Bounded geometry for PCF-special subvarieties

TL;DR

This work studies the distribution of postcritically finite (PCF) maps inside the moduli space $\\mathrm{M}_d$ of degree $d$ rational maps. By combining arithmetic equidistribution (Yuan–Zhang) with complex-dynamical bifurcation theory, it establishes uniform geometric bounds: for any subvariety $X\\subset\\mathrm{M}_d$ of degree at most $D$, the closure of $X\\cap \\mathrm{PCF}_d$ has a uniformly bounded number and size of irreducible components, and there are only finitely many positive-dimensional PCF-special subvarieties of degree at most $D$. The authors extend these results to points of small critical height in $\\mathrm{M}_d(\\overline{\\mathbb{Q}})$ and provide a relative framework for maximally varying families, constructing nontrivial bifurcation measures on fiber powers to force height gaps and derive uniformity. The approach yields Bogomolov-type finiteness statements in the dynamical setting, offering a uniform trajectory-based analogue to the classical André–Oort theory without requiring a full classification of PCF-special subvarieties.

Abstract

For each integer $d\geq 2$, let $M_d$ denote the moduli space of maps $f: \mathbb{P}^1\to \mathbb{P}^1$ of degree $d$. We study the geometric configurations of subsets of postcritically finite (or PCF) maps in $M_d$. A complex-algebraic subvariety $Y \subset M_d$ is said to be PCF-special if it contains a Zariski-dense set of PCF maps. Here we prove that there are only finitely many positive-dimensional irreducible PCF-special subvarieties in $M_d$ with degree $\leq D$. In addition, there exist constants $N = N(D,d)$ and $B = B(D,d)$ so that for any complex algebraic subvariety $X \subset M_d$ of degree $\leq D$, the Zariski closure $\overline{X\cap\mathrm{PCF}}~$ has at most $N$ irreducible components, each with degree $\leq B$. We also prove generalizations of these results for points with small critical height in $M_d(\bar{\mathbb{Q}})$.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Example 1.3
• Example 1.4
• Theorem 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8
• Remark 1.11
• Proposition 2.1