Non-autonomous iteration of polynomials in the complex plane

Abstract

We consider a sequence $(p_n{)}_{n=1}^{\infty }$ of polynomials with uniformly bounded zeros and $\hat{A}°p_1\ge 1$, $\hat{A}°p_n\ge 2$ for $n\ge 2$, satisfying certain asymptotic conditions. We prove that the function sequence ${\left(\frac{1}{\hat{A}°p_n\cdot ...\cdot \hat{A}°p_1}{\log }^{+}|p_n\circ ...\circ p_1|\right)}_{n=1}^{\infty }$ 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 $n$.

25 pages, 3 figures. Title changed as focus shifted from Kalmár-Walsh sequences to more general ones. Previous results are extended. The section on Chebyshev polynomials on Julia sets cut out with a view to separate future development. Minor corrections for clarity in the final version