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Register a new userPost-Critically Finite Maps on $mathbb{P}^n$ for $nge2$ are Sparse
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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.
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A rational function of degree at least two with coefficients in an algebraically closed field is post-critically finite (PCF) if all of its critical points have finite forward orbit under iteration. We show that the collection of PCF rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers, once the well-understood family known as flexible Lattes maps is excluded. As a consequence, there are only finitely many conjugacy classes of non-Lattes PCF rational maps of a given degree defined over any given number field. The key ingredient of the proof is a non-archimedean version of Fatous classical result that every attracting cycle of a rational function over the complex numbers attracts a critical point.
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