Post-critically finite maps on ℙⁿ for

Let f : P n → f:{\mathbb P}^n\to {\mathbb P}^n be a morphism of degree alttext="d greater-than-or-equal-to 2"> d ≥ 0 encoding="application/x-tex">\ell \ge 0 such that the critical locus alttext="upper C r i t Subscript f"> Crit encoding="application/x-tex">\operatorname {Crit}_f satisfies k plus script left-parenthesis f right-parenthesis subset-of-or-equal-to right-parenthesis"> + stretchy="false">( stretchy="false">) ⊆<!-- ⊆ encoding="application/x-tex">f^{k+\ell }(\operatorname {Crit}_f)\subseteq {f^\ell (\operatorname {Crit}_f)} smallest l"> encoding="application/x-tex">\ell called tail-length. We prove for 3"> 3 3 alttext="n encoding="application/x-tex">n\ge , set PCF maps with tail-length at most alttext="2"> encoding="application/x-tex">2 not Zariski dense in parameter space all maps. In particular, periodic loci, i.e., equals = =0 are dense.