Discreteness of postcritically finite maps in p-adic moduli space

Abstract

Let $p\ge 2$ be a prime number and let ${\mathbb{C}}_p$ be the completion of an algebraic closure of the $p$-adic rational field ${\mathbb{Q}}_p$. Let $f_c(z)$ be a one-parameter family of rational functions of degree $d\ge 2$, where the coefficients are meromorphic functions defined at all parameters $c$ in some open disk $D\subseteq {\mathbb{C}}_p$. Assuming an appropriate stability condition, we prove that the parameters $c$ for which $f_c$ is postcritically finite (PCF) are isolated from one another in the $p$-adic disk $D$, except in certain trivial cases. In particular, all PCF parameters of the family $f_c(z)=z^d+c$ are $p$-adically isolated.

22 pages. Added Examples 7.1, 7.2, 7.3 to illustrate the necessity of certain assumptions in our main theorems