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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, 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., Academic Press, Orlando, FL, 2 (1986), 181-201.  Google Scholar

[2]

A. Douady and J. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Éc. Norm. Sup. (4), 18 (1985), 287-343.  Google Scholar

[3]

K. Katagata, On a certain kind of polynomials of degree 4 with disconnected Julia set, Discrete Contin. Dyn. Syst. , 20 (2008), 975-987. 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, London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, 348 (2008), 217-250. doi: 10.1017/CBO9780511735233.012.  Google Scholar

[5]

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

show all references

References:
[1]

P. Blanchard, Disconnected Julia sets, chaotic dynamics and fractals, Notes Rep. Math. Sci. Engrg., Academic Press, Orlando, FL, 2 (1986), 181-201.  Google Scholar

[2]

A. Douady and J. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. Éc. Norm. Sup. (4), 18 (1985), 287-343.  Google Scholar

[3]

K. Katagata, On a certain kind of polynomials of degree 4 with disconnected Julia set, Discrete Contin. Dyn. Syst. , 20 (2008), 975-987. 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, London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, 348 (2008), 217-250. doi: 10.1017/CBO9780511735233.012.  Google Scholar

[5]

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

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