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
Received August 15, 2013; revised September 15, 2013; accepted September 23, 2013
Copyright © 2013 Patricia Domínguez, Iván Hernández. This is an open access article distributed under the Creative Commons At-
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


r
J
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


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
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 ,.
f
HolX YfDYf
HolX XHolX



c
The set of singular values of


,
f
HolX Y is
 
SVfCfA f, where is the set of

Cf
critical values and


A
f is the set of asymptotic val-
ues.
Let


f
Hol X, the sequence formed by its iterates
will be defined and denoted by ,
0:Idf1
:
nn
f
ff

,
n

. The study makes sense and is no n-trivial when X
is either the Riemann sphere , the complex plane
ˆ

or the complex plane minus one point, this is


0.
Taking
X

 and ˆ Y

 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
f consists of all points
X
z
 (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
  
c

J
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
fz


  studied in [2]. The stable set (Fatou set)
and the chaotic set (Julia set) for the parameters 4


,
1

 can be seen in Figure 1.
Examples of functions in class can be found in the 
family

2
e
z
c
c
fRz  

, where is a rational func-

Rz
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

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-

main for the function 4, 11
4ez
fz


  , 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


r
J
f as the set of those points of


J
f which do
not belong to the boundary of any component of the Fa-
tou set


F
f. The points of

r
J
f

r
are called buried
points and the components of
J
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
 


J
f, then


J
f is
disconnected.
Proposition 2.3. Let f be in class or . If 


)( fJr, then


r
J
f is completely invariant, dense
in


J
f and uncountably infinite.
Proposition 2.4. If f

 has no wandering do-
mains and


r
Jf


U
, then there is a periodic Fatou
component such that

J
fU .
3. Some Conditions When for
Functions in Class

r
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 20

ˆ
Jf. The Julia set


J
f

contains buried com-
ponents if and only if i)
J
f is disconnected and ii)

F
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



J
f is
not connected and that

F
f has no completely in-
variant component. Then the residual Julia set


r
J
f
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 


J
f
is not connected and

F
f has no completely invariant
component. Then the residual Julia set

r

J
f is non-
empty, this is .

r
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
U


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
H
of connectivity at least five. Then sin-
gleton 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.
U
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

r
Jf
U
U
U

f
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, .

r
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


r
Jf

.
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

ˆ

A
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,


n
f
A

 for all nN, where N
depends on

.
Then


r
Jf

.
Sketch of Proof B.
Take any point





ˆ
zBAJ f and a neigh-
bourhood


ˆ
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


r
J
f   since it lies in the complement of A. There-
fore


r
Jf

.
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Open Access AM
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