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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
References:
[1]

M. Aspenberg and W. Bergweiler, Entire functions with Julia sets of positive measure, Math. Ann., 352 (2012), 27-54.  Google Scholar

[2]

L. Carleson and T. Gamelin, "Complex Dynamics," Springer-Verlag, New York 1991.  Google Scholar

[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.  Google Scholar

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A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble), 42 (1992), 989-1020.  Google Scholar

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C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc., 300 (1987), 329-342.  Google Scholar

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C. McMullen, "Complex Dynamics and Renormalization," Ann. of Math. Studies, Vol. 135, 1994.  Google Scholar

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C. McMullen, Self-similarity of Siegel disk and Hausdorff dimension of Julia set, Acta Mathematica, 180 (1998), 247-292.  Google Scholar

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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.  Google Scholar

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L. Rempe, Rigidity of escaping dynamics for transdental entire functions, Acta Mathematica, 203 (2009), 235-267.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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H. Schubert, Area of Fatou sets of trigonometric functions, Proc. Amer. Math. Soc., 136 (2008), 1251-1259.  Google Scholar

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G. Zhang, On the dynamics of $e^{2\pi i\theta}sin(z)$, Illinois J. Math., 49 (2005), 1171-1179.  Google Scholar

show all references

References:
[1]

M. Aspenberg and W. Bergweiler, Entire functions with Julia sets of positive measure, Math. Ann., 352 (2012), 27-54.  Google Scholar

[2]

L. Carleson and T. Gamelin, "Complex Dynamics," Springer-Verlag, New York 1991.  Google Scholar

[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.  Google Scholar

[4]

A. E. Eremenko and M. Yu. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. Fourier (Grenoble), 42 (1992), 989-1020.  Google Scholar

[5]

C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions, Trans. Amer. Math. Soc., 300 (1987), 329-342.  Google Scholar

[6]

C. McMullen, "Complex Dynamics and Renormalization," Ann. of Math. Studies, Vol. 135, 1994.  Google Scholar

[7]

C. McMullen, Self-similarity of Siegel disk and Hausdorff dimension of Julia set, Acta Mathematica, 180 (1998), 247-292.  Google Scholar

[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.  Google Scholar

[9]

L. Rempe, Rigidity of escaping dynamics for transdental entire functions, Acta Mathematica, 203 (2009), 235-267.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

H. Schubert, Area of Fatou sets of trigonometric functions, Proc. Amer. Math. Soc., 136 (2008), 1251-1259.  Google Scholar

[13]

G. Zhang, On the non-escaping set of $e^{2\pi i\theta}sin(z)$, Israel J. Math., 165 (2008), 233-252.  Google Scholar

[14]

G. Zhang, On the dynamics of $e^{2\pi i\theta}sin(z)$, Illinois J. Math., 49 (2005), 1171-1179.  Google Scholar

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