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

doi:10.4236/am.2013.411A2004

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Applied Mathematics, 2013, 4, 18-21

Published Online November 2013 (http://www.scirp.org/journal/am)

http://dx.doi.org/10.4236/am.2013.411A2004

Open Access AM

Conditions Where the Chaotic Set Has a Non-Empty

Residual Julia Set for Two Classes of Meromorphic

Functions*

Patricia Domínguez, Iván Hernández

Facultad de Ciencias Fsico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Puebla, Mexico

Email: [email protected].mx, [email protected]

Received August 15, 2013; revised September 15, 2013; accepted September 23, 2013

tribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop-

erly cited.

ABSTRACT

We define the Fatou and Julia sets for two classes of meromorphic functions. The Ju lia 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 compo-

nents is called th e residual Julia set, denoted by





f, and is defined to be the subset of those poin ts 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





are called buried points and the components of





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 .



Jf

Keywords: Fatou Set; Julia Set; Residual Julia Set; Buried Points; Buried Components

1. Introduction

Let X,Y be Riemann surfaces (complex 1-manifolds) and

Df be an arbitrary non-empty open subset of X. W e d ef i ne





 

,: isanalyti

and ,.

HolX YfDYf

HolX XHolX







The set of singular values of





HolX Y is

 

SVfCfA f, where is the set of



critical values and





f is the set of asymptotic val-

ues.

Let





Hol X, the sequence formed by its iterates

will be defined and denoted by ,

0:Idf1



,



. The study makes sense and is no n-trivial when X

is either the Riemann sphere , the complex plane



or the complex plane minus one point, this is





0.

Taking



 and ˆ



 we deal with the fol-

lowing classes of meromorphic maps.



:is transcendentalmeromorphic with at least onenot omitted polefX Yf.



:isa compact countableset andismeromorphicfY BYBf .

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 .



If f is a map in any of the classes above the Fatou set





f consists of all points

 (or ) such

that the sequence of iterates of f is well defined and

forms a normal family in a neig hbo urh ood of z. The Julia

set is the complement of the Fatou set, denoted by

zYB

 





fFf. The Fatou and the Julia sets are also

known as the stable and the chaotic sets respectively. In

*The authors were supported by CONACYT grant 128005.

P. DOMÍNGUEZ, I. HERNÁNDEZ 19

the Julia set or chaotic set is easy to find fractals, exam-

ples of this fact are below. The fractals are typically self-

similar patterns, where self-similar means they are “the

same from near as from far” [1].

Examples of functions in class live in the family



,ez







 studied in [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



family



fRz 





, 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.

0c

Class was initially studied by Baker, Kotus and

Yi Nian [3-6]. The class  has been introduced and

studied by Bolsch in [7-9].



Many properties of



f and



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 is open and the Julia set





f is closed; the Julia set is perfect and non-empty;

the sets



f and



f are completely invariant

under f; and finally the repelling periodic points are

dense in





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-



main for the function 4, 11

4ez





  , see [2] for

details.

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 in-

finitely connected.



In [6] the authors proved that for functions in class

Figure 1. The chaotic set, which is a fractal, with colors and

the Fatou set on black.

 with a finite set of singular values there are neither

wandering components nor Baker domains. The same

statement works for functions in class and the proofs

are similar to those in [6]. 

We define the residual Julia set of f denoted by





f as the set of those points of





f which do

not belong to the boundary of any component of the Fa-

tou set





f. The points of



are called buried

points and the components of

f are called buried

components. This is the Residual Julia set (buried points

and buried components) which are in the chaotic set.

This concept was first introduced in the context of

Kleinian groups by Abikoff in [10,11]. In [12], McMull en

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 Sec-

tion 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 .



2. 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].





Proposition 2.1. Let f be in class or . If the

Fatou set of f has a completely invariant component, then

the residual Julia set is empty.



Proposition 2.2. Let f be in class or . If there

exists a buried component of

 





f, then





f is

disconnected.

Proposition 2.3. Let f be in class or . If 





)( fJr, then





f is completely invariant, dense





f and uncountably infinite.

Proposition 2.4. If f



 has no wandering do-

mains and









, then there is a periodic Fatou

component such that



fU .

3. Some Conditions When for

Functions in Class



Jf



In this section we will extend some results related with

the residual Julia set for functions in class to func-

tions in class . Qiao in [14] proved the following

theorem for rational fu nctio ns .





Theorem 3.1. Let f be a rational function and

Open Access AM

P. DOMÍNGUEZ, I. HERNÁNDEZ



Jf. The Julia set







contains buried com-

ponents if and only if i)

f is disconnected and ii)



f has no completely invariant component.

Baker and Domínguez in [13] gave the following re-

sult which was step towards a generalisation of Theorem

3.1 for functions in class .



Theorem 3.2. Let f be a meromorphic function in

with no wandering domains. Assume that the







f is

not connected and that



f has no completely in-

variant component. Then the residual Julia set





is non-empty.

If we removed the hypothesis of no wandering do-

mains of Theorem 3.2 and extend it to functions in class

the statement is as follows.

Theorem A. Let f be a function in class . If 





is not connected and



f has no completely invariant

component. Then the residual Julia set





f is non-

empty, this is .



Jf

In order to prove Theorem A we need to state some

results for functions in classes  and . The fol-

lowing lemma was given in [16] for functions in class

, since the proof works for functions in class we

do not write it.





Lemma 3.3. If or and U is a multiply

connected periodic Fatou component such that

, then is completely invariant.

f





UJf

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

of connectivity at least five. Then sin-

gleton components are dense and buried in



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.

No wandering case.

Let . Assume that there are not wandering do-

mains in the Fatou set and that . By Proposi-

tion 2.4 there is a periodic Fatou component such

that . The component is multiply con-

nected 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.

f



UJ



Jf



Wandering case.

We assume that the Fatou set has wandering compo-

nents. 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 un-

countable 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 com-

ponents is countable. Thus the Julia set has uncountably

many buried components. Therefore, .



Jf

2) If we take a multiply-connected Fatou compo-

nent of connectivity n, , , then the proof

follows as the proof of Theorem 3.4 in [17]. Thus sin-

gleton buried components are dense in the Julia set.

Therefore,

U5nn







.

The following theorem is an extension of Proposition

6.1 given in [16], since the proof given in [16] extends

easily to our case, functions in class , we shall give

just a sketch of it. 

Theorem B. Let f



, and





B a closed

set with non-empty interior. Suppose the following two

conditions are satisfied:













ˆBA Jf



 .

 All the Fatou components of f eventually iterate inside

A and never leave again. That is, if  is a Fatou

component,







 for all nN, where N

depends on



Then







.

Sketch of Proof B.

Take any point











zBAJ f

and a neigh-

bourhood





VB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







 since it lies in the complement of A. There-

fore







.

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P. DOMÍNGUEZ, I. HERNÁNDEZ

Open Access AM

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