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

Mark Comerford, Christopher Staniszewski

TL;DR

This work studies non-autonomous iteration of bounded polynomial sequences and proves that a bounded sequence of quadratic polynomials can generate, on a common Fatou component, every function in the classical Schlicht class $\mathcal{S}$ as a limit of iterates. The authors construct a two-phase method: Phase I builds polynomial compositions that approximate prescribed univalent germs on a scaled Siegel disc, while Phase II corrects residual errors on a slightly smaller domain via quasiconformal interpolation and the Polynomial Implementation Lemma. The combination of high-iterate near-identity dynamics on Siegel discs and careful hyperbolic-geometry bookkeeping yields a robust mechanism to realize all Schlicht functions as local limits, demonstrating a rich range of limit behaviors in non-autonomous settings. The results establish a new link between non-autonomous polynomial dynamics and univalent function theory, with implications for the study of normal families and domain variation across iterates.

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.

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

TL;DR

This work studies non-autonomous iteration of bounded polynomial sequences and proves that a bounded sequence of quadratic polynomials can generate, on a common Fatou component, every function in the classical Schlicht class as a limit of iterates. The authors construct a two-phase method: Phase I builds polynomial compositions that approximate prescribed univalent germs on a scaled Siegel disc, while Phase II corrects residual errors on a slightly smaller domain via quasiconformal interpolation and the Polynomial Implementation Lemma. The combination of high-iterate near-identity dynamics on Siegel discs and careful hyperbolic-geometry bookkeeping yields a robust mechanism to realize all Schlicht functions as local limits, demonstrating a rich range of limit behaviors in non-autonomous settings. The results establish a new link between non-autonomous polynomial dynamics and univalent function theory, with implications for the study of normal families and domain variation across iterates.

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.

Paper Structure

This paper contains 24 sections, 49 theorems, 190 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Non-Autonomous Iteration
  3. The Schlicht Class
  4. Statement of the Main Theorem
  5. Related Results
  6. Acknowledgements
  7. Background
  8. Classical Results on $\mathcal{S}$
  9. The Hyperbolic Metric
  10. Star-Shaped Domains
  11. The Carathéodory Topology
  12. The Polynomial Implementation Lemma
  13. Setup
  14. Statement and Proof of the Polynomial Implementation Lemma
  15. Phase I
  16. ...and 9 more sections

Key Result

Theorem 1.1

For any $m \le n \in \mathbb N$, $Q_{m,n}({\mathcal{J}}_m) = {\mathcal{J}}_n$ and $Q_{m,n}({\mathcal{F}}_m) = {\mathcal{F}}_n$, with Fatou components of ${\mathcal{F}}_m$ being mapped surjectively onto those of ${\mathcal{F}}_n$ by $Q_{m,n}$.

Figures (7)

  • Figure 1: The filled Julia set $\mathcal{K}_\lambda$ for $P_\lambda$ with Siegel disc highlighted
  • Figure 2: Supports of Dilatations Converging to Zero Almost Everywhere
  • Figure 3: The filled Julia Set $\mathcal{K}$ for $P$ with the Green's Lines $\partial V_h = \{z: G(z)=h \}$ and $\partial V_{2h} = \{z: G(z)=2h \}$
  • Figure 4: Finding a lower bound for $\rho_{{\tilde{V}_{2h}}}(0,z_0)$
  • Figure 5: The Setup for Phase II in Rotated Logarithmic Coordinates
  • ...and 2 more figures

Theorems & Definitions (100)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • ...and 90 more