Remarks on iterated cubic maps

Asymptotics for a class of iterated random cubic operators

We consider a class of cubic stochastic operators that are motivated by models for evolution of frequencies of genetic types in populations. We take populations with three mutually exclusive genetic types. The long term dynamics of single maps, starting with a generic initial condition where in particular all genetic types occur with positive frequency, is asymptotic to equilibria where either only one genetic type survives, or where all three genetic types occur. We consider a family of independent and identically distributed maps from this class and study its long term dynamics, in particula…

Finite index theorems for iterated Galois groups of cubic polynomials

Let $K$ be a number field or a function field. Let $f\in K(x)$ be a rational function of degree $d\geq 2$, and let $β\in\mathbb{P}^1(K)$. For all $n\in\mathbb{N}\cup\{\infty\}$, the Galois groups $G_n(β)=\text{Gal}(K(f^{-n}(β))/K)$ embed into $\text{Aut}(T_n)$, the automorphism group of the $d$-ary rooted tree of level $n$. A major problem in arithmetic dynamics is the arboreal finite index problem: determining when $[\text{Aut}(T_\infty):G_\infty]<\infty$. When $f$ is a cubic polynomial and $K$ is a function field of transcendence degree $1$ over an algebraic extension of $\mathbb{Q}$, we res…

广义逻辑斯谛映射与收敛性

Generalized Logistic Maps and Convergence

本文处理了三个三次递推式，其中两个推广了离散混沌理论中的迭代映射 $x \mapsto x (1-x)$，并揭示了这些递推式所发展的渐近级数中存在一个依赖于初始条件但本质固有的常数特征。