# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] J. Milnor, "Dynamics in One Complex Variable," 3rd edition, Princeton University Press, 2006.  Google Scholar [5] J. Milnor, On rational maps with two critical points, Experimental Mathematics, 9 (2000), 481-522.  Google Scholar [6] K. Pilgrim and L. Tan, Rational maps with disconnected Julia set, Astérique, 261 (2000), 349-384.  Google Scholar [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. Google Scholar [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.  Google Scholar [9] M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup., 20 (1987), 1-29.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### References:
 [1] A. F. Beardon, "Iteration of Rational Functions," Springer-Verlag, New York, 1991.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] J. Milnor, "Dynamics in One Complex Variable," 3rd edition, Princeton University Press, 2006.  Google Scholar [5] J. Milnor, On rational maps with two critical points, Experimental Mathematics, 9 (2000), 481-522.  Google Scholar [6] K. Pilgrim and L. Tan, Rational maps with disconnected Julia set, Astérique, 261 (2000), 349-384.  Google Scholar [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. Google Scholar [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.  Google Scholar [9] M. Shishikura, On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup., 20 (1987), 1-29.  Google Scholar [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.  Google Scholar [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.  Google Scholar
 [1] Guizhen Cui, Yan Gao. Wandering continua for rational maps. Discrete & 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 & 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 & 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 & 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 & 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 & 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 & 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 & 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] 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 [11] Carlos Cabrera, Peter Makienko, Peter Plaumann. Semigroup representations in holomorphic dynamics. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1333-1349. doi: 10.3934/dcds.2013.33.1333 [12] John Erik Fornæss. Periodic points of holomorphic twist maps. Discrete & 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 & 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 & 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 & 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 & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021153 [19] Hiroki Sumi, Mariusz Urbański. Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 313-363. doi: 10.3934/dcds.2011.30.313 [20] Rich Stankewitz. Density of repelling fixed points in the Julia set of a rational or entire semigroup, II. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2583-2589. doi: 10.3934/dcds.2012.32.2583

2020 Impact Factor: 1.392