P. DOMÍNGUEZ, I. HERNÁNDEZ

20

ˆ

Jf. The Julia set

f

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

r

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

f

is not connected and

f has no completely invariant

component. Then the residual Julia set

r

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

of connectivity at least five. Then sin-

gleton components are dense and buried in

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,

U5nn

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

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

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

VBA

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

f

since it lies in the complement of A. There-

fore

r

Jf

.

REFERENCES

[1] J. F. Gouyet, “Physics and Fractal Structures,” Masson

Springer, Paris, New York, 1996.

[2] M. A. Mo n tes de Oc a Ba lderas, G. J. F Sienra Loera and J.

E. King Dávalos, “Baker Domains for Period Two for the

Family ,ez

fz

,” International Journal of Bifur-

cation and Chaos, in Press, 2013.

[3] I. N. Baker, J. Kotus and Y. N. Lü, “Iterates of Mero-

morphic Functions II: Examples of Wandering Domains,”

Journal of the London Mathematical Society, Vol. 42, No.

2, 1990, pp. 267-278.

http://dx.doi.org/10.1112/jlms/s2-42.2.267

[4] I. N. Baker, J. Kotus and Y. N. Lü, “Iterates of Mero-

morphic Functions: I,” Ergodic Theory and Dynamical

Systems, Vol. 11, No. 2, 1991, pp. 241-248.

[5] I. N. Baker, J. Kotus and Y. N. Lü, “Iterates of Mero-

Open Access AM