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Let $f:{\mathbb P}^n\to{\mathbb P}^n$ be a morphism of degree $d\ge2$. Themap $f$ is said to be post-critically finite (PCF) if there exist integers$k\ge1$ and $\ell\ge0$ 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 $d\ge3$and $n\ge2$, the set of PCF maps $f$ with tail-length at most $2$ is notZariski dense in the the parameter space of all such maps. In particular, mapswith periodic critical loci, i.e., with $\ell=0$, are not Zariski dense.

Post-Critically Finite Maps on $\mathbb{P}^n$ for $n\ge2$ are Sparse