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

TL;DR

This paper proves that for higher-dimensional projective spaces, post-critically finite maps with small tail-length are rare and not dense in the parameter space, highlighting the scarcity of such maps with periodic critical loci.

Contribution

It establishes the non-density of PCF maps with small tail-length in the parameter space for degrees at least 3 and dimensions at least 2.

Findings

PCF maps with tail-length ≤ 2 are not Zariski dense for d ≥ 3, n ≥ 2.

Maps with periodic critical loci are not Zariski dense.

The result extends understanding of the scarcity of special dynamical maps in higher dimensions.

Abstract

Let $f:{\mathbb P}^n\to{\mathbb P}^n$ be a morphism of degree $d\ge2$. The map $f$ 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 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.