July  2011, 29(3): 929-952. doi: 10.3934/dcds.2011.29.929

On the topology of wandering Julia components

1. 

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, China

2. 

Département de Mathématiques, Université d'Angers, Angers, 49045, France

Received  December 2009 Revised  May 2010 Published  November 2010

It is known that for a rational map $f$ with a disconnected Julia set, the set of wandering Julia components is uncountable. We prove that all but countably many of them have a simple topology, namely having one or two complementary components. We show that the remaining countable subset $\Sigma$ is backward invariant. Conjecturally $\Sigma$ does not contain an infinite orbit. We give a very strong necessary condition for $\Sigma$ to contain an infinite orbit, thus proving the conjecture for many different cases. We provide also two sufficient conditions for a Julia component to be a point. Finally we construct several examples describing different topological structures of Julia components.
Citation: Guizhen Cui, Wenjuan Peng, Lei Tan. On the topology of wandering Julia components. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 929-952. doi: 10.3934/dcds.2011.29.929
References:
[1]

A. F. Beardon, "Iteration of Rational Functions," Springer-Verlag, New York, 1991.

[2]

O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. London Math. Soc., 99 (2009), 275-296. doi: 10.1112/plms/pdn055.

[3]

C. T. McMullen, Automorphisms of rational maps, in "Holomorphic Functions and Moduli I" (eds. D. Drasin, C. J. Earle, F. W. Gehring, I. Kra and A. Marden), Springer, (1988), 31-60.

[4]

J. Milnor, "Dynamics in One Complex Variable," 3rd edition, Princeton University Press, 2006.

[5]

J. Milnor, On rational maps with two critical points, Experimental Mathematics, 9 (2000), 481-522.

[6]

K. Pilgrim and L. Tan, Rational maps with disconnected Julia set, Astérique, 261 (2000), 349-384.

[7]

K. Pilgrim and L. Tan, On disc-annulus surgery of rational maps, in "Proceedings of the International Conference in Dynamical Systems in Honor of Professor Liao Shan-tao 1998" (eds. Y. Jiang and L. Wen), World Scientific, (1999), 237-250.

[8]

W. Qiu and Y. Yin, Proof of the Branner-Hubbard conjecture on Cantor Julia sets, Sci. China Ser. A, 52 (2009), 45-65. doi: 10.1007/s11425-008-0178-9.

[9]

M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup., 20 (1987), 1-29.

[10]

D. Sullivan, Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains, Ann. of Math., 122 (1985), 401-418. doi: 10.2307/1971308.

[11]

Y. Zhai, A generalized version of Branner-Hubbard conjecture for rational functions, Acta Mathematica Sinica, 26 (2010), 2199-2208. doi: 10.1007/s10114-010-7632-7.

show all references

References:
[1]

A. F. Beardon, "Iteration of Rational Functions," Springer-Verlag, New York, 1991.

[2]

O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. London Math. Soc., 99 (2009), 275-296. doi: 10.1112/plms/pdn055.

[3]

C. T. McMullen, Automorphisms of rational maps, in "Holomorphic Functions and Moduli I" (eds. D. Drasin, C. J. Earle, F. W. Gehring, I. Kra and A. Marden), Springer, (1988), 31-60.

[4]

J. Milnor, "Dynamics in One Complex Variable," 3rd edition, Princeton University Press, 2006.

[5]

J. Milnor, On rational maps with two critical points, Experimental Mathematics, 9 (2000), 481-522.

[6]

K. Pilgrim and L. Tan, Rational maps with disconnected Julia set, Astérique, 261 (2000), 349-384.

[7]

K. Pilgrim and L. Tan, On disc-annulus surgery of rational maps, in "Proceedings of the International Conference in Dynamical Systems in Honor of Professor Liao Shan-tao 1998" (eds. Y. Jiang and L. Wen), World Scientific, (1999), 237-250.

[8]

W. Qiu and Y. Yin, Proof of the Branner-Hubbard conjecture on Cantor Julia sets, Sci. China Ser. A, 52 (2009), 45-65. doi: 10.1007/s11425-008-0178-9.

[9]

M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup., 20 (1987), 1-29.

[10]

D. Sullivan, Quasiconformal homeomorphisms and dynamics I: Solution of the Fatou-Julia problem on wandering domains, Ann. of Math., 122 (1985), 401-418. doi: 10.2307/1971308.

[11]

Y. Zhai, A generalized version of Branner-Hubbard conjecture for rational functions, Acta Mathematica Sinica, 26 (2010), 2199-2208. doi: 10.1007/s10114-010-7632-7.

[1]

Guizhen Cui, Yan Gao. Wandering continua for rational maps. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1321-1329. doi: 10.3934/dcds.2016.36.1321

[2]

Jun Hu, Oleg Muzician, Yingqing Xiao. Dynamics of regularly ramified rational maps: Ⅰ. Julia sets of maps in one-parameter families. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3189-3221. doi: 10.3934/dcds.2018139

[3]

Rich Stankewitz, Hiroki Sumi. Random backward iteration algorithm for Julia sets of rational semigroups. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2165-2175. doi: 10.3934/dcds.2015.35.2165

[4]

Yan Gao, Luxian Yang, Jinsong Zeng. Subhyperbolic rational maps on boundaries of hyperbolic components. Discrete and Continuous Dynamical Systems, 2022, 42 (1) : 319-326. doi: 10.3934/dcds.2021118

[5]

Weiyuan Qiu, Fei Yang, Yongcheng Yin. Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3375-3416. doi: 10.3934/dcds.2016.36.3375

[6]

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 and Continuous Dynamical Systems, 2019, 39 (9) : 5185-5206. doi: 10.3934/dcds.2019211

[7]

Cezar Joiţa, William O. Nowell, Pantelimon Stănică. Chaotic dynamics of some rational maps. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 363-375. doi: 10.3934/dcds.2005.12.363

[8]

Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1205-1244. doi: 10.3934/dcds.2011.29.1205

[9]

Jeffrey Diller, Han Liu, Roland K. W. Roeder. Typical dynamics of plane rational maps with equal degrees. Journal of Modern Dynamics, 2016, 10: 353-377. doi: 10.3934/jmd.2016.10.353

[10]

Carlos Cabrera, Peter Makienko, Peter Plaumann. Semigroup representations in holomorphic dynamics. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1333-1349. doi: 10.3934/dcds.2013.33.1333

[11]

Inês Cruz, M. Esmeralda Sousa-Dias. Reduction of cluster iteration maps. Journal of Geometric Mechanics, 2014, 6 (3) : 297-318. doi: 10.3934/jgm.2014.6.297

[12]

John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 1047-1056. doi: 10.3934/dcds.2005.13.1047

[13]

Song Shao, Xiangdong Ye. Non-wandering sets of the powers of maps of a star. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1175-1184. doi: 10.3934/dcds.2003.9.1175

[14]

Marco Abate, Francesca Tovena. Formal normal forms for holomorphic maps tangent to the identity. Conference Publications, 2005, 2005 (Special) : 1-10. doi: 10.3934/proc.2005.2005.1

[15]

Eugen Mihailescu, Mariusz Urbański. Holomorphic maps for which the unstable manifolds depend on prehistories. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 443-450. doi: 10.3934/dcds.2003.9.443

[16]

Rich Stankewitz, Hiroki Sumi. Backward iteration algorithms for Julia sets of Möbius semigroups. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6475-6485. doi: 10.3934/dcds.2016079

[17]

Eriko Hironaka, Sarah Koch. A disconnected deformation space of rational maps. Journal of Modern Dynamics, 2017, 11: 409-423. doi: 10.3934/jmd.2017016

[18]

Richard Sharp, Anastasios Stylianou. Statistics of multipliers for hyperbolic rational maps. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1225-1241. doi: 10.3934/dcds.2021153

[19]

Kingshook Biswas. Complete conjugacy invariants of nonlinearizable holomorphic dynamics. Discrete and Continuous Dynamical Systems, 2010, 26 (3) : 847-856. doi: 10.3934/dcds.2010.26.847

[20]

Hiroki Sumi, Mariusz Urbański. Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 313-363. doi: 10.3934/dcds.2011.30.313

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (81)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]