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The Julia sets of certain classes of permutable entire functions

Authors So, Hon Chung
Issue Date 2007
Summary Let ƒ(z) and g(z) be permutable functions, i.e. ƒ(g(z)) = g(ƒ(z)). Fatou and Julia proved that the Julia sets of two permutable rational functions with degree at least 2 are the same independently. Then, I.N. Baker mentioned a open problem, ”Let ƒ(z) and g(z) be permutable nonlinear entire functions. Are their Julia sets the same, J(ƒ) = J(g)?”. Recently, it is proved that certain classes of permutable entire functions are true for the above problem. In this thesis, we will prove that if there exists some open set U such that U ∩ J(ƒ) = U ∩ J(g) is nonempty set, then J(ƒ) = J(g). The next result is that if ƒ = p(z)eα(z) + a and g = q(z)eβ(z) + a where p(z) and q(z) are non-constant polynomials not in the form (z − a)n , n > 0 and α(z) and β(z) are nonconstant entire functions, then J(ƒ) = J(g). At last, we will give some different versions of Ng’s and Liao and Yang’s results.
Note Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2007
Language English
Format Thesis
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