Degree gaps for multipliers and the dynamical Andre-Oort conjecture

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.

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 …

The Dynamical Andre-Oort Conjecture for cubic polynomials

In the moduli space of degree d polynomials, the special subvarieties are those cut out by critical orbit relations, and then the special points are the post-critically finite polynomials. It was conjectured that in the moduli space of degree d polynomials, the subvarieties containing a Zariski-dense set of special points are exactly these special subvarieties. In this article, we prove the first non-trivial case for this conjecture: the case of cubic polynomials.

On the geometric André-Oort conjecture for variations of Hodge structures

Let $\mathbb{V}$ be a polarized variation of integral Hodge structure on a smooth complex quasi-projective variety $S$. In this paper, we show that the union of the non-factor special subvarieties for $(S, \mathbb{V})$, which are of Shimura type with dominant period maps, is a finite union of special subvarieties of $S$. This generalizes previous results of Clozel and Ullmo arXiv:math/0404131, Ullmo \cite{Ullmo07} on the distribution of the non-factor (in particular, strongly) special subvarieties in a Shimura variety to the non-classical setting and also answers positively the geometric part …

曲线的 DAO

DAO for curves

本文证明了 Baker–DeMarco 关于代数曲线参数化有理映射族的 DAO 猜想，并进一步给出其 Bogomolov 型推广。