# American Institute of Mathematical Sciences

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

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

 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.
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 - A, 2013, 33 (5) : 1883-1890. doi: 10.3934/dcds.2013.33.1883
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##### References:
 [1] M. Aspenberg and W. Bergweiler, Entire functions with Julia sets of positive measure,, Math. Ann., 352 (2012), 27.   Google Scholar [2] L. Carleson and T. Gamelin, "Complex Dynamics,", Springer-Verlag, (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.   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.   Google Scholar [5] C. McMullen, Area and Hausdorff dimension of Julia sets of entire functions,, Trans. Amer. Math. Soc., 300 (1987), 329.   Google Scholar [6] C. McMullen, "Complex Dynamics and Renormalization,", Ann. of Math. Studies, 135 (1994).   Google Scholar [7] C. McMullen, Self-similarity of Siegel disk and Hausdorff dimension of Julia set,, Acta Mathematica, 180 (1998), 247.   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.  doi: 10.1017/S0143385710000386.  Google Scholar [9] L. Rempe, Rigidity of escaping dynamics for transdental entire functions,, Acta Mathematica, 203 (2009), 235.   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.  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.  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.   Google Scholar [13] G. Zhang, On the non-escaping set of $e^{2\pi i\theta}sin(z)$,, Israel J. Math., 165 (2008), 233.   Google Scholar [14] G. Zhang, On the dynamics of $e^{2\pi i\theta}sin(z)$,, Illinois J. Math., 49 (2005), 1171.   Google Scholar
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