Misiurewicz points and dynamical units by Vefa Goksel from UMASS Amherst, by zoom.

Abstract: We study the dynamics of the unicritical polynomial family $f_{d,c}(z) = z^d+c$. The $c$-values for which $f_{d,c}$ has a strictly preperiodic postcritical orbit are called Misiurewicz points. The arithmetic properties of these special points have found applications in both arithmetic and complex dynamics. In our recent work, we investigate some new such properties. In particular, we consider the algebraic integers obtained by taking the difference of two distinct Misiurewicz points, and we ask when these algebraic integers are algebraic units. This question naturally arises from a well-known analogous question in arithmetic geometry, and it is also evocative of the study of dynamical units introduced by Morton and Silverman. We propose a conjectural answer to this question, which we prove in many cases. This is a joint work with Rob Benedetto.