Density of repelling periodic points for cubic rational maps
Abstract
We prove the repelling-density conjecture for cubic rational maps over Cp: 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≥5 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.
Related
- Ideal Classes of Order 2n in a Family of Totally Real Cubic Fields and Rational Points on Associated Elliptic Curves
- On the density of rational lines on diagonal cubic hypersurfaces, II
- Fine points of the moduli space of cubic rational maps
- Potential density of rational points on sextic double solids
- On potential density of rational points on hyperkahler varieties
BibTeX
@misc{truong2026density,
title = {{Density of repelling periodic points for cubic rational maps}},
author = {Nha Xuan Truong},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21559},
eprint = {2610.21559},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21559},
}
RIS
TY - UNPB
TI - Density of repelling periodic points for cubic rational maps
AU - Nha Xuan Truong
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21559
UR - https://arxiv.org/abs/2610.21559
AB - We prove the repelling-density conjecture for cubic rational maps over 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≥5 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.
ER -