Density of repelling periodic points for cubic rational maps

Abstract

We prove the repelling-density conjecture for cubic rational maps over $\mathbf C_p$: for every prime $p$, the classical Julia set is the closure of the classical repelling periodic points. By Bézivin's theorem, it suffices to prove that a nonempty classical Julia set contains a classical repelling periodic point. The case $p\geq5$ follows from known results for tame maps, so we treat residue characteristics two and three. We argue by contraposition, assuming that the map has no classical repelling periodic points. Maps with potential good reduction already have empty classical Julia set. Otherwise, we conjugate the map to an explicit normalized form and analyze its coefficients. In each case, we obtain coefficient restrictions and establish a uniform bound on the spherical derivatives of all iterates. This bound implies equicontinuity on the classical projective line and hence emptiness of the classical Julia set.