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 & Continuous Dynamical Systems - A, 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. Google Scholar

[2]

G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997. Google Scholar

[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. Google Scholar

[4]

W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc., 29 (1993), 151-188. doi: 10.1090/S0273-0979-1993-00432-4. Google Scholar

[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. Google Scholar

[6] B. Branner and N. Fagella, Quasiconformal Surgery in Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, 141, Cambridge University Press, Cambridge, 2014. Google Scholar
[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. Google Scholar

[8] E. de Faria and W. de Melo, Mathematical Tools for One-Dimensional Dynamics, Cambridge University Press, New York, 2008. doi: 10.1017/CBO9780511755231. Google Scholar
[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. Google Scholar

[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. Google Scholar

[11] J. Milnor, Dynamics in one Complex Variable, Third edition, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, NJ, 2006. Google Scholar
[12]

S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics 66, 2000. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

show all references

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. Google Scholar

[2]

G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Conformal Invariants, Inequalities, and Quasiconformal Maps, John Wiley & Sons, New York, 1997. Google Scholar

[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. Google Scholar

[4]

W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc., 29 (1993), 151-188. doi: 10.1090/S0273-0979-1993-00432-4. Google Scholar

[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. Google Scholar

[6] B. Branner and N. Fagella, Quasiconformal Surgery in Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics, 141, Cambridge University Press, Cambridge, 2014. Google Scholar
[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. Google Scholar

[8] E. de Faria and W. de Melo, Mathematical Tools for One-Dimensional Dynamics, Cambridge University Press, New York, 2008. doi: 10.1017/CBO9780511755231. Google Scholar
[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. Google Scholar

[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. Google Scholar

[11] J. Milnor, Dynamics in one Complex Variable, Third edition, Annals of Mathematics Studies, 160, Princeton University Press, Princeton, NJ, 2006. Google Scholar
[12]

S. Morosawa, Y. Nishimura, M. Taniguchi and T. Ueda, Holomorphic Dynamics, Cambridge Studies in Advanced Mathematics 66, 2000. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

Figure 1.  The definition of the quasiregular map g near infinity
Figure 2.  The definition of the quasiregular map $ g $ near $ R_{m(j)} $
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