On Possible Limit Functions on a Fatou Component in non-Autonomous Iteration (Research on the Theory of Random Dynamical Systems and Fractal Geometry)

Abstract

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 ought to observe a far greater range of behaviour. We show this is indeed the case and we exhibit a 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 suitably normalized univalent functions on the unit disc. The main idea behind this is to make use of dynamics on Siegel discs where high iterates of a single polynomial with a Siegel disc approximate the identity arbitrarily closely on compact subsets of the Siegel disc.