Discreteness of postcritically finite maps in $p$-adic moduli space

Abstract:

Let $p \geq 2$ 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 $d\geq 2$, 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.