\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Transcendental entire functions whose Julia sets contain any infinite collection of quasiconformal copies of quadratic Julia sets

Abstract Full Text(HTML) Figure(2) Related Papers Cited by
  • We prove that for any infinite collection of quadratic Julia sets, there exists a transcendental entire function whose Julia set contains quasiconformal copies of the given quadratic Julia sets. In order to prove the result, we construct a quasiregular map with required dynamics and employ the quasiconformal surgery to obtain the desired transcendental entire function. In addition, the transcendental entire function has order zero.

    Mathematics Subject Classification: Primary: 37F10; Secondary: 30D05, 37F50.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  The definition of the quasiregular map g near infinity

    Figure 2.  The definition of the quasiregular map $ g $ near $ R_{m(j)} $

  • [1] L. V. Ahlfors, Lectures on Quasiconformal Mappings, Second edition, with supplemental chapters by C.J. Earle, I. Kra, M. Shishikura and J.H. Hubbard, University Lecture Series, 38, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/ulect/038.
    [2] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997.
    [3] G. D. AndersonM. K. Vamanamurthy and M. Vuorinen, Distortion functions for plane quasiconformal mappings, Israel J. Math., 62 (1988), 1-16.  doi: 10.1007/BF02767349.
    [4] W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc., 29 (1993), 151-188.  doi: 10.1090/S0273-0979-1993-00432-4.
    [5] W. Bergweiler, An entire function with simply and multiply connected wandering domains, Pure Appl. Math. Q., 7 (2011), 107-120.  doi: 10.4310/PAMQ.2011.v7.n1.a6.
    [6] B. Branner and  N. FagellaQuasiconformal Surgery in Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, 141, Cambridge University Press, Cambridge, 2014. 
    [7] A. Douady and J. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Éc. Norm. Sup. (4), 18 (1985), 287–343. doi: 10.24033/asens.1491.
    [8] E. de Faria and  W. de MeloMathematical Tools for One-Dimensional Dynamics, Cambridge University Press, New York, 2008.  doi: 10.1017/CBO9780511755231.
    [9] K. Katagata, Entire functions whose Julia sets include any finitely many copies of quadratic Julia sets, Nonlinearity, 30 (2017), 2360-2380.  doi: 10.1088/1361-6544/aa6c01.
    [10] M. Kisaka and M. Shishikura, On multiply connected wandering domains of entire functions, Transcendental Dynamics and Complex Analysis, 217–250, London Math. Soc. Lecture Note Ser. 348, Cambridge Univ. Press, Cambridge, 2008 doi: 10.1017/CBO9780511735233.012.
    [11] J. MilnorDynamics in one Complex Variable, Third edition, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, NJ, 2006. 
    [12] S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics 66, 2000.
    [13] J. Osborne, Connectedness properties of the set where the iterates of an entire function are bounded, Math. Proc. Cambridge Philos. Soc., 155 (2013), 391-410.  doi: 10.1017/S0305004113000455.
    [14] S. Rickman, Quasiregular Mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 26. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78201-5.
    [15] N. Steinmetz, Rational Iteration, Complex analytic dynamical systems. De Gruyter Studies in Mathematics, 16. Walter de Gruyter & Co., Berlin, 1993. doi: 10.1515/9783110889314.
  • 加载中

Figures(2)

SHARE

Article Metrics

HTML views(256) PDF downloads(253) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return