A BOGOMOLOV PROPERTY FOR THE CANONICAL HEIGHT OF MAPS WITH SUPERATTRACTING PERIODIC POINTS

Looper’s Theorem 1.5 explicitly proves the Bogomolov property over Kab for polynomials with a finite superattracting periodic point and a nonarchimedean place of bad reduction, and gives a correct proof using Berkovich geometry, equidistribution, and a global capacity product formula (see the statement and proof path in Theorem 1.5 and equation (4.4) in the paper ). The candidate solution’s core step is flawed: it asserts uniform ramification growth at every prime above v and deduces that almost all v-adic conjugates concentrate in a single disk, leading to μn,v(D)→1—claims not justified by the paper’s results (compare Corollary 3.2, which provides growth along some primes, not all, and the actual capacity-based contradiction used in the proof ).