Non-autonomous iteration of polynomials in the complex plane
Marta Kosek, Malgorzata Stawiska
TL;DR
The article develops a potential-theoretic framework for non-autonomous polynomial iteration by introducing guided sequences of polynomials with uniformly bounded zeros and controlled growth. It proves uniform convergence of the normalized log-escape function $\frac{1}{\deg p_n\cdots\deg p_1}\log^+|p_n\circ\cdots\circ p_1|$ and defines the non-autonomous Julia set $\mathcal{K}[(p_n)]$, which is non-empty, compact, and regular; Klimek’s metric is used to study convergence of preimage sets and Green functions. The work unifies KW, Chebyshev, and $\mathcal{B}$-type sequences under guided sequences and provides a detailed analysis of autonomous versus non-autonomous Julia sets, including a thorough case study for Chebyshev polynomials on $[-1,1]$. The results extend complex dynamics to non-autonomous settings, offering tools for potential theory in approximation and interpolation contexts and clarifying the geometric structure of non-autonomous Julia sets. The toy example based on $t_n=\frac{1}{2^{n-1}}T_n$ illustrates how classical polynomials inform the non-autonomous theory.
Abstract
We consider a sequence $(p_n)_{n=1}^\infty$ of polynomials with uniformly bounded zeros and $°p_1\geq 1$, $°p_n\geq 2$ for $n\geq 2$, satisfying certain asymptotic conditions. We prove that the function sequence $\left(\frac{1}{°p_n\cdot...\cdot °p_1}\log^+|p_n\circ...\circ p_1|\right)_{n=1}^\infty$ 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 $n$.