A case of the Dynamical Andre-Oort Conjecture

2014/04/24 by Ghioca, Dragos, Krieger, Holly, Nguyen, Khoa · 1 citation
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1404.6011

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.

Cited by

• Rigidity in Complex Dynamics: Multiplier Spectrum and Dynamical André-Oort Conjecture

Related

• DAO for curves
• The Dynamical Andre-Oort Conjecture for cubic polynomials
• The arithmetic of polynomial dynamical pairs
• The Dynamical Mordell-Lang Conjecture
• Degree gaps for multipliers and the dynamical Andre-Oort conjecture