Post-critically finite maps on \mathbb{P}^n for n\ge2 are sparse.

Let f:{\mathbb P}^n\to {\mathbb P}^n be a morphism of degree d\ge 2. The map f is said to be post-critically finite (PCF) if there exist integers k\ge 1 and \ell \ge 0 such that the critical locus \operatorname {Crit}_f satisfies f^{k+\ell }(\operatorname {Crit}_f)\subseteq {f^\ell (\operatorname {Crit}_f)}. The smallest such \ell is called the tail-length. We prove that for d\ge 3 and n\ge 2, the set of PCF maps f with tail-length at most 2 is not Zariski dense in the the parameter space of all such maps. In particular, maps with periodic critical loci, i.e., with \ell =0, are not Zariski dense. [ABSTRACT FROM AUTHOR]