Abstract
For a polynomial of degree at least two, the Julia set and thefilled-in Julia set are either connected or have uncountably manycomponents. In the case that the Julia set of a polynomial of degree 4is neither connected nor totally disconnected, there exists ahomeomorphism between the set of all components of the filled-inJulia set and some subset of the corresponding symbol space.Furthermore the polynomial is topologically conjugate to the shiftmap via the homeomorphism. Moreover there exists ahomeomorphism between the Julia sets of the polynomial and that of acertain polynomial semigroup.
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