$K^{\mathrm{ab}}$ has the Bogomolov property for canonical heights

Abstract

We show that for any number field $K$ and any rational map $f$ in $K(x)$ of degree at least $2$ that is not conjugate to a power, (signed) Chebyshev or CM Lattès map, $K^{\mathrm{ab}}$ has the strong Bogomolov property for the canonical height of $f$. Combined with former results of Amoroso-Zannier, Pottmeyer and Ratazzi, this implies that $K^{\mathrm{ab}}$ has the Bogomolov property for canonical heights of any rational map of degree at least $2$. We also classify the pairs $(f,\alpha )$ whose backward orbit contains infinitely many abelian points. This settles the Andrews--Petsche conjecture in the greater generality of rational maps over a number field and 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 Yuan's equidistribution results, followed by a classification of $(f,f)$-preperiodic curves due to Pakovich and Beaumont followed by an additional application of equidistribution on a parametrized preperiodic curve.