Discreteness of postcritically finite maps in 𝑝-adic moduli space

Abstract

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