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Nonhydrostatic Pollard-like internal geophysical waves
Escape quartered theorem and the connectivity of the Julia sets of a family of rational maps
1. | Department of Mathematics, Nanjing University, Nanjing 210093, China |
2. | Department of Applied Mathematics, Hunan Agricultural University, Changsha 410128, China |
3. | Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China |
$ \begin{equation*} f_{\lambda}(z) = \frac{z^{2n} - \lambda^{3n+1}}{z^n(z^{2n} - \lambda^{n - 1})} \end{equation*} $ |
$ \lambda \in \mathbb{C}^* - \{\lambda: \lambda^{2n + 2} = 1\} $ |
$ n\geq 2 $ |
$ P_{-n}(z) = z^{-n} $ |
$ \lambda \neq 0 $ |
$ J(f_\lambda) $ |
$ f_\lambda $ |
$ 0\leftrightarrow\infty $ |
$ \lambda $ |
$ J(f_\lambda) $ |
$ f_{\lambda} $ |
$ J(f_{\lambda}) $ |
$ J(f_{\lambda}) $ |
$ 0\leftrightarrow\infty $ |
$ f_\lambda $ |
References:
[1] |
A. F. Beardon, Iteration of Rational Functions, Grad. Texts in Math., 132, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-4422-6. |
[2] |
M. Bonk, M. Lyubich and S. Merenkov,
Quasisymmetries of Sierpiński carpet Julia sets, Adv. Math., 301 (2016), 383-422.
doi: 10.1016/j.aim.2016.06.007. |
[3] |
L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer, New York, N.Y., 1993.
doi: 10.1007/978-1-4612-4364-9. |
[4] |
R. L. Devaney, N. Fagella, A. Garijo and X. Jarque,
Sierpiński curve Julia sets for quadratic rational maps, Ann. Acad. Sci. Fenn. Math., 39 (2014), 3-22.
doi: 10.5186/aasfm.2014.3903. |
[5] |
R. L. Devaney, D. Look and D. Uminsky,
The escape trichotomy for singularly perturbed rational maps, Indiana. Univ. Math. J., 54 (2005), 1621-1634.
doi: 10.1512/iumj.2005.54.2615. |
[6] |
R. L. Devaney,
Singular perturbations of complex polynomials, Bull. Amer. Math. Soc., 50 (2013), 391-429.
doi: 10.1090/S0273-0979-2013-01410-1. |
[7] |
R. L. Devaney and E. D. Russell, Connectivity of Julia sets for singularly perturbed rational maps, Chaos, CNN, Memristors and Beyond, World Scientific, 2013,239–245.
doi: 10.1142/9789814434805_0018. |
[8] |
A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École. Norm. Sup., 18 (1985), 287–343.
doi: 10.24033/asens.1491. |
[9] |
J. Fu and F. Yang,
On the dynamics of a family of singularly perturbed rational maps, J. Math. Anal. Appl., 424 (2015), 104-121.
doi: 10.1016/j.jmaa.2014.10.090. |
[10] |
J. Fu and Y. Zhang, Connectivity of the Julia sets of singularly perturbed rational maps, Proc. Indian Acad. Sci. Math. Sci., 129 (2019), 32.
doi: 10.1007/s12044-019-0478-8. |
[11] |
Y. Fu and F. Yang, Area and Hausdorff dimension of Sierpiński carpet Julia sets, to appear in Math. Z., (2019). arXiv: 1812.03016.
doi: 10.1007/s00209-019-02319-4. |
[12] |
A. Garijo and S. Godillon,
On McMullen-like mappings, J. Fractal Geom., 2 (2015), 249-279.
doi: 10.4171/JFG/21. |
[13] |
A. Garijo, S. M. Marotta and E. D. Russell,
Singular perturbations in the quadratic family with multiple poles, J. Difference Equ. Appl., 19 (2013), 124-145.
doi: 10.1080/10236198.2011.630668. |
[14] |
P. Haïsinsky and K. Pilgrim,
Quasisymmetrically inequivalent hyperbolic Julia sets, Rev. Mat. Iberoam., 28 (2012), 1025-1034.
doi: 10.4171/RMI/701. |
[15] |
J. Hu, O. Muzician and Y. Xiao,
Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families, Discrete Contin. Dyn. Syst., 38 (2018), 3189-3221.
doi: 10.3934/dcds.2018139. |
[16] |
J. Hu and Y. Xiao,
No Herman rings for regularly ramified rational maps, Proc. Amer. Math. Soc., 147 (2019), 1587-1596.
doi: 10.1090/proc/14347. |
[17] |
C. T. McMullen, Automorphisms of rational maps, In Holomorphic functions and moduli I, Mathematical Sciences Research Institute Publications, Springer-Verlag, New York, NY, 10 (1988), 31–60.
doi: 10.1007/978-1-4613-9602-4_3. |
[18] |
J. Milnor, Dynamics in one Complex Variable, Third Edition, Annals of Mathematics Studies,
160, Princeton Univ. Press, Princeton, NJ, 2006. |
[19] |
M. Pilgrim and L. Tan, Rational maps with disconnected Julia set, in Géométrie Complexe Et Systèmes Dynamiques, Astérisque, 261 (2000), 349–384. |
[20] |
F. Przytycki,
On the hyperbolic Hausdorff dimension of the boundary of a basin of attraction for a holomorphic map and of quasirepellers, Bull. Pol. Acad. Sci. Math., 54 (2006), 41-52.
doi: 10.4064/ba54-1-4. |
[21] |
W. Qiu, X. Wang and Y. Yin,
Dynamics of McMullen maps, Adv. Math., 229 (2012), 2525-2577.
doi: 10.1016/j.aim.2011.12.026. |
[22] |
W. Qiu and F. Yang, Hausdorff dimension and quasi-symmetric uniformization of Cantor circle Julia sets, arXiv: 1811.10042, 2018. |
[23] |
W. Qiu, F. Yang and Y. Yin,
Rational maps whose Julia sets are Cantor circles, Ergodic Theory Dynam. Systems, 35 (2015), 499-529.
doi: 10.1017/etds.2013.53. |
[24] |
W. Qiu, F. Yang and Y. Yin,
Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps, Discrete Contin. Dynam. Sys., 36 (2016), 3375-3416.
doi: 10.3934/dcds.2016.36.3375. |
[25] |
W. Qiu, F. Yang and J. Zeng,
Quasisymmetric geometry of the carpet Julia sets, Fund. Math., 244 (2019), 73-107.
doi: 10.4064/fm494-12-2017. |
[26] |
D. Sullivan, Conformal dynamical systems, Geometric Dynamics (Rio de Janeiro, 1981), 725–752, Lecture Notes in Math., 1007, Springer, Berlin, 1983.
doi: 10.1007/BFb0061443. |
[27] |
Y. Wang and F. Yang, Julia sets as buried Julia components, arXiv: 1707.04852, 2017. |
[28] |
G. T. Whyburn,
Topological characterization of the Sierpiński curves, Fund. Math., 45 (1958), 320-324.
doi: 10.4064/fm-45-1-320-324. |
[29] |
Y. Xiao and W. Qiu,
The rational maps $F_\lambda(z)=z^m+\lambda/z^d$ have no Herman rings, Proc. Indian Acad. Sci. Math. Sci., 120 (2010), 403-407.
doi: 10.1007/s12044-010-0044-x. |
[30] |
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. |
[31] |
Y. Xiao and F. Yang,
Singular perturbations with multiple poles of the simple polynomials, Qual. Theory Dyn. Syst., 16 (2017), 731-747.
doi: 10.1007/s12346-016-0205-0. |
[32] |
Y. Xiao and F. Yang,
Singular perturbations of unicritical polynomials with two parameters, Ergod. Th. Dynam. Sys., 37 (2017), 1997-2016.
doi: 10.1017/etds.2015.114. |
[33] |
F. Yang,
Rational maps without Herman rings, Proc. Amer. Math. Sci, 145 (2017), 1649-1659.
doi: 10.1090/proc/13336. |
[34] |
F. Yang,
A criterion to generate carpet Julia sets, Proc. Amer. Math. Soc., 146 (2018), 2129-2141.
doi: 10.1090/proc/13924. |
show all references
References:
[1] |
A. F. Beardon, Iteration of Rational Functions, Grad. Texts in Math., 132, Springer-Verlag, New York, 1991.
doi: 10.1007/978-1-4612-4422-6. |
[2] |
M. Bonk, M. Lyubich and S. Merenkov,
Quasisymmetries of Sierpiński carpet Julia sets, Adv. Math., 301 (2016), 383-422.
doi: 10.1016/j.aim.2016.06.007. |
[3] |
L. Carleson and T. W. Gamelin, Complex Dynamics, Universitext: Tracts in Mathematics, Springer, New York, N.Y., 1993.
doi: 10.1007/978-1-4612-4364-9. |
[4] |
R. L. Devaney, N. Fagella, A. Garijo and X. Jarque,
Sierpiński curve Julia sets for quadratic rational maps, Ann. Acad. Sci. Fenn. Math., 39 (2014), 3-22.
doi: 10.5186/aasfm.2014.3903. |
[5] |
R. L. Devaney, D. Look and D. Uminsky,
The escape trichotomy for singularly perturbed rational maps, Indiana. Univ. Math. J., 54 (2005), 1621-1634.
doi: 10.1512/iumj.2005.54.2615. |
[6] |
R. L. Devaney,
Singular perturbations of complex polynomials, Bull. Amer. Math. Soc., 50 (2013), 391-429.
doi: 10.1090/S0273-0979-2013-01410-1. |
[7] |
R. L. Devaney and E. D. Russell, Connectivity of Julia sets for singularly perturbed rational maps, Chaos, CNN, Memristors and Beyond, World Scientific, 2013,239–245.
doi: 10.1142/9789814434805_0018. |
[8] |
A. Douady and J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École. Norm. Sup., 18 (1985), 287–343.
doi: 10.24033/asens.1491. |
[9] |
J. Fu and F. Yang,
On the dynamics of a family of singularly perturbed rational maps, J. Math. Anal. Appl., 424 (2015), 104-121.
doi: 10.1016/j.jmaa.2014.10.090. |
[10] |
J. Fu and Y. Zhang, Connectivity of the Julia sets of singularly perturbed rational maps, Proc. Indian Acad. Sci. Math. Sci., 129 (2019), 32.
doi: 10.1007/s12044-019-0478-8. |
[11] |
Y. Fu and F. Yang, Area and Hausdorff dimension of Sierpiński carpet Julia sets, to appear in Math. Z., (2019). arXiv: 1812.03016.
doi: 10.1007/s00209-019-02319-4. |
[12] |
A. Garijo and S. Godillon,
On McMullen-like mappings, J. Fractal Geom., 2 (2015), 249-279.
doi: 10.4171/JFG/21. |
[13] |
A. Garijo, S. M. Marotta and E. D. Russell,
Singular perturbations in the quadratic family with multiple poles, J. Difference Equ. Appl., 19 (2013), 124-145.
doi: 10.1080/10236198.2011.630668. |
[14] |
P. Haïsinsky and K. Pilgrim,
Quasisymmetrically inequivalent hyperbolic Julia sets, Rev. Mat. Iberoam., 28 (2012), 1025-1034.
doi: 10.4171/RMI/701. |
[15] |
J. Hu, O. Muzician and Y. Xiao,
Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families, Discrete Contin. Dyn. Syst., 38 (2018), 3189-3221.
doi: 10.3934/dcds.2018139. |
[16] |
J. Hu and Y. Xiao,
No Herman rings for regularly ramified rational maps, Proc. Amer. Math. Soc., 147 (2019), 1587-1596.
doi: 10.1090/proc/14347. |
[17] |
C. T. McMullen, Automorphisms of rational maps, In Holomorphic functions and moduli I, Mathematical Sciences Research Institute Publications, Springer-Verlag, New York, NY, 10 (1988), 31–60.
doi: 10.1007/978-1-4613-9602-4_3. |
[18] |
J. Milnor, Dynamics in one Complex Variable, Third Edition, Annals of Mathematics Studies,
160, Princeton Univ. Press, Princeton, NJ, 2006. |
[19] |
M. Pilgrim and L. Tan, Rational maps with disconnected Julia set, in Géométrie Complexe Et Systèmes Dynamiques, Astérisque, 261 (2000), 349–384. |
[20] |
F. Przytycki,
On the hyperbolic Hausdorff dimension of the boundary of a basin of attraction for a holomorphic map and of quasirepellers, Bull. Pol. Acad. Sci. Math., 54 (2006), 41-52.
doi: 10.4064/ba54-1-4. |
[21] |
W. Qiu, X. Wang and Y. Yin,
Dynamics of McMullen maps, Adv. Math., 229 (2012), 2525-2577.
doi: 10.1016/j.aim.2011.12.026. |
[22] |
W. Qiu and F. Yang, Hausdorff dimension and quasi-symmetric uniformization of Cantor circle Julia sets, arXiv: 1811.10042, 2018. |
[23] |
W. Qiu, F. Yang and Y. Yin,
Rational maps whose Julia sets are Cantor circles, Ergodic Theory Dynam. Systems, 35 (2015), 499-529.
doi: 10.1017/etds.2013.53. |
[24] |
W. Qiu, F. Yang and Y. Yin,
Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps, Discrete Contin. Dynam. Sys., 36 (2016), 3375-3416.
doi: 10.3934/dcds.2016.36.3375. |
[25] |
W. Qiu, F. Yang and J. Zeng,
Quasisymmetric geometry of the carpet Julia sets, Fund. Math., 244 (2019), 73-107.
doi: 10.4064/fm494-12-2017. |
[26] |
D. Sullivan, Conformal dynamical systems, Geometric Dynamics (Rio de Janeiro, 1981), 725–752, Lecture Notes in Math., 1007, Springer, Berlin, 1983.
doi: 10.1007/BFb0061443. |
[27] |
Y. Wang and F. Yang, Julia sets as buried Julia components, arXiv: 1707.04852, 2017. |
[28] |
G. T. Whyburn,
Topological characterization of the Sierpiński curves, Fund. Math., 45 (1958), 320-324.
doi: 10.4064/fm-45-1-320-324. |
[29] |
Y. Xiao and W. Qiu,
The rational maps $F_\lambda(z)=z^m+\lambda/z^d$ have no Herman rings, Proc. Indian Acad. Sci. Math. Sci., 120 (2010), 403-407.
doi: 10.1007/s12044-010-0044-x. |
[30] |
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. |
[31] |
Y. Xiao and F. Yang,
Singular perturbations with multiple poles of the simple polynomials, Qual. Theory Dyn. Syst., 16 (2017), 731-747.
doi: 10.1007/s12346-016-0205-0. |
[32] |
Y. Xiao and F. Yang,
Singular perturbations of unicritical polynomials with two parameters, Ergod. Th. Dynam. Sys., 37 (2017), 1997-2016.
doi: 10.1017/etds.2015.114. |
[33] |
F. Yang,
Rational maps without Herman rings, Proc. Amer. Math. Sci, 145 (2017), 1649-1659.
doi: 10.1090/proc/13336. |
[34] |
F. Yang,
A criterion to generate carpet Julia sets, Proc. Amer. Math. Soc., 146 (2018), 2129-2141.
doi: 10.1090/proc/13924. |




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