A case of the Dynamical Andre-Oort Conjecture
Number Theory
2014-04-25 v1 Complex Variables
Dynamical Systems
Abstract
We prove a special case of the Dynamical Andre-Oort Conjecture formulated by Baker and DeMarco. For any integer d>1, we show that for a rational plane curve C parametrized by (t, h(t)) for some non-constant polynomial h with complex coefficients, if there exist infinitely many points (a,b) on the curve C such that both z^d+a and z^d+b are postcritically finite maps, then h(z)=uz for a (d-1)-st root of unity u. As a by-product of our proof, we show that the Mandelbrot set is not the filled Julia set of any polynomial with complex coefficients.
Keywords
Cite
@article{arxiv.1404.6011,
title = {A case of the Dynamical Andre-Oort Conjecture},
author = {Dragos Ghioca and Holly Krieger and Khoa Nguyen},
journal= {arXiv preprint arXiv:1404.6011},
year = {2014}
}