DISCRETENESS OF POSTCRITICALLY FINITE MAPS IN p-ADIC MODULI SPACE

Let p >= 2 be a prime number and let C-p be the completion of an algebraic closure of the p-adic rational field Q(p). Let fc(z) be a one-parameter family of rational functions of degree d >= 2, where the coefficients are meromorphic functions defined at all parameters c in some open disk D subset of C-p. Assuming an appropriate stability condition, we prove that the parameters c for which fc 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.