The possibilities for limit functions on a Fatou component for the iteration of a single polynomial or rational function are well understood and quite restricted. In non-autonomous iteration, where one considers compositions of arbitrary polynomials with suitably bounded degrees and coefficients, one should observe a far greater range of behavior. We show this is indeed the case and we exhibit a bounded sequence of quadratic polynomials which has a bounded Fatou component on which one obtains as limit functions every member of the classical Schlicht family of normalized univalent functions on the unit disc. The proof is based on quasiconformal surgery and the use of high iterates of a quadratic polynomial with a Siegel disc which closely approximate the identity on compact subsets. Careful bookkeeping using the hyperbolic metric is required to control the errors in approximating the desired limit functions and ensure that these errors ultimately tend to zero.

We consider properties of Julia sets arising from composition sequences for arbitrarily chosen polynomials with uniformly bounded degrees and coefficients. This is a slight generalization of the situation first considered by Fornaess and Sibony in Random iterations of rational functions. Ergod. Th. & Dynam. Sys.11 (1991), 687–708. The classical definition of hyperbolicity has a natural extension to our setting, and we show that many of the classical results concerning hyperbolic Julia sets can indeed be generalized. We introduce two important ideas in order to prove these results—the use of Cantor diagonalization, and how the Julia set moves in the Hausdorff topology as the sequence generating it changes.