The Dynamical Schinzel-Zassenhaus Conjecture and the Transfinite Diameter of Trees

In 2019, Dimitrov proved the Schinzel-Zassenhaus Conjecture. Harry Schmidt and I extended his general strategy to cover some dynamical variants of this conjecture. One common tool in both results is Dubinin's Theorem on the transfinite diameter of hedgehogs, a star-shaped tree in the plane. I will report on joint work in progress with Harry Schmidt. We find new upper bounds for the transfinite diameter of some finite topological trees. Our trees arise from the Hubbard tree of a postcritically finite polynomial and reflect its dynamical properties. As a consequence, we prove new lower bounds for the Call-Silverman or canonical height for a class of postcritically finite polynomials.