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

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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

Abstract
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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 and Continuous Dynamical Systems, 2013, 33 (5) : 1883-1890. doi: 10.3934/dcds.2013.33.1883

References:

[1]	M. Aspenberg and W. Bergweiler, Entire functions with Julia sets of positive measure, Math. Ann., 352 (2012), 27-54.
[2]	L. Carleson and T. Gamelin, "Complex Dynamics," Springer-Verlag, New York 1991.
[3]	B. Karpińska, Area and Hausdorff dimension of the set of accessible points of the Julia sets of $\lambda e^z$ and $\lambda\sin z$, Fund. Math., 159 (1999), 269-287.
[4]	A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble), 42 (1992), 989-1020.
[5]	C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc., 300 (1987), 329-342.
[6]	C. McMullen, "Complex Dynamics and Renormalization," Ann. of Math. Studies, Vol. 135, 1994.
[7]	C. McMullen, Self-similarity of Siegel disk and Hausdorff dimension of Julia set, Acta Mathematica, 180 (1998), 247-292.
[8]	W. De Melo, P. Salomão, and E. Vargas, A full family of multimodel family of mappings on the circle, Ergodic Theory and Dynamical Systems, 31 (2011), 1325-1344. doi: 10.1017/S0143385710000386.
[9]	L. Rempe, Rigidity of escaping dynamics for transdental entire functions, Acta Mathematica, 203 (2009), 235-267.
[10]	L. Rempe and S. van Strien, Absence of line fields and Mane's theorem for non-recurrent transcendental functions, Trans Amer. Math. Soc., 363 (2011), 203-228. doi: 10.1090/S0002-9947-2010-05125-6.
[11]	D. Schleicher, The dynamical fine structure of iterated cosine maps and a dimension paradox, Duke Math. J., 136 (2007), 343-356. doi: 10.1215/S0012-7094-07-13625-1.
[12]	H. Schubert, Area of Fatou sets of trigonometric functions, Proc. Amer. Math. Soc., 136 (2008), 1251-1259.
[13]	G. Zhang, On the non-escaping set of $e^{2\pi i\theta}sin(z)$, Israel J. Math., 165 (2008), 233-252.
[14]	G. Zhang, On the dynamics of $e^{2\pi i\theta}sin(z)$, Illinois J. Math., 49 (2005), 1171-1179.

show all references

References:

[1]	M. Aspenberg and W. Bergweiler, Entire functions with Julia sets of positive measure, Math. Ann., 352 (2012), 27-54.
[2]	L. Carleson and T. Gamelin, "Complex Dynamics," Springer-Verlag, New York 1991.
[3]	B. Karpińska, Area and Hausdorff dimension of the set of accessible points of the Julia sets of $\lambda e^z$ and $\lambda\sin z$, Fund. Math., 159 (1999), 269-287.
[4]	A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble), 42 (1992), 989-1020.
[5]	C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc., 300 (1987), 329-342.
[6]	C. McMullen, "Complex Dynamics and Renormalization," Ann. of Math. Studies, Vol. 135, 1994.
[7]	C. McMullen, Self-similarity of Siegel disk and Hausdorff dimension of Julia set, Acta Mathematica, 180 (1998), 247-292.
[8]	W. De Melo, P. Salomão, and E. Vargas, A full family of multimodel family of mappings on the circle, Ergodic Theory and Dynamical Systems, 31 (2011), 1325-1344. doi: 10.1017/S0143385710000386.
[9]	L. Rempe, Rigidity of escaping dynamics for transdental entire functions, Acta Mathematica, 203 (2009), 235-267.
[10]	L. Rempe and S. van Strien, Absence of line fields and Mane's theorem for non-recurrent transcendental functions, Trans Amer. Math. Soc., 363 (2011), 203-228. doi: 10.1090/S0002-9947-2010-05125-6.
[11]	D. Schleicher, The dynamical fine structure of iterated cosine maps and a dimension paradox, Duke Math. J., 136 (2007), 343-356. doi: 10.1215/S0012-7094-07-13625-1.
[12]	H. Schubert, Area of Fatou sets of trigonometric functions, Proc. Amer. Math. Soc., 136 (2008), 1251-1259.
[13]	G. Zhang, On the non-escaping set of $e^{2\pi i\theta}sin(z)$, Israel J. Math., 165 (2008), 233-252.
[14]	G. Zhang, On the dynamics of $e^{2\pi i\theta}sin(z)$, Illinois J. Math., 49 (2005), 1171-1179.

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