The Dynamical Andre-Oort Conjecture: Unicritical Polynomials

TL;DR

This paper proves equidistribution of post-critically finite unicritical polynomials and classifies certain algebraic curves with dense sets of parameters where both polynomials are post-critically finite, advancing the dynamical Andre-Oort conjecture.

Contribution

It establishes the first complete case of the dynamical Andre-Oort phenomenon for unicritical polynomials, combining equidistribution and combinatorial analysis.

Findings

Classified algebraic curves with dense post-critically finite parameters.

Proved equidistribution of post-critically finite maps in unicritical polynomial families.

Connected dynamical properties to classical André-Oort results.

Abstract

We establish the equidistribution with respect to the bifurcation measure of post-critically finite maps in any one-dimensional algebraic family of unicritical polynomials. Using this equidistribution result, together with a combinatorial analysis of certain algebraic correspondences on the complement of the Mandelbrot set $M_2$ (or generalized Mandelbrot set $M_d$ for degree $d>2$), we classify all complex plane curves $C$ with Zariski-dense subsets of points $(a,b)\in C$, such that both $z^d+a$ and $z^d+b$ are simultaneously post-critically finite for a fixed degree $d\geq 2$. Our result is analogous to the famous result of Andre regarding plane curves which contain infinitely many points with both coordinates CM parameters in the moduli space of elliptic curves, and is the first complete case of the dynamical Andre-Oort phenomenon studied by Baker and DeMarco.