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May  2013, 33(5): 1883-1890. doi: 10.3934/dcds.2013.33.1883

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

Tao Chen 1, , Yunping Jiang 2, and  Gaofei Zhang 3,

1. 

Department of Mathematics, Graduate Center, CUNY, 365 Fifth Avenue, New York, NY 10016, United States
2. 

Department of Mathematics, Queens College, Flushing, NY 11367, United States
3. 

Department of Mathematics, Nanjing University, Nanjing 210090, China

Received  September 2011 Revised  August 2012 Published  December 2012

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.
Keywords: Escaping set, univalent and Koebe Distortion., line field, Julia set.
Mathematics Subject Classification: Primary: 37F10; Secondary: 37F1.
Citation: Tao Chen, Yunping Jiang, Gaofei Zhang. No invariant line fields on escaping sets of the family $\lambda e^{iz}+\gamma e^{-iz}$. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1883-1890. doi: 10.3934/dcds.2013.33.1883
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show all references

References:
[1]

Math. Ann., 352 (2012), 27-54.  Google Scholar

[2]

Springer-Verlag, New York 1991.  Google Scholar

[3]

Fund. Math., 159 (1999), 269-287.  Google Scholar

[4]

Ann. Inst. Fourier (Grenoble), 42 (1992), 989-1020.  Google Scholar

[5]

Trans. Amer. Math. Soc., 300 (1987), 329-342.  Google Scholar

[6]

Ann. of Math. Studies, Vol. 135, 1994.  Google Scholar

[7]

Acta Mathematica, 180 (1998), 247-292.  Google Scholar

[8]

Ergodic Theory and Dynamical Systems, 31 (2011), 1325-1344. doi: 10.1017/S0143385710000386.  Google Scholar

[9]

Acta Mathematica, 203 (2009), 235-267.  Google Scholar

[10]

Trans Amer. Math. Soc., 363 (2011), 203-228. doi: 10.1090/S0002-9947-2010-05125-6.  Google Scholar

[11]

Duke Math. J., 136 (2007), 343-356. doi: 10.1215/S0012-7094-07-13625-1.  Google Scholar

[12]

Proc. Amer. Math. Soc., 136 (2008), 1251-1259.  Google Scholar

[13]

Israel J. Math., 165 (2008), 233-252.  Google Scholar

[14]

Illinois J. Math., 49 (2005), 1171-1179.  Google Scholar

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