A Conditional Uniform Boundedness Result for a Julia Set with a finite superattracting periodic point and a non-Archimedean height

A Bogomolov property for the canonical height of maps with superattracting periodic points

We prove that if is a polynomial over a number field with a finite superattracting periodic point and a nonarchimedean place of bad reduction, then there is an such that only finitely many have canonical height less than with respect to.We also prove a conditional uniform boundedness result for the preperiodic points of such polynomials, as well as a uniform lower bound on the canonical height of non-preperiodic points in the function field setting.We further prove unconditional analogues of these results in the function field setting.We prove that if is a polynomial over a number field with a finite superattracting periodic point and a nonarchimedean place of bad reduction, then there is an such that only finitely many have canonical height less than with respect to.