Post-critically finite maps on $\mathbb {P}^n$ for $n\ge 2$ are sparse

Abstract:

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.