On postcritically finite unicritical polynomials

In this article, we first study arithmetical properties of post-critically finite unicritical polynomials f(a) : z -> az(D) + 1 with D >= 2. In particular, we answer a question of Milnor, showing that there exist non-Galois conjugate parameters a(1) is an element of C and a(2) is an element of C such that f(a1) and f(a2) have critical orbits periodic with the same period. We also answer a question of Baker and DeMarco, proving that the set of parameters a is an element of C such that 0 and 1 are simultaneously (pre)periodic for q(a) : w bar right arrow w(2) + a is equal to {0, -1, -2}.