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 & 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. [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.

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. [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 $R_{m(j)}$
 [1] Tien-Cuong Dinh, Nessim Sibony. Rigidity of Julia sets for Hénon type maps. Journal of Modern Dynamics, 2014, 8 (3&4) : 499-548. doi: 10.3934/jmd.2014.8.499 [2] Weiyuan Qiu, Fei Yang, Yongcheng Yin. Quasisymmetric geometry of the Cantor circles as the Julia sets of rational maps. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3375-3416. doi: 10.3934/dcds.2016.36.3375 [3] 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 - A, 2019, 39 (9) : 5185-5206. doi: 10.3934/dcds.2019211 [4] 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 - A, 2018, 38 (7) : 3189-3221. doi: 10.3934/dcds.2018139 [5] Luiz Henrique de Figueiredo, Diego Nehab, Jorge Stolfi, João Batista S. de Oliveira. Rigorous bounds for polynomial Julia sets. Journal of Computational Dynamics, 2016, 3 (2) : 113-137. doi: 10.3934/jcd.2016006 [6] Michał Misiurewicz, Sonja Štimac. Lozi-like maps. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2965-2985. doi: 10.3934/dcds.2018127 [7] S. R. Bullett and W. J. Harvey. Mating quadratic maps with Kleinian groups via quasiconformal surgery. Electronic Research Announcements, 2000, 6: 21-30. [8] Xu Zhang, Guanrong Chen. Polynomial maps with hidden complex dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2941-2954. doi: 10.3934/dcdsb.2018293 [9] Christopher Cleveland. Rotation sets for unimodal maps of the interval. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 617-632. doi: 10.3934/dcds.2003.9.617 [10] Evelyn Sander. Hyperbolic sets for noninvertible maps and relations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 339-357. doi: 10.3934/dcds.1999.5.339 [11] Boju Jiang, Jaume Llibre. Minimal sets of periods for torus maps. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 301-320. doi: 10.3934/dcds.1998.4.301 [12] Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1205-1244. doi: 10.3934/dcds.2011.29.1205 [13] Héctor E. Lomelí. Heteroclinic orbits and rotation sets for twist maps. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 343-354. doi: 10.3934/dcds.2006.14.343 [14] Song Shao, Xiangdong Ye. Non-wandering sets of the powers of maps of a star. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1175-1184. doi: 10.3934/dcds.2003.9.1175 [15] Bruce Kitchens, Michał Misiurewicz. Omega-limit sets for spiral maps. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 787-798. doi: 10.3934/dcds.2010.27.787 [16] Youngna Choi. Attractors from one dimensional Lorenz-like maps. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 715-730. doi: 10.3934/dcds.2004.11.715 [17] Can Gao, Joachim Krieger. Optimal polynomial blow up range for critical wave maps. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1705-1741. doi: 10.3934/cpaa.2015.14.1705 [18] Romain Aimino, Huyi Hu, Matthew Nicol, Andrei Török, Sandro Vaienti. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 793-806. doi: 10.3934/dcds.2015.35.793 [19] Mary Wilkerson. Thurston's algorithm and rational maps from quadratic polynomial matings. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2403-2433. doi: 10.3934/dcdss.2019151 [20] José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781

2017 Impact Factor: 1.179