# American Institute of Mathematical Sciences

September  2019, 39(9): 5319-5337. doi: 10.3934/dcds.2019217

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

 National Institute of Technology, Ichinoseki College, Takanashi, Hagisho, Ichinoseki, Iwate 021-8511, Japan

Received  October 2018 Published  May 2019

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.

Citation: Koh Katagata. Transcendental entire functions whose Julia sets contain any infinite collection of quasiconformal copies of quadratic Julia sets. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5319-5337. doi: 10.3934/dcds.2019217
##### References:
 [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. Anderson, M. 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. Fagella, Quasiconformal 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 Melo, Mathematical 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. Milnor, Dynamics 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.

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##### References:
 [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. Anderson, M. 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. Fagella, Quasiconformal 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 Melo, Mathematical 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. Milnor, Dynamics 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.
The definition of the quasiregular map g near infinity
The definition of the quasiregular map $g$ near $R_{m(j)}$
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