Non-autonomous iteration of polynomials in the complex plane

TL;DR

This paper studies the dynamics of non-autonomous polynomial iterations in the complex plane, proving convergence of certain normalized logarithmic sequences and analyzing the properties of the associated Julia sets.

Contribution

It establishes uniform convergence of normalized logarithmic compositions and characterizes the non-autonomous Julia sets as compact and regular with respect to Green functions.

Findings

Normalized logarithmic sequences converge uniformly in or polynomial compositions.

Non-autonomous Julia sets are compact and regular with respect to Green functions.

A specific example with Chebyshev polynomials illustrates the theory.

Abstract

We consider a sequence $(p_n)_{n=1}^\infty$ of polynomials with uniformly bounded zeros and $\deg p_1\geq 1$, $\deg p_n\geq 2$ for $n\geq 2$, satisfying certain asymptotic conditions. We prove that the function sequence $\left(\frac{1}{\deg p_n\cdot...\cdot \deg p_1}\log^+|p_n\circ...\circ p_1|\right)_{n=1}^\infty$ is uniformly convergent in $\mathbb{C}$. The non-autonomous filled Julia set $\mathcal{K}[(p_{n})_{n=1}^\infty]$ generated by the polynomial sequence $(p_{n})_{n=1}^\infty$ is defined and shown to be compact and regular with respect to the Green function. Our toy example is generated by $t_n=\frac{1}{2^{n-1}}T_n,\ n\in\{1,2,...\}$, where $T_n$ is the classical Chebyshev polynomial of degree $n$.