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Dicritical nilpotent holomorphic foliations
Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families
1. | Department of Mathematics, Brooklyn College of CUNY, 2900 Bedford Avenue, Brooklyn, NY 11210, USA |
2. | Ph.D. Program in Mathematics, Graduate Center of CUNY, 365 Fifth Avenue, New York, NY 10016, USA |
3. | Mathematics Department, BMCC of CUNY, 199 Chambers Street, New York, NY 10007, USA |
4. | College of Mathematics and Econometrics, Hunan University, Changsha 410082, China |
In [
References:
[1] |
A. Beardon,
Iteration of Rational Functions, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-4422-6. |
[2] |
R. L. Devaney,
Dynamics of $ z^n+λ /z^n$; Why is the case $ n = 2$ crazy, Contemp. Math., 573 (2012), 49-65.
doi: 10.1090/conm/573/11379. |
[3] |
R. L. Devaney, D. M. Look and D. Uminsky,
The escape trichotomy for singularly perturbed rational maps, Indiana University Mathematics Journal, 54 (2005), 1621-1634.
doi: 10.1512/iumj.2005.54.2615. |
[4] |
R. L. Devaney and E. D. Russell, Connectivity of Julia sets for singularly perturbed rational maps, in Chaos, CNN, Memristors and Beyond, World Scientific, (2013), 239--245.
doi: 10.1142/9789814434805_0018. |
[5] |
H. M. Farkas and I. Kra,
Riemann Surfaces, Springer-Verlag, 1980. |
[6] |
J. Hu, F. G. Jimenez and O. Muzician,
Rational maps with half symmetries, Julia sets, and Multibrot sets in parameter planes, Contemp. Math., 573 (2012), 119-146.
doi: 10.1090/conm/573/11393. |
[7] |
C. McMullen, Automorphisms of rational maps, in Holomorphic Functions and Moduli, Vol. Ⅰ (Berkeley, CA, 1986), 31-60, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988.
doi: 10.1007/978-1-4613-9602-4_3. |
[8] |
J. Milnor,
Dynamics in one Complex Variable - Introductory Lectures, Friedr. Vieweg & Sohn, Braunschweig, 1999. |
[9] |
——,
On rational maps with two critical points, Experimental Mathematics, 9 (2000), 481-522.
doi: 10.1080/10586458.2000.10504657. |
[10] |
S. Morosawa,
Julia sets of sub-hyperbolic rational functions, Complex Variables Theory and Application, 41 (2000), 151-162.
doi: 10.1080/17476930008815244. |
[11] |
M. Stiemer,
Rational maps with Fatou components of arbitrary connectivity number, Computational Methods and Function Theory, 7 (2007), 415-427.
doi: 10.1007/BF03321654. |
[12] |
G. T. Whyburn,
Topological characterization of the Sierpinski curve, Fund. Math., 45 (1958), 320-324.
doi: 10.4064/fm-45-1-320-324. |
[13] |
Y. Xiao and W. Qiu,
The rational maps $ F_{λ }(z)=z^m+\frac{λ }{z^d}$ have no Herman rings, Proc. Indian Acad. Sci. (Math. Sci.), 120 (2010), 403-407.
doi: 10.1007/s12044-010-0044-x. |
[14] |
Y. Xiao, W. Qiu and Y. Yin,
On the dynamics of generalized McMullen maps, Ergod. Th. & Dynam. Sys., 34 (2014), 2093-2112.
doi: 10.1017/etds.2013.21. |
[15] |
Y. Xiao and F. Yang,
Singular perturbations of the unicritical polynomials with two parameters, Ergod. Th. & Dynam. Sys., 37 (2017), 1997-2016.
doi: 10.1017/etds.2015.114. |
show all references
References:
[1] |
A. Beardon,
Iteration of Rational Functions, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-4422-6. |
[2] |
R. L. Devaney,
Dynamics of $ z^n+λ /z^n$; Why is the case $ n = 2$ crazy, Contemp. Math., 573 (2012), 49-65.
doi: 10.1090/conm/573/11379. |
[3] |
R. L. Devaney, D. M. Look and D. Uminsky,
The escape trichotomy for singularly perturbed rational maps, Indiana University Mathematics Journal, 54 (2005), 1621-1634.
doi: 10.1512/iumj.2005.54.2615. |
[4] |
R. L. Devaney and E. D. Russell, Connectivity of Julia sets for singularly perturbed rational maps, in Chaos, CNN, Memristors and Beyond, World Scientific, (2013), 239--245.
doi: 10.1142/9789814434805_0018. |
[5] |
H. M. Farkas and I. Kra,
Riemann Surfaces, Springer-Verlag, 1980. |
[6] |
J. Hu, F. G. Jimenez and O. Muzician,
Rational maps with half symmetries, Julia sets, and Multibrot sets in parameter planes, Contemp. Math., 573 (2012), 119-146.
doi: 10.1090/conm/573/11393. |
[7] |
C. McMullen, Automorphisms of rational maps, in Holomorphic Functions and Moduli, Vol. Ⅰ (Berkeley, CA, 1986), 31-60, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988.
doi: 10.1007/978-1-4613-9602-4_3. |
[8] |
J. Milnor,
Dynamics in one Complex Variable - Introductory Lectures, Friedr. Vieweg & Sohn, Braunschweig, 1999. |
[9] |
——,
On rational maps with two critical points, Experimental Mathematics, 9 (2000), 481-522.
doi: 10.1080/10586458.2000.10504657. |
[10] |
S. Morosawa,
Julia sets of sub-hyperbolic rational functions, Complex Variables Theory and Application, 41 (2000), 151-162.
doi: 10.1080/17476930008815244. |
[11] |
M. Stiemer,
Rational maps with Fatou components of arbitrary connectivity number, Computational Methods and Function Theory, 7 (2007), 415-427.
doi: 10.1007/BF03321654. |
[12] |
G. T. Whyburn,
Topological characterization of the Sierpinski curve, Fund. Math., 45 (1958), 320-324.
doi: 10.4064/fm-45-1-320-324. |
[13] |
Y. Xiao and W. Qiu,
The rational maps $ F_{λ }(z)=z^m+\frac{λ }{z^d}$ have no Herman rings, Proc. Indian Acad. Sci. (Math. Sci.), 120 (2010), 403-407.
doi: 10.1007/s12044-010-0044-x. |
[14] |
Y. Xiao, W. Qiu and Y. Yin,
On the dynamics of generalized McMullen maps, Ergod. Th. & Dynam. Sys., 34 (2014), 2093-2112.
doi: 10.1017/etds.2013.21. |
[15] |
Y. Xiao and F. Yang,
Singular perturbations of the unicritical polynomials with two parameters, Ergod. Th. & Dynam. Sys., 37 (2017), 1997-2016.
doi: 10.1017/etds.2015.114. |











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