$\Kab$ has the Bogomolov property for canonical heights

Abstract

We show that for any number field $K$ and any rational map $f$ in $K(x)$ that is not conjugate to a power, (signed) Chebyshev or Latt\`es map, then $K^{\mathrm{ab}}$ has the strong Bogomolov property for the canonical height of $f$. We also classify the pairs $(f,\alpha)$ whose backward orbit contains infinitely many abelian points. This settles the Andrews--Petsche conjecture to rational maps over a number field and to infinite abelian subsets of backward orbits. The authors were led to the main idea of the proof in conversation with \emph{Astra}. The key insight consists of applying the equidistribution results \cite{Yua08}, followed by a classification of $(f,f)$-preperiodic curves \cite{Pak23,Pak20, Bea25} followed by an additional application of equidistribution on a parametrized preperiodic curve.