Post-Critically Finite Maps on $mathbb{P}^n$ for $nge2$ are Sparse

Let $f:{mathbb P}^nto{mathbb P}^n$ be a morphism of degree $dge2$. The map $f$ is said to be post-critically finite (PCF) if there exist integers $kge1$ and $ellge0$ 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 $dge3$ and $nge2$, 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.