The Carathéodory Convergence of Fatou Components of Polynomials to Baker Domains or Wandering Domains

Abstract

Let f be a rational function or a transcendental entire function. Consider X is a complex sphere \( \widehat{\mathbb{C}} \) if f is a rational function and X is a complex plain ℂ if f is a transcendental entire function. The maximal open subset of X on which the family {f n } is normal is called the Fatou set of f and we denote it by F(f). The complement of F(f)in \( \widehat{\mathbb{C}} \) is called the Julia set of f and we denote it by J(f). We remark that, in this note, ∞ is always contained in the Julia sets of transcendental entire functions. Fatou sets and Julia sets are completely invariant. Hence a component of F(f) is mapped into some component of F(f) by f. We say that a component of D of F(f) is a cyclic component if f p (D) is contained in D for some natural number p. If p = 1, then it is called invariant. All the kinds of cyclic components are already known. They are an attracting component, a parabolic component, a Siege disk, a Herman ring and a Baker domain. The first three have relationship with cyclic points and the first four have relationship with singular values. On the other hand, Baker domains have relationship with neither cyclic point nor singular value. Baker domains are never appeared in the case of rational functions. Some facts on Baker domains are stated in §3. We say that a component D is a wandering domain if f n (D) ≠ f m (D) for all natural numbers n and m with n ≠ m. Due to Sullivan’s theorem, we know that there exists no wandering domain in the case of rational functions. On the contrary, in the case of transcendental entire functions, there may exist wandering domains if it has infinitely many singular values.