The Dynamical Andre-Oort Conjecture: Unicritical Polynomials

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 $d\ge 2$. 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.

References in corpus (5)

• Preperiodic points for families of rational map
• Profinite iterated monodromy groups arising from quadratic morphisms with infinite postcritical orbits
• Rational Parameter Rays of The Multibrot Sets
• Finiteness and liftability of postcritically finite quadratic morphisms in arbitrary characteristic
• Bifurcations, intersections, and heights

Cited by in corpus (2)

• The Dynamical Andre-Oort Conjecture for cubic polynomials
• Geometry of PCF parameters in spaces of quadratic polynomials