September  2019, 39(9): 5185-5206. doi: 10.3934/dcds.2019211

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

* Corresponding author: Youming Wang

Received  August 2018 Revised  April 2019 Published  May 2019

Fund Project: This work is supported by the NSFC (grant Nos. 11671092, 11671191, 11871208), by the Scientific Research Foundation of Hunan Provincial Education Department (grant No. 16C0763), by Natural Science Foundation of Hunan Province (grant No. 2018JJ2159) and by the Fundamental Research Funds for the Central Universities (grant Nos. 0203-14380022 and 0203-14380025).

In this paper, we investigate the dynamics of the following family of rational maps
$ \begin{equation*} f_{\lambda}(z) = \frac{z^{2n} - \lambda^{3n+1}}{z^n(z^{2n} - \lambda^{n - 1})} \end{equation*} $
with one parameter
$ \lambda \in \mathbb{C}^* - \{\lambda: \lambda^{2n + 2} = 1\} $
, where
$ n\geq 2 $
. This family of rational maps is viewed as a singular perturbation of the bi-critical map
$ P_{-n}(z) = z^{-n} $
if
$ \lambda \neq 0 $
is small. It is proved that the Julia set
$ J(f_\lambda) $
is either a quasicircle, a Cantor set of circles, a Sierpiński carpet or a degenerate Sierpiński carpet provided the free critical orbits of
$ f_\lambda $
are attracted by the super-attracting cycle
$ 0\leftrightarrow\infty $
. Furthermore, we prove that there exists suitable
$ \lambda $
such that
$ J(f_\lambda) $
is a Cantor set of circles but the dynamics of
$ f_{\lambda} $
on
$ J(f_{\lambda}) $
is not topologically conjugate to that of any known rational maps with only one or two free critical orbits (including McMullen maps and the generalized McMullen maps). The connectivity of
$ J(f_{\lambda}) $
is also proved if the free critical orbits are not attracted by the cycle
$ 0\leftrightarrow\infty $
. Finally we give an estimate of the Hausdorff dimension of the Julia set of
$ f_\lambda $
in some special cases.
Citation: Youming Wang, Fei Yang, Song Zhang, Liangwen Liao. Escape quartered theorem and the connectivity of the Julia sets of a family of rational maps. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5185-5206. doi: 10.3934/dcds.2019211
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.  Google Scholar

[2]

M. BonkM. Lyubich and S. Merenkov, Quasisymmetries of Sierpiński carpet Julia sets, Adv. Math., 301 (2016), 383-422.  doi: 10.1016/j.aim.2016.06.007.  Google Scholar

[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.  Google Scholar

[4]

R. L. DevaneyN. FagellaA. 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.  Google Scholar

[5]

R. L. DevaneyD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

A. Garijo and S. Godillon, On McMullen-like mappings, J. Fractal Geom., 2 (2015), 249-279.  doi: 10.4171/JFG/21.  Google Scholar

[13]

A. GarijoS. 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.  Google Scholar

[14]

P. Haïsinsky and K. Pilgrim, Quasisymmetrically inequivalent hyperbolic Julia sets, Rev. Mat. Iberoam., 28 (2012), 1025-1034.  doi: 10.4171/RMI/701.  Google Scholar

[15]

J. HuO. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. Milnor, Dynamics in one Complex Variable, Third Edition, Annals of Mathematics Studies, 160, Princeton Univ. Press, Princeton, NJ, 2006.  Google Scholar

[19]

M. Pilgrim and L. Tan, Rational maps with disconnected Julia set, in Géométrie Complexe Et Systèmes Dynamiques, Astérisque, 261 (2000), 349–384.  Google Scholar

[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.  Google Scholar

[21]

W. QiuX. Wang and Y. Yin, Dynamics of McMullen maps, Adv. Math., 229 (2012), 2525-2577.  doi: 10.1016/j.aim.2011.12.026.  Google Scholar

[22]

W. Qiu and F. Yang, Hausdorff dimension and quasi-symmetric uniformization of Cantor circle Julia sets, arXiv: 1811.10042, 2018. Google Scholar

[23]

W. QiuF. 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.  Google Scholar

[24]

W. QiuF. 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.  Google Scholar

[25]

W. QiuF. Yang and J. Zeng, Quasisymmetric geometry of the carpet Julia sets, Fund. Math., 244 (2019), 73-107.  doi: 10.4064/fm494-12-2017.  Google Scholar

[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.  Google Scholar

[27]

Y. Wang and F. Yang, Julia sets as buried Julia components, arXiv: 1707.04852, 2017. Google Scholar

[28]

G. T. Whyburn, Topological characterization of the Sierpiński curves, Fund. Math., 45 (1958), 320-324.  doi: 10.4064/fm-45-1-320-324.  Google Scholar

[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.  Google Scholar

[30]

Y. XiaoW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

F. Yang, Rational maps without Herman rings, Proc. Amer. Math. Sci, 145 (2017), 1649-1659.  doi: 10.1090/proc/13336.  Google Scholar

[34]

F. Yang, A criterion to generate carpet Julia sets, Proc. Amer. Math. Soc., 146 (2018), 2129-2141.  doi: 10.1090/proc/13924.  Google Scholar

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.  Google Scholar

[2]

M. BonkM. Lyubich and S. Merenkov, Quasisymmetries of Sierpiński carpet Julia sets, Adv. Math., 301 (2016), 383-422.  doi: 10.1016/j.aim.2016.06.007.  Google Scholar

[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.  Google Scholar

[4]

R. L. DevaneyN. FagellaA. 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.  Google Scholar

[5]

R. L. DevaneyD. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

A. Garijo and S. Godillon, On McMullen-like mappings, J. Fractal Geom., 2 (2015), 249-279.  doi: 10.4171/JFG/21.  Google Scholar

[13]

A. GarijoS. 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.  Google Scholar

[14]

P. Haïsinsky and K. Pilgrim, Quasisymmetrically inequivalent hyperbolic Julia sets, Rev. Mat. Iberoam., 28 (2012), 1025-1034.  doi: 10.4171/RMI/701.  Google Scholar

[15]

J. HuO. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. Milnor, Dynamics in one Complex Variable, Third Edition, Annals of Mathematics Studies, 160, Princeton Univ. Press, Princeton, NJ, 2006.  Google Scholar

[19]

M. Pilgrim and L. Tan, Rational maps with disconnected Julia set, in Géométrie Complexe Et Systèmes Dynamiques, Astérisque, 261 (2000), 349–384.  Google Scholar

[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.  Google Scholar

[21]

W. QiuX. Wang and Y. Yin, Dynamics of McMullen maps, Adv. Math., 229 (2012), 2525-2577.  doi: 10.1016/j.aim.2011.12.026.  Google Scholar

[22]

W. Qiu and F. Yang, Hausdorff dimension and quasi-symmetric uniformization of Cantor circle Julia sets, arXiv: 1811.10042, 2018. Google Scholar

[23]

W. QiuF. 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.  Google Scholar

[24]

W. QiuF. 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.  Google Scholar

[25]

W. QiuF. Yang and J. Zeng, Quasisymmetric geometry of the carpet Julia sets, Fund. Math., 244 (2019), 73-107.  doi: 10.4064/fm494-12-2017.  Google Scholar

[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.  Google Scholar

[27]

Y. Wang and F. Yang, Julia sets as buried Julia components, arXiv: 1707.04852, 2017. Google Scholar

[28]

G. T. Whyburn, Topological characterization of the Sierpiński curves, Fund. Math., 45 (1958), 320-324.  doi: 10.4064/fm-45-1-320-324.  Google Scholar

[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.  Google Scholar

[30]

Y. XiaoW. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

F. Yang, Rational maps without Herman rings, Proc. Amer. Math. Sci, 145 (2017), 1649-1659.  doi: 10.1090/proc/13336.  Google Scholar

[34]

F. Yang, A criterion to generate carpet Julia sets, Proc. Amer. Math. Soc., 146 (2018), 2129-2141.  doi: 10.1090/proc/13924.  Google Scholar

Figure 1.  The Julia sets of $ f_\lambda $ for different $ \lambda $'s when $ n = 4 $. Top left: $ \lambda = 0.8 + 0.3 \rm{i} $ and $ J(f_\lambda) $ is a quasicircle; Top right: $ \lambda = 0.4 $ and $ J(f_\lambda) $ is a Cantor set of circles; Bottom left: $ \lambda = 0.7 $ and $ J(f_\lambda) $ is a Sierpiński carpet; Bottom right: $ \lambda = 0.92 + 0.01 \rm{i} $ and $ J(f_\lambda) $ is a degenerate Sierpiński carpet
Figure 2.  The non-escaping loci of $ f_\lambda $, where $ n = 3 $ and $ 4 $. Left: $ n = 3 $, the McMullen domain does not exist and the Julia set $ J(f_\lambda) $ cannot be a Cantor set of circles; Right: $ n = 4 $, there is a punctured domain centered at origin which corresponds to the McMullen domain (the big white part in the center)
Figure 3.  The above and below pictures illustrate the mapping relations of $ h_\lambda $ (see (1)) and $ f_\lambda $ respectively when $ D_0 $ contains one of the free critical values but contains no free critical points. One can observe clearly that $ f_{\lambda} $ and $ h_{\lambda} $ are not topologically conjugate on their corresponding Julia sets
Figure 4.  The Julia sets of $ f_\lambda $ with $ n = 4 $, $ \lambda = 0.4 $ and $ F(z) = z^3 + 0.01/z^3 $. Both of them are Cantor circles. But $ f_{\lambda} $ and $ F $ are not topologically conjugate on their corresponding Julia sets
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