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We partition the fast escaping set of a transcendental entire function into two subsets, the maximally fast escaping set and the non-maximally fast escaping set. These sets are shown to have strong dynamical properties. We show that the intersection of the Julia set with the non-maximally fast escaping set is never empty. The proof uses a new covering result for annuli, which is of wider interest.It was shown by Rippon and Stallard that the fast escaping set has no bounded components. In contrast, by studying a function considered by Hardy, we give an example of a transcendental entire function for which the maximally and non-maximally fast escaping sets each have uncountably many singleton components.

We show that if the growth of a transcendental entire function f is sufficiently regular, then the Julia set and the escaping set of f have Hausdorff dimension 2., Comment: 21 pages

We study transcendental entire mapsfof finite order, such that all the singularities off−1are contained in a compact subset of the immediate basinBof an attracting fixed point off. Then the Julia set offconsists of disjoint curves tending to infinity (hairs), attached to the unique point accessible fromB(endpoint of the hair). We prove that the Hausdorff dimension of the set of endpoints of the hairs is equal to 2, while the union of the hairs without endpoints has Hausdorff dimension 1, which generalizes the result for exponential maps. Moreover, we show that for every transcendental entire map of finite order from class(i.e. with bounded set of singularities) the Hausdorff dimension of the Julia set is equal to 2.

Meromorphic solutions of the Schroder equation f(sz) = R(f(z)) are studied, where |s| > 1 and R(w) is a rational function with deg[R] ≥ 2. We will show that, if arg[s] /∈ 2πQ, then f(z) has any direction as Borel, and besides, without exceptional values other than Picard values, which depend on R(w). Further the case arg[s] ∈ 2πQ is also considered. We investigate therelation between Julia directions of f(z) and the Julia set of R(w).

Letfbe a transcendental entire function such that the finite singularities of f−1lie in a bounded set. We show that the Hausdorff dimension of the Julia set of such a function is strictly greater than one.

The purpose of this note is to present a few facts about the Julia set of a rational function that are well known to the experts in the subject of complex dynamics but whose documented exposition in the literature seems to need a little clarification. For example, under conditions set out in Theorem 1, the Julia set of a rational function can be expressed as the limit of a sequence of finite sets. In particular, for certain choices of a point α, the Julia set is the limit as n increases of the inverse image sets R−n(α). This formulation is widely exploited in the backwards iteration algorithm to produce computer illustrations of Julia sets (see for example Section 5·4 of [14]).

Let U be an invariant component of the Fatou set of an entire transcendental function f such that the iterates of f tend to ∞ in U. Let P(f) be the closure of the set of the forward orbits of all critical and asymptotic values of f. We show that there exists a sequence pn∈P(f) such that dist(pn, U) = o(|pn|), where dist(·, ·) denotes Euclidean distance. On the other hand, we give an example where dist (P(f), U) > 0. In this example, U is bounded by a Jordan curve.

Letfbe a rational function of degree at least two. We shall prove that the Julia setJ(f) offis uniformly perfect. This means that there is a constantc∈(0, 1) depending onfonly such that wheneverz∈J(f) and 0

Let . The set \[ F(f)=\{z:(f^n)\hbox{ is normal in some neighbourhood of }z\} \] is the Fatou set or the set of normality and its complement .

In 1922–23, Julia and Fatou proved that any 2 rational functions f and g of degree at least 2 such that f(g(z)) = g(f(z)), have the same Julia set. Baker then asked whether the result remains true for nonlinear entire functions. In this paper, we shall show that the answer to Baker's question is true for almost all nonlinear entire functions. The method we use is useful for solving functional equations. It actually allows us to find out all the entire functions g which permute with a given f which belongs to a very large class of entire functions.

In this paper, we study the residual Julia sets of meromorphic functions. In fact, we prove that if a meromorphic function f belongs to the class S and its Julia set is locally connected, then the residual Julia set of f is empty if and only if its Fatou set F(f) has a completely invariant component or consists of only two components. We also show that if f is a meromorphic function which is not of the form fi + (z i fi) ik e g(z) , where k is a natural number, fi is a complex number and g is an entire function, then f has buried components provided that f has no completely invariant components and its Julia set J(f) is disconnected. Moreover, if F(f) has an inflnitely connected component, then the singleton buried components are dense in J(f). This generalizes a result of Baker and Dom¶‡nguez. Finally, we give some examples of meromorphic functions with buried points but without any buried components.