The Julia sets of certain classes of permutable entire functions

Abstract

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)e^α(z) + a and g = q(z)e^β(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.

Date of Award	2007
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology