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We prove that if $f$ is a polynomial over a number field $K$ with a finitesuperattracting periodic point and a non-archimedean place of bad reduction,then there is an $\epsilon>0$ such that only finitely many $P\in K^{\text{ab}}$have canonical height less than $\epsilon$ with respect to $f$. The keyingredient is the geometry of the filled Julia set at a place of bad reduction.We also prove a conditional uniform boundedness result for the $K$-rationalpreperiodic points of such polynomials, as well as a uniform lower bound on thecanonical height of non-preperiodic points in $K$. We further proveunconditional analogues of these results in the function field setting.

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