Bounded critical Fatou components are Jordan domains for polynomials

We prove that any bounded Fatou component of a polynomial of degree at least two, which is not(eventually) a Siegel disk, is a Jordan domain.

Bounded Fatou Components of Transcendental Entire Functions with Order Less than 1/2

Let f be a transcendental entire function with order ρ 〈 1/2 and let a be a sufficiently large constant. We prove that if there exists r0 〉 1 such that, for all r 〉 r0 and any small ε 〉0,M（r^σ,f）≥M（r,f）σ＋ε.then every component of the Fatou set F（f） is bounded.

Fatou Components and Julia Sets of Singularly Perturbed Rational Maps with Positive Parameter

In this paper, we discuss the rational maps Fλ（z）=z^n＋λ/z^n,n≥2with the positive real parameter ）λ. It is shown that the immediately attracting basin Bλ of ∞ for Fλ is always a Jordan domain if the Julia set of Fλ is not a Cantor set. Fuhermore, Bλ is a quasidisk if there is no parabolic fixed point on the boundary of Bλ. It is also shown that if the Julia set of Fλ is connected, then it is locally connected and all Fatou components are Jordan domains. Finally, a complete description to the problem when the Julia set is a Sierpirlski curve is given.

ON THE INVERSE IMAGE SET OF RATIONAL FUNCTIONS

This article studies the inverse image of rational functions. Several theorems are obtained on the Julia set expressed by the inverse image, and a mistake is pointed out in H.Brolin＇ theorem incidentally.