Conditions Where the Chaotic Set Has a Non-Empty Residual Julia Set for Two Classes of Meromorphic Functions *

We define the Fatou and Julia sets for two classes of meromorphic functions. The Julia set is the chaotic set where the fractals appear. The chaotic set can have points and components which are buried. The set of these points and components is called the residual Julia set, denoted by   r J f , and is defined to be the subset of those points of the Julia set, chaotic set, which do not belong to the boundary of any component of the Fatou set (stable set). The points of   r J f are called buried points and the components of   r J f are called buried components. In this paper we extend some results related with the residual Julia set of transcendental meromorphic functions to functions which are meromorphic outside a compact countable set of essential singularities. We give some conditions where .   r J f  


Introduction
Let X,Y be Riemann surfaces (complex 1-manifolds) and D f be an arbitrary non-empty open subset of X.We define i sa n a l y t i and , . f , where is the set of  

C f critical values and  
A f is the set of asymptotic values.Let

 
f Hol X  , the sequence formed by its iterates will be defined and denoted by , 0 : Id f  1 : , n   .The study makes sense and is non-trivial when X is either the Riemann sphere , the complex plane   or the complex plane minus one point, this is Taking X   and Ŷ   we deal with the following classes of meromorphic maps.

 
: is transcendental meromorphic with at least one not omitted pole The set B is formed by the essential singularities of f, where f is non-constant.We assume B to have at least two elements and f to have poles.With this assumption we have .
(or ) such that the sequence of iterates of f is well defined and forms a normal family in a neighbourhood of z.The Julia set is the complement of the Fatou set, denoted by . The Fatou and the Julia sets are also known as the stable and the chaotic sets respectively.In the Julia set or chaotic set is easy to find fractals, examples of this fact are below.The fractals are typically selfsimilar patterns, where self-similar means they are "the same from near as from far" [1].
Examples of functions in class live in the family [2].The stable set (Fatou set) and the chaotic set (Julia set) for the parameters 4    can be seen in Figure 1.
Examples of functions in class can be found in the , where is a rational func- tion, , and .We do not have any picture of the Fatou and Julia set but the Julia set should be a fractal for some parameters c and ϵ sufficiently small.
It was proved in [5], for functions in class , and in [8], for functions in class  , that a periodic Fatou component (of arbitrary period) is simply, doubly or infinitely connected.


In [6] the authors proved that for functions in class This concept was first introduced in the context of Kleinian groups by Abikoff in [10,11].In [12], McMullen defined a buried component of a rational function to be a component of the Julia set which does not meet the boundary of any component of the Fatou set.Similarly, for a buried point of the Julia set.McMullen gave an example of a rational function with buried components.
Baker and Domínguez in [13] extended some results of Qiao [14] (for rational functions) to have buried points or buried components to functions in class .In Section 2 we prove that the same results can be extended to functions in class .
  Finally, Section 3 contains Theorems A and B which assure with some conditions that the residual Julia set is not empty for functions in classes and . 

Basic Results of the Residual Julia Set for Functions in Classes and  
In this Section we will state some basic results about the residual Julia set which hold for functions in classes and .The proofs of the these results can be found in [13] and [15].

  r J f   
In this section we will extend some results related with the residual Julia set for functions in class to functions in class .Qiao in [14] proved the following theorem for rational functions.
Baker and Domínguez in [13] gave the following result which was step towards a generalisation of Theorem   r J f   In order to prove Theorem A we need to state some results for functions in classes  and .The following lemma was given in [16] for functions in class , since the proof works for functions in class we do not write it.
The following result was given in [17] for functions in class . Theorem 3.4.Let .Suppose that the Fatou set has no completely invariant domain and the Julia set is disconnected in such a way that the Fatou set has a component f   H of connectivity at least five.Then singleton components are dense and buried in   J f .Proof of Theorem A. If is a component of the Fatou set, then it can be either periodic, preperiodic or wandering.We will split the proof in two cases the no wandering case and the wandering case.

Let
. Assume that there are not wandering domains in the Fatou set and that .By Proposition 2.4 there is a periodic Fatou component such that .The component is multiply connected since the Julia set, by hypothesis, is not connected.By Lemma 3.3 the component must be completely invariant which gives us a contradiction.Therefore, the residual Julia set is not empty.

Wandering case.
We assume that the Fatou set has wandering components.We prove the result in two cases: 1) f has only finite connected Fatou components and 2) f has at least one infinitely connected Fatou component.
1) Since the Julia set is disconnected it consists of uncountable many components.Now as the connectivity of each component of the Fatou set of f is finite, then the number of the boundary components of all Fatou components is countable.Thus the Julia set has uncountably many buried components.Therefore, .
  r J f   2) If we take a multiply-connected Fatou component of connectivity n, , , then the proof follows as the proof of Theorem 3.4 in [17].Thus singleton buried components are dense in the Julia set.Therefore, The following theorem is an extension of Proposition 6.1 given in [16], since the proof given in [16]


Many properties of  J f and   F f are much the same for all classes above but different proofs are needed and some discrepancies arise.For functions in classes or  we recall some properties of the Fatou and Julia sets: the Fatou set    F f is open and the Julia set   J f is closed; the Julia set is perfect and non-empty; the sets   J f and   F f are completely invariant under f; and finally the repelling periodic points are dense in   J f .A Fatou component for a function in class or can be periodic, pre-periodic or wandering.The possible dynamics of a periodic component of the Fatou set is either attracting, parabolic, Siegel disc, Herman ring or Baker domain.Figure 1 is an example of a Baker do-

Figure 1 .
Figure 1.The chaotic set, which is a fractal, with colors and the Fatou set on black.

1 .Proposition 2 . 2 .
Let f be in class or .If the Fatou set of f has a completely invariant component, then the residual Julia set is empty. Let f be in class or.If there exists a buried component of

J f and uncountably infinite. Proposition 2 . 4 .
If f   has no wandering domains and

  Theorem 3 . 1 .
Let f be a rational function and  A of z.Since periodic points are dense in Julia, then V must contain a periodic point  of the Julia set.Under iteration the point  has to come back to itself infinitely often.By hypothesis, points on the boundary of any Fatou component must iterate inside A and never leave again.Then points in the Julia which leaves A infinitely often are not in the boundary of a Fatou component, thus   r J f   since it lies in the complement of A. Therefore   r J f   .


All the Fatou components of f eventually iterate inside A and never leave again.That is, if  is a Fatou component,