On Possible Limit Functions on a Fatou Component in non-Autonomous Iteration

TL;DR

This paper demonstrates that in non-autonomous polynomial iteration, the set of possible limit functions on a Fatou component is much broader than in autonomous cases, including all classical univalent functions.

Contribution

It constructs a bounded sequence of quadratic polynomials with a Fatou component whose limit functions encompass the entire Schlicht family, expanding understanding of non-autonomous dynamics.

Findings

Limit functions include all classical univalent functions.

Bounded quadratic polynomial sequences can produce diverse limit behaviors.

Quasiconformal surgery and Siegel disc approximations are effective tools.

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 should observe a far greater range of behaviour. 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.