Non-autonomous iteration of polynomials in the complex plane

We consider a sequence $(p_n{)}_{n=1}^{\mathrm{\infty }}$ of polynomials with uniformly bounded zeros and $\mathrm{°}{p}_1\ge 1$, $\mathrm{°}{p}_n\ge 2$ for $n\ge 2$, satisfying certain asymptotic conditions. We prove that the function sequence ${(\frac{1}{\mathrm{°}{p}_n\cdot \mathrm{.}\mathrm{.}\mathrm{.}\cdot \mathrm{°}{p}_1}{\log }^{+}\mathrm{\mid }{p}_n\circ \mathrm{.}\mathrm{.}\mathrm{.}\circ p_1\mathrm{\mid })}_{n=1}^{\mathrm{\infty }}$ is uniformly convergent in $\mathbb{C}$. The non-autonomous filled Julia set $\mathcal{K}[(p_n{)}_{n=1}^{\mathrm{\infty }}]$ generated by the polynomial sequence $(p_n{)}_{n=1}^{\mathrm{\infty }}$ is defined and shown to be compact and regular with respect to the Green function. Our toy example is generated by $t_n=\frac{1}{2^{n-1}}{T}_n,n\in \{1,2,\mathrm{.}\mathrm{.}\mathrm{.}\}$, where $T_n$ is the classical Chebyshev polynomial of degree $n$.