Georgia Institute of Technology College of Sciences

Unicritical polynomials, typically written in the form $z^d+c$, have been widely studied in arithmetic and complex dynamics and are characterized by their one finite critical point. The behavior of a map's critical points under iteration often determines the dynamics of the entire map. Rational maps where the critical points have a finite forward orbit are called post-critically finite (PCF), and these are of great interest in arithmetic dynamics. They are viewed as a dynamical analogue of abelian varieties with complex multiplication and often display interesting dynamical behavior. The family of (single-cycle normalized) dynamical Belyi polynomials have two fixed critical points, so they are PCF by construction, and these maps provide a new testing ground for conjectures and ideas related to post-critically finite polynomials. Using this family, we can begin to explore properties of polynomial maps with two critical points. In this talk we will discuss applications of this family in arithmetic dynamics; in particular, how this family can be used to determine more general reduction properties of PCF polynomials.