Unlikely intersection problems for restricted lifts of p-th power

Unlikely Intersection For Two-Parameter Families of Polynomials

Let $c_1, c_2, c_3$ be distinct complex numbers, and let $d\ge 3$ be an integer. We show that the set of all pairs $(a,b)\in \mathbb{C}\times \mathbb{C}$ such that each $c_i$ is preperiodic for the action of the polynomial $x^d+ax+b$ is not Zariski dense in the affine plane.

马尔可夫曲面自同构的不可能交集问题

Unlikely intersections problem for automorphisms of Markov surfaces

本文研究马尔可夫曲面正熵自同构的不可能交集问题，证明特定参数下两自同构共享Zariski稠密周期点集当且仅当共享公共迭代，方法涉及算术等分布、层流流与拟Fuchsian表示理论。

A metric of mutual energy and unlikely intersections for dynamical systems

We introduce a metric of mutual energy for adelic measures associated to the Arakelov-Zhang pairing. Using this metric and potential theoretic techniques involving discrete approximations to energy integrals, we prove an effective bound on a problem of Baker and DeMarco on unlikely intersections of dynamical systems, specifically, for the set of complex parameters $c$ for which $z=0$ and $1$ are both preperiodic under iteration of $f_c(z)=z^2 + c$.

Unlikely intersections over finite fields: polynomial orbits in small subgroups

We estimate the frequency of polynomial iterations which falls in a given multiplicative subgroup of a finite field of $p$ elements. We also give a lower bound on the size of the subgroup which is multiplicatively generated by the first $N$ elements in an orbit. We derive these from more general results about sequences of compositions or a fixed set of polynomials.

Homogeneous Dynamics and Unlikely Intersections

This is a preliminary version of a monograph on homogeneous dynamics and application to some problems of unlikely intersections in Shimura varieties. It consists of four articles, which can be read independently. The first one, by the two first named authors, discuss the main application, to some refinement of the André-Pink-Zannier conjecture about closures of subsets of restricted Hecke orbits in Shimura varieties. The second article, by the first and last named author, establishes results about the dynamics of sequences translates of some measures in spaces of $S$-arithmetic lattices. These…

单临界多项式的一致不寻常交集

Uniform unlikely intersections for unicritical polynomials

本文证明对单临界多项式族 $f_t(z)=z^d+t$，存在仅依赖 $d$ 的常数 $C(d)$，使得满足 $a^d\neq b^d$ 的任意两点 $a,b$ 同时前周期的参数 $t$ 数量有界。