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

Previous Article
Continuous limit and the moments system for the globally coupled phase oscillators
DCDS Home
This Issue
Next Article
Almost periodic and almost automorphic solutions of linear differential equations

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
Related Papers

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.

[1]	François Berteloot, Tien-Cuong Dinh. The Mandelbrot set is the shadow of a Julia set. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6611-6633. doi: 10.3934/dcds.2020262
[2]	Luke G. Rogers, Alexander Teplyaev. Laplacians on the basilica Julia set. Communications on Pure and Applied Analysis, 2010, 9 (1) : 211-231. doi: 10.3934/cpaa.2010.9.211
[3]	Artem Dudko. Computability of the Julia set. Nonrecurrent critical orbits. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2751-2778. doi: 10.3934/dcds.2014.34.2751
[4]	Koh Katagata. On a certain kind of polynomials of degree 4 with disconnected Julia set. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 975-987. doi: 10.3934/dcds.2008.20.975
[5]	Volodymyr Nekrashevych. The Julia set of a post-critically finite endomorphism of $\mathbb{PC}^2$. Journal of Modern Dynamics, 2012, 6 (3) : 327-375. doi: 10.3934/jmd.2012.6.327
[6]	Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583
[7]	Yu-Hao Liang, Wan-Rou Wu, Jonq Juang. Fastest synchronized network and synchrony on the Julia set of complex-valued coupled map lattices. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 173-184. doi: 10.3934/dcdsb.2016.21.173
[8]	Lan Wen. On the preperiodic set. Discrete and Continuous Dynamical Systems, 2000, 6 (1) : 237-241. doi: 10.3934/dcds.2000.6.237
[9]	James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667
[10]	Nancy Guelman, Jorge Iglesias, Aldo Portela. Examples of minimal set for IFSs. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5253-5269. doi: 10.3934/dcds.2017227
[11]	Mark Comerford. Non-autonomous Julia sets with measurable invariant sequences of line fields. Discrete and Continuous Dynamical Systems, 2013, 33 (2) : 629-642. doi: 10.3934/dcds.2013.33.629
[12]	Yu Zhang, Tao Chen. Minimax problems for set-valued mappings with set optimization. Numerical Algebra, Control and Optimization, 2014, 4 (4) : 327-340. doi: 10.3934/naco.2014.4.327
[13]	Sanjit Chatterjee, Chethan Kamath, Vikas Kumar. Private set-intersection with common set-up. Advances in Mathematics of Communications, 2018, 12 (1) : 17-47. doi: 10.3934/amc.2018002
[14]	Maxim Arnold, Walter Craig. On the size of the Navier - Stokes singular set. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1165-1178. doi: 10.3934/dcds.2010.28.1165
[15]	Michel Crouzeix. The annulus as a K-spectral set. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2291-2303. doi: 10.3934/cpaa.2012.11.2291
[16]	Stefanie Hittmeyer, Bernd Krauskopf, Hinke M. Osinga, Katsutoshi Shinohara. How to identify a hyperbolic set as a blender. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6815-6836. doi: 10.3934/dcds.2020295
[17]	Mohsen Tourang, Mostafa Zangiabadi, Chaoqian Li. Generalized minimal Gershgorin set for tensors. Journal of Industrial and Management Optimization, 2022 doi: 10.3934/jimo.2022106
[18]	Roger Metzger, Carlos Arnoldo Morales Rojas, Phillipe Thieullen. Topological stability in set-valued dynamics. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1965-1975. doi: 10.3934/dcdsb.2017115
[19]	Yaotang Li, Suhua Li. Exclusion sets in the Δ-type eigenvalue inclusion set for tensors. Journal of Industrial and Management Optimization, 2019, 15 (2) : 507-516. doi: 10.3934/jimo.2018054
[20]	Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Tao Chen Yunping Jiang Gaofei Zhang
on Google Scholar
Tao Chen Yunping Jiang Gaofei Zhang

[Back to Top]