Non-autonomous iteration of polynomials in the complex plane

We consider a sequence $$ of polynomials with uniformly bounded zeros and $$, $$ for $$, satisfying certain asymptotic conditions. We prove that the function sequence $$ is uniformly convergent in $$. The non-autonomous filled Julia set $$ generated by the polynomial sequence $$ is defined and shown to be compact and regular with respect to the Green function. Our toy example is generated by $$, where $$ is the classical Chebyshev polynomial of degree $$.