Advanced Search
Article Contents
Article Contents

On the topology of wandering Julia components

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 37F10, 37F20.


    \begin{equation} \\ \end{equation}
  • [1]

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


    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.


    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.


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


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


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


    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.


    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.


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


    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.


    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.

  • 加载中

Article Metrics

HTML views() PDF downloads(93) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint