American Institute of Mathematical Sciences

Advanced
April  2016, 36(4): 2103-2112. doi: 10.3934/dcds.2016.36.2103

Quartic Julia sets including any two copies of quadratic Julia sets

Koh Katagata 1,

1. 

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

Received  January 2015 Revised  July 2015 Published  September 2015

If the Julia set of a quartic polynomial with certain conditions is neither connected nor totally disconnected, there exists a homeomorphism between the set of all components of the filled-in Julia set and some subset of the corresponding symbol space. The question is to determine the quartic polynomials exhibiting such a dynamics and describe the topology of the connected components of their filled-in Julia sets. In this paper, we answer the question, namely we show that for any two quadratic Julia sets, there exists a quartic polynomial whose Julia set includes copies of the two quadratic Julia sets.
Keywords: polynomial-like maps, Julia sets, quasiconformal/quasiregular maps..
Mathematics Subject Classification: Primary: 37F10; Secondary: 30D05, 37F5.
Citation: Koh Katagata. Quartic Julia sets including any two copies of quadratic Julia sets. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2103-2112. doi: 10.3934/dcds.2016.36.2103
References:
[1]

P. Blanchard, Disconnected Julia sets, chaotic dynamics and fractals,, Notes Rep. Math. Sci. Engrg., 2 (1986), 181. Google Scholar

[2]

A. Douady and J. Hubbard, On the dynamics of polynomial-like mappings,, Ann. Sci. Éc. Norm. Sup. (4), 18 (1985), 287. Google Scholar

[3]

K. Katagata, On a certain kind of polynomials of degree 4 with disconnected Julia set,, Discrete Contin. Dyn. Syst. , 20 (2008), 975. doi: 10.3934/dcds.2008.20.975. Google Scholar

[4]

M. Kisaka and M. Shishikura, On multiply connected wandering domains of entire functions,, Transcendental dynamics and complex analysis, 348 (2008), 217. doi: 10.1017/CBO9780511735233.012. Google Scholar

[5]

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

show all references

References:
[1]

P. Blanchard, Disconnected Julia sets, chaotic dynamics and fractals,, Notes Rep. Math. Sci. Engrg., 2 (1986), 181. Google Scholar

[2]

A. Douady and J. Hubbard, On the dynamics of polynomial-like mappings,, Ann. Sci. Éc. Norm. Sup. (4), 18 (1985), 287. Google Scholar

[3]

K. Katagata, On a certain kind of polynomials of degree 4 with disconnected Julia set,, Discrete Contin. Dyn. Syst. , 20 (2008), 975. doi: 10.3934/dcds.2008.20.975. Google Scholar

[4]

M. Kisaka and M. Shishikura, On multiply connected wandering domains of entire functions,, Transcendental dynamics and complex analysis, 348 (2008), 217. doi: 10.1017/CBO9780511735233.012. Google Scholar

[5]

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

[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]

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
[7]

Michał Misiurewicz, Sonja Štimac. Lozi-like maps. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 2965-2985. doi: 10.3934/dcds.2018127
[8]

S. R. Bullett and W. J. Harvey. Mating quadratic maps with Kleinian groups via quasiconformal surgery. Electronic Research Announcements, 2000, 6: 21-30.

[9]

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
[10]

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
[11]

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
[12]

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
[13]

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
[14]

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
[15]

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
[16]

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
[17]

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
[18]

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
[19]

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
[20]

Mary Wilkerson. Thurston's algorithm and rational maps from quadratic polynomial matings. Discrete & Continuous Dynamical Systems - S, 2019, 12 (8) : 2403-2433. doi: 10.3934/dcdss.2019151

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences