critically finite (PCF) rational maps are rational maps of the Riemann sphere with all their critical points with finite orbit under iteration. In the space of rational maps of the projective line with a given degree, post-critically finite maps play a central role. They can be thought as the equivalent of CM elliptic curves in the modular curve; in particular, they are Zariski dense.

In this talk, I will first introduce certain arithmetic and dynamical motivations to the study of PCF maps and discuss some of their properties. I will also discuss their non-Zariski density among endomorphisms of the projective space of dimension k>1. This is a result we proved recently in a joint work with Johan Taflin and Gabriel Vigny.