Dynamical Canonical Heights and Finite Trees

Abstract

We establish lower bounds for the canonical height of a wandering point that decays like the square of the field degree. Our methods and results apply to centered, postcritically finite, hyperbolic polynomials of prime power degree whose coefficients are algebraic integers. Our approach is ultimately inspired by an idea of Dimitrov that led to his proof of the Schinzel--Zassenhaus Conjecture. The height lower bound is derived from the lower bound of the local canonical height at an archimedean place. In previous work, the authors used a result of Dubinin on the transfinite diameter of a star-shaped tree. In this paper, we develop tools to bound the transfinite diameter of more general finite trees in the complex plane. We construct these trees using the Hubbard tree of a postcritically finite polynomial. Thurston's notion of core entropy helps us analyze combinatorial properties of the Hubbard tree.