Misiurewicz polynomials and dynamical units, part II

Abstract Fix an integer d 2 d ≥ 2. The parameters c₀ Q c 0 ∈ Q ¯ for which the unicritical polynomial f₃, ₂ (z) =zᵈ+c Cz f d, c (z) = z d + c ∈ C z has finite postcritical orbit, also known as Misiurewicz parameters, play a significant role in complex dynamics. Recent work of Buff, Epstein, and Koch proved the first known cases of a long-standing dynamical conjecture of Milnor using their arithmetic properties, about which relatively little is otherwise known. Continuing our work from a companion paper, we address further arithmetic properties of Misiurewicz parameters, especially the nature of the algebraic integers obtained by evaluating the polynomial defining one such parameter at a different Misiurewicz parameter. In the most challenging such combinations, we describe a connection between such algebraic integers and the multipliers of associated periodic points. As part of our considerations, we also introduce a new class of polynomials we call p - special, which may be of independent number theoretic interest.