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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 and Continuous Dynamical Systems, 2008, 20 (4) : 975-987. doi: 10.3934/dcds.2008.20.975
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