The dynamical André–Oort conjecture: Unicritical polynomials

Abstract

We establish equidistribution with respect to the bifurcation measure of postcritically finite (PCF) 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 ${\mathcal{M}}_2$ (or generalized Mandelbrot set ${\mathcal{M}}_d$ for degree $d>2$), we classify all curves $C\subset {\mathbb{A}}^2$ defined over $\mathbb{C}$ with Zariski-dense subsets of points $(a,b)\in C$, such that both $z^d+a$ and $z^d+b$ are simultaneously PCF for a fixed degree $d\ge 2$. Our result is analogous to the famous result of André regarding plane curves which contain infinitely many points with both coordinates being complex multiplication parameters in the moduli space of elliptic curves and is the first complete case of the dynamical André–Oort phenomenon studied by Baker and DeMarco.