Post-critically finite maps on ℙⁿ for 𝕟≥2 are sparse

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