Heights and totally p-adic numbers

Abstract: We will study the behaviour of canonical height functions associated to rational functions on totally p-adic fields. For each prime p, including p$=\infty$, we will classify all rational functions $f$ such that there is a positive constant $c$ with the property that the associated canonical height $\widehat{h}_f$ on the maximal totally p-adic field admits only finitely many values less than $c$. This might be seen as a dynamical version of results of Schinzel and Bombieri and Zannier.