In this note, we study the Julia sets of subhyperbolic rational functions. The boundaries of simply c o~e c t e d attracting components of the Fatou sets of such functions are closed curves. We show that, under certain conditions, they are Jordan curves. Making use of this result, we see that the Fatou set of Newton's method for z 3 -1 = 0 consists of Jordan domains.

We consider the subset of the Julia set called the residual Julia set, which comes from an analogy of the residual limit set of a Kleinian group. We give a necessary and sufficient condition in order that the residual Julia set is empty in the case of hyperbolic rational functions.

The concept of semihyperbolicity introduced by Carleson, Jones and Yoccoz for polynomials is carried over to transcendental entire functions. For certain classes of semihyperbolic entire functions it is shown that there are no wandering domains and that the Julia sets are locally connected.

In this note, we consider the dynamics of the family of transcendental entire functions f(z; a, b, c) = abz + e bz + c. In particular, while studying the case when they have a Baker domains, we investigate the area of Julia sets and the Hausdorff convergence of Julia sets. Consider the family f (z; a, b, c) = abz + e bz + c. These f (z; a, b, c) with a, b = 0 are structurally infinite [7], for they have infinitely many critical values. For the sake of simplicity, we assume that a, b and c are real. Theorem 1. If ab ≥ 1, a > 0, c ≤ −1 and a log a − a + c < 0, then f = f (z; a, b, c) has a Baker domain B = B a,b,c , which is completely invariant.In particular, the Julia set is the boundary of B.Proof. We set D 0 = {z|Re z < 0}. Elementary calculation shows that D 0 is mapped by f to a proper subdomain in D 0 , and hence D 0 is invariant. The Fatou component B containing D 0 is a Baker domain, for if z ∈ D 0 , then Re f (z) < abRe z.

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