American Institute of Mathematical Sciences

Advanced
October  2008, 20(4): 975-987. doi: 10.3934/dcds.2008.20.975

On a certain kind of polynomials of degree 4 with disconnected Julia set

Koh Katagata 1,

1. 

Interdisciplinary Graduate School of Science and Engineering, Shimane University, Matsue 690-8504, Japan

Received  November 2006 Revised  August 2007 Published  January 2008

For a polynomial of degree at least two, the Julia set and the filled-in Julia set are either connected or have uncountably many components. In the case that the Julia set of a polynomial of degree 4 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. Furthermore the polynomial is topologically conjugate to the shift map via the homeomorphism. Moreover there exists a homeomorphism between the Julia sets of the polynomial and that of a certain polynomial semigroup.
Keywords: Symbolic dynamics., Polynomial semigroups, Disconnected Julia sets.
Mathematics Subject Classification: Primary: 37F10; Secondary: 37F9.
Citation: Koh Katagata. On a certain kind of polynomials of degree 4 with disconnected Julia set. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 975-987. doi: 10.3934/dcds.2008.20.975
[1]

Nathaniel D. Emerson. Dynamics of polynomials with disconnected Julia sets. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 801-834. doi: 10.3934/dcds.2003.9.801
[2]

Hiroki Sumi. Dynamics of postcritically bounded polynomial semigroups I: Connected components of the Julia sets. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1205-1244. doi: 10.3934/dcds.2011.29.1205
[3]

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
[4]

Rich Stankewitz, Hiroki Sumi. Random backward iteration algorithm for Julia sets of rational semigroups. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2165-2175. doi: 10.3934/dcds.2015.35.2165
[5]

Rich Stankewitz, Hiroki Sumi. Backward iteration algorithms for Julia sets of Möbius semigroups. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6475-6485. doi: 10.3934/dcds.2016079
[6]

Hiroki Sumi, Mariusz Urbański. Measures and dimensions of Julia sets of semi-hyperbolic rational semigroups. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 313-363. doi: 10.3934/dcds.2011.30.313
[7]

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, 2018, 38 (7) : 3189-3221. doi: 10.3934/dcds.2018139
[8]

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
[9]

Robert L. Devaney, Daniel M. Look. Buried Sierpinski curve Julia sets. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 1035-1046. doi: 10.3934/dcds.2005.13.1035
[10]

Danilo Antonio Caprio. A class of adding machines and Julia sets. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 5951-5970. doi: 10.3934/dcds.2016061
[11]

Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete & Continuous Dynamical Systems, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725
[12]

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
[13]

Tarik Aougab, Stella Chuyue Dong, Robert S. Strichartz. Laplacians on a family of quadratic Julia sets II. Communications on Pure & Applied Analysis, 2013, 12 (1) : 1-58. doi: 10.3934/cpaa.2013.12.1
[14]

Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293
[15]

Ali Messaoudi, Rafael Asmat Uceda. Stochastic adding machine and $2$-dimensional Julia sets. Discrete & Continuous Dynamical Systems, 2014, 34 (12) : 5247-5269. doi: 10.3934/dcds.2014.34.5247
[16]

Ranjit Bhattacharjee, Robert L. Devaney, R.E. Lee Deville, Krešimir Josić, Monica Moreno-Rocha. Accessible points in the Julia sets of stable exponentials. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 299-318. doi: 10.3934/dcdsb.2001.1.299
[17]

Jim Wiseman. Symbolic dynamics from signed matrices. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 621-638. doi: 10.3934/dcds.2004.11.621
[18]

George Osipenko, Stephen Campbell. Applied symbolic dynamics: attractors and filtrations. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 43-60. doi: 10.3934/dcds.1999.5.43
[19]

Michael Hochman. A note on universality in multidimensional symbolic dynamics. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 301-314. doi: 10.3934/dcdss.2009.2.301
[20]

Koh Katagata. Transcendental entire functions whose Julia sets contain any infinite collection of quasiconformal copies of quadratic Julia sets. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5319-5337. doi: 10.3934/dcds.2019217

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences