| 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, with so much freedom in choosing the particular polynomial sequence, one expects to observe a far greater range of behaviour. We show this is indeed the case and that it is possible to obtain the \emph{whole} of the classical Schlicht family of normalized univalent functions on the unit disc as limit functions on a \emph{single} Fatou component for a \emph{single} bounded sequence of quadratic polynomials.
The main ideas behind this are quasiconformal surgery and the feature of dynamics on Siegel discs where suitable high iterates of a single polynomial with a Siegel disc U approximate the identity closely on compact subsets of U. This allows us both to approximate many functions from the Schlicht family on a Fatou component and to correct the small but inevitable errors arising from these approximations. Do almost nothing and you can do almost anything! |
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