曲线的 DAO

The Dynamical Andre-Oort Conjecture: Unicritical Polynomials

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 …

复动力学中的刚性：乘子谱与动力André-Oort猜想

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

本研究介绍一维复动力学刚性问题的近期进展，证明了曲线的动力André-Oort猜想及乘子谱的一般单射性，其证明结合代数几何、Arakelov几何与复动力学思想。

A case of the Dynamical Andre-Oort Conjecture

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.