No invariant line fields on escaping sets of the family $\lambda e^{iz}+\gamma e^{-iz}$

Tao Chen; Yunping Jiang; Gaofei Zhang

doi:10.3934/dcds.2013.33.1883

Article Contents

2013, Volume 33, Issue 5: 1883-1890. Doi: 10.3934/dcds.2013.33.1883

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No invariant line fields on escaping sets of the family $\lambda e^{iz}+\gamma e^{-iz}$

1.
Department of Mathematics, Graduate Center, CUNY, 365 Fifth Avenue, New York, NY 10016
2.
Department of Mathematics, Queens College, Flushing, NY 11367
3.
Department of Mathematics, Nanjing University, Nanjing 210090

Received: August 31, 2011

Revised: July 31, 2012

Published: April 2013

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Abstract

Consider the family $f_{\lambda, \gamma}(z) = \lambda e^{iz}+\gamma e^{-iz}$ where $\lambda$ and $\gamma$ are non-zero complex numbers. It contains the sine family $\lambda \sin z$ and is a natural extension of the sine family. We give a direct proof of that the escaping set $I_{\lambda, \gamma}$ of $f_{\lambda, \gamma}$ supports no $f_{\lambda,\gamma}$-invariant line fields.

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Mathematics Subject Classification: Primary: 37F10; Secondary: 37F15.

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