Misiurewicz polynomials and dynamical units, part II

Abstract

AbstractFix an integer $$d\ge 2$$ d ≥ 2 . The parameters $$c_0\in \overline{\mathbb {Q}}$$ c 0 ∈ Q ¯ for which the unicritical polynomial $$f_{d,c}(z)=z^d+c\in \mathbb {C}[z]$$ 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.