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Abstract Fix an integer$$d\ge 2$$ $d\ge 2$. The parameters$$c_0\in \overline{\mathbb {Q}}$$ $c_0\in \overline{Q}$for which the unicritical polynomial$$f_{d,c}(z)=z^d+c\in \mathbb {C}[z]$$ $f_{d,c}\left(z\right)=z^d+c\in C\left[z\right]$has finite postcritical orbit, also known asMisiurewiczparameters, 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 callp-special, which may be of independent number theoretic interest. 

We study the dynamics of the unicritical polynomial family [Formula: see text]. The [Formula: see text]-values for which [Formula: see text] has a strictly preperiodic postcritical orbit are called Misiurewicz parameters, and they are the roots of Misiurewicz polynomials. The arithmetic properties of these special parameters have found applications in both arithmetic and complex dynamics. In this paper, we investigate some new such properties. In particular, when [Formula: see text] is a prime power and [Formula: see text] is a Misiurewicz parameter, we prove certain arithmetic relations between the points in the postcritical orbit of [Formula: see text]. We also consider the algebraic integers obtained by evaluating a Misiurewicz polynomial at a different Misiurewicz parameter, and we ask when these algebraic integers are algebraic units. This question naturally arises from some results recently proven by Buff, Epstein, and Koch and by the second author. We propose a conjectural answer to this question, which we prove in many cases.