A case of the Dynamical Andre-Oort Conjecture

TL;DR

This paper proves a special case of the Dynamical Andre-Oort Conjecture, linking postcritically finite maps on rational curves to specific polynomial forms, and shows the Mandelbrot set cannot be a filled Julia set of any polynomial.

Contribution

It establishes a new case of the Dynamical Andre-Oort Conjecture and demonstrates a property of the Mandelbrot set related to polynomial Julia sets.

Findings

If infinitely many points on a rational curve satisfy certain postcritically finite conditions, then the polynomial h(z) is a scalar multiple of z by a root of unity.

The Mandelbrot set cannot be realized as the filled Julia set of any complex polynomial.

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.