Bulletin of the
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Bull. Korean Math. Soc. 2011; 48(4): 713-724

Printed July 1, 2011

https://doi.org/10.4134/BKMS.2011.48.4.713

Copyright © The Korean Mathematical Society.

Dynamics of transcendental entire functions with Siegel disks and its applications

Koh Katagata

Ichinoseki National College of Technology

Abstract

We study the dynamics of transcendental entire functions with Siegel disks whose singular values are just two points. One of the two singular values is not only a superattracting fixed point with multiplicity more than two but also an asymptotic value. Another one is a critical value with free dynamics under iterations. We prove that if the multiplicity of the superattracting fixed point is large enough, then the restriction of the transcendental entire function near the Siegel point is a quadratic-like map. Therefore the Siegel disk and its boundary correspond to those of some quadratic polynomial at the level of quasiconformality. As its applications, the logarithmic lift of the above transcendental entire function has a wandering domain whose shape looks like a Siegel disk of a quadratic polynomial.

Keywords: Siegel disks, wandering domains

MSC numbers: Primary 37F50; Secondary 32A10, 37F99

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