Invariant Properties of Fatou Set, Julia Set and Escaping Set of Holomorphic Semigroup

Abstract

In this paper, we prove that escaping set of transcendental semigroup is S-forward invariant. We also prove that if holomorphic semigroup is abelian, then Fatou set, Julia set and escaping set are S-completely invariant. We see certain cases and conditions that the holomorphic semigroup dynamics exhibits same dynamical behavior just like the classical complex dynamics. Frequently, we also examine certain amount of connection and contrast between classical complex dynamics and holomorphic semigroup dynamics.

Full text

arXiv:1803.09662v1 [math.DS] 23 Mar 2018
Invariant Properties of Fatou Set, Julia Set and
Escaping Set of Holomorphic Semigroup
Bishnu Hari Subedi and Ajaya Singh
Abstract. In this paper, we prove that escaping set of transcendental semigroup
is S-forward invariant. We also prove that if holomorphic semigroup is abelian,
then Fatou set, Julia set and escaping set are S-completely invariant. We see
certain cases and conditions that the holomorphic semigroup dynamics exhibits
same dynamical behavior just like the classical complex dynamics. Frequently,
we also examine certain amount of connection and contrast between classical
complex dynamics and holomorphic semigroup dynamics.
1. Introduction
It is quite natural to extend the Fatou-Julia-Eremenko theory of the iteration
of single holomorphic map in complex plane Cor extended complex plane C∞to
composite of the family of holomorphic maps. Let Fbe a space of holomorphic
maps on Cor C∞. For any map φ∈F,Cor C∞is naturally partitioned into
two subsets: the set of normality and its complement. We say that a family Fis
normal if each sequence from the family has a subsequence which either converges
uniformly on compact subsets of Cor C∞or diverges uniformly to ∞. The set of
normality or Fatou set F(φ) of the map φ∈Fis the largest open set on which the
iterates φn=φ◦φ◦...◦φ(n-fold composition of φwith itself) is a normal family.
The complement J(φ) is the Julia set. A maximally connected subset of Fatou set
F(f) is a Fatou component.
Semigroup Sis a very classical algebraic structure with binary composition that
satisfies associative law. It naturally arose from the general mapping of a set into
itself. So a set of holomorphic maps on Cor C∞naturally forms a semigroup. Here,
we take a set Aof holomorphic maps and construct a semigroup Sconsisting of
all elements that can be expressed as a finite composition of elements in A. We
say such a semigroup Sby holomorphic semigroup generated by the set A. For
our simplicity, we denote the class of all rational maps of C∞by Rand class of
2010 Mathematics Subject Classification. 37F10, 30D05.
Key words and phrases. Transcendental semigroup, Fatou set, Julia set, escaping set, S-
completely invariant set.
This research work of the first author is supported by PhD faculty fellowship from University
Grants Commission, Nepal.
1

2 B. H. SUBEDI AND A. SINGH
all transcendental entire maps of Cby E. Our particular interest is to study of
dynamics of the families of above two classes of holomorphic maps. For a collection
F={fα}α∈∆of such maps, let
S=hfαi
be a holomorphic semigroup generated by them. Here Fis either a collection R
of rational maps or a collection Eof transcendental entire maps. ∆ represents an
index set to which αbelongs is finite or infinite. Here, each f∈Sis a holomorphic
function and Sis closed under functional composition. Thus f∈Sis constructed
through the composition of finite number of functions fαk,(k= 1,2,3,...,m).
That is, f=fα1◦fα2◦fα3◦ · · · ◦ fαm. In particular, if fα∈R, we say S=hfαia
rational semigroup and if fα∈E, we say S=hfαiatranscendental semigroup.
A semigroup generated by finitely many holomorphic functions fi,(i= 1,2,...,
n) is called finitely generated holomorphic semigroup. We write S=hf1, f2,...,fni.
If Sis generated by only one holomorphic function f, then Sis cyclic semigroup.
We write S=hfi. In this case, each g∈Scan be written as g=fn, where fnis
the nth iterates of fwith itself. Note that in our study of semigroup dynamics, we
say S=hfiatrivial semigroup.
The following result will be clear from the definition of holomorphic semigroup.
It shows that every element of holomorphic semigroup can be written as finite
composition of the sequence of fα
Proposition 1.1.Let S=hfαibe an arbitrary holomorphic semigroup. Then
for every f∈S,fm(for all m∈N) can be written as fm=fα1◦fα2◦fα3◦ · · · ◦ fαp
for some p∈N.
Let fbe a holomorphic map. We say that fiteratively divergent at z∈Cif
fn(z)→αas n→ ∞, where αis an essential singularity of f. A sequence (fk)k∈Nof
holomorphic maps is said to be iteratively divergent at zif fn
k(z)→αkas n→ ∞
for all k∈N, where αkis an essential singularity of fkfor each k. Semigroup S
is iteratively divergent at zif fn(z)→αfas n→ ∞, where αfis an essential
singularity of each f∈S. Otherwise, a function f, sequence (fk)k∈Nand semigroup
Sare said to be iteratively bounded at z.
Based on the definition of classical complex dynamics (that is, on the Fatou-
Julia-Eremenko theory of a complex analytic function), the Fatou set, Julia set and
escaping set in the settings of holomorphic semigroup are defined as follows.
Definition 1.1 (Fatou set, Julia set and escaping set).Fatou set of the
semigroup Sis defined by
F(S) = {z∈C:Sis normal in a neighborhood of z}
and the Julia set J(S)of Sis the compliment of F(S). If Sis a transcendental
semigroup, the escaping set of Sis defined by
I(S) = {z∈C:Sis iteratively divergent at z}
We call each point of the set I(S)by escaping point. Any maximally connected
subset Uof the Fatou set F(S)is called Fatou component.

INVARIANT PROPERTIES OF FATOU SET, JULIA SET AND ESCAPING SET ... 3
It is obvious that F(S) is the largest open subset of Con which the family F
in S(or semigroup Sitself) is normal. Hence its compliment J(S) is a smallest
closed set for any semigroup S. Whereas the escaping set I(S) is neither an open
nor a closed set (if it is non-empty) for any semigroup S.
If S=hfi, then F(S), J(S) and I(S) are respectively the Fatou set, Julia set
and escaping set in classical complex dynamics. In this situation we simply write:
F(f), J(f) and I(f).
The fundamental contrast between classical complex dynamics and semigroup
dynamics appears by different algebraic structure of corresponding semigroups. In
fact, non-trivial semigroup (rational or transcendental) need not be, and most often
will not be abelian. However, trivial semigroup is cyclic and therefore abelian. As
we discussed before, classical complex dynamics is a dynamical study of trivial
(cyclic) semigroup whereas semigroup dynamics is a dynamical study of non-trivial
semigroup.
The following characterization of escaping set will be clear from the definition
1.1 of escaping set and proposition 1.1, which can be an alternative definition.
Theorem 1.1.If a complex number z∈Cis escaping point of any transcen-
dental semigroup S, then every sequence in Shas a subsequence which diverges to
∞at z.
The following immediate relations hold for any f∈Sfrom the definition 1.1.
Indeed, it shows certain connection between classical complex dynamics and semi-
group dynamics.
Theorem 1.2.Let Sbe a semigroup. Then
(1) F(S)⊂F(f)for all f∈Sand hence F(S)⊂Tf∈SF(f).
(2) J(f)⊂J(S)for all f∈S.
(3) I(S)⊂I(f)for all f∈Sand hence I(S)⊂Tf∈SI(f)in the case of
transcendental semigroup S.
Hinkkanen and Martin proved the following results ([4, Lemma 3.1 and Corollary
3.1]).
Theorem 1.3.Let Sbe a rational semigroup. Then Julia set J(S)is perfect
and J(S) = Sf∈SJ(f)
K. K. Poon proved the following results ([11, Theorems 4.1 and 4.2]).
Theorem 1.4.Let Sbe a transcendental semigroup. Then Julia set J(S)is
perfect and J(S) = Sf∈SJ(f)
From the theorem 1.2 ((1) and (3)), we can say that the Fatou set and the
escaping set may be empty. For example, the escaping set of semigroup S=hf, gi
generated by functions f(z) = ezand g(z) = e−zis empty (the particular function
h=g◦fk∈S(say) is iteratively bounded at any z∈I(f)). There are several
transcendental semigroups where Fatou set and escaping set are non-empty (see for
instance [9, Examples 3.2 and 3.3] and [7, Examples 2.6 and 2.7])

4 B. H. SUBEDI AND A. SINGH
2. Invariant features of Fatou set, Julia set and escaping set
The main contrast between classical complex dynamics and semigroup dynamics
will appear in the invariant features of Fatou set, Julia set and escaping set. Note
that invariant feature is considered a very basic and fundamental structure of these
sets.
Definition 2.1 (Forward, backward and completely invariant set).For
a given semigroup S, a set U⊂Cis said to be S-forward invariant if f(U)⊂Ufor
all f∈S. It is said to be S-backward invariant if f−1(U) = {z∈C:f(z)∈U} ⊂ U
for all f∈Sand it is called S-completely invariant if it is both S-forward and S-
backward invariant.
If Sis a rational semigroup, then Hinkkanen and Martin [4, Theorem 2.1] proved
the following result.
Theorem 2.1.The Fatou set F(S)of Sis S-forward invariant and Julia set
J(S)of Sis S-backward invariant.
If Sis a transcendental semigroup, then K. K. Poon [11, Theorem 2.1] proved
the following result.
Theorem 2.2.The Fatou set F(S)of Sis S-forward invariant and Julia set
J(S)of Sis S-backward invariant.
Dinesh Kumar and Sanjay Kumar [7, Theorem 4.1] proved the following result
that shows escaping set I(S) is also S-forward invariant. Here, we provide another
proof based on our definition 1.1 of escaping set.
Theorem 2.3.The escaping set I(S)of transcendental semigroup Sis S-
forward invariant.
Before proving this theorem 2.3, we define and discuss some special subsets of
semigrioup S.
Definition 2.2 (Subsemigroup, left ideal, right ideal and ideal).A non-
empty subset Tof semigroup Sis a subsemigroup of Sif f◦g∈Tfor all f, g ∈T.
A non-empty subset Iof Sis called a left (or right) ideal of Sif f◦g∈I(or
g◦f∈I)for all f∈Sand g∈I.Iis called an ideal of Sif it is both left and
right ideal of S.
Every (left or right) ideal is a subsemigroup but converse may not hold. The
left ideal, right ideal and an ideal of a semigroup Scan be constructed easily as
shown in the following proposition.
Proposition 2.1.For any f∈S, the set S◦f={g◦f:g∈S}is a left
ideal of S, the set f◦S={f◦g:g∈S}is an right ideal of Sand the set
S◦f◦S={h◦f◦g:h, g ∈S}is an ideal of S. For any n∈N,(g◦f)n=h◦f
for some h∈Sand (h◦f◦g)n=p◦f◦qfor some p, q ∈S.

INVARIANT PROPERTIES OF FATOU SET, JULIA SET AND ESCAPING SET ... 5
Proof of the theorem 2.1. Let z∈I(S). Then by the definition 1.1,
semigroup Sis iteratively divergent at zand for any g∈S, the subsemigroup
S◦g={f◦g:f∈S}is also iteratively divergent at z. That is, (f◦g)n(z)→
∞as n→ ∞ for all f∈S. By the propositon 2.1, for all n∈N, we have
(f◦g)n=hni◦gfor some hni∈Swhere nidepends on n. Therefore, (f◦g)n(z)→ ∞
as n→ ∞ ⇒ (hni◦g)(z) = hni(g(z)) → ∞. Since hni∈Sis transcendental entire
function with hni(g(z)) → ∞, we must have hni=pmifor some p∈Sand mi∈N.
Thus, from (f◦g)n(z)→ ∞ as n→ ∞ we get pmi(g(z)) → ∞ as mi→ ∞. This
proves that Sdiverges at g(z), so g(z)∈I(S) for all g∈S. Hence I(S) is S-forward
invariant. 
In the case of rational semigroup, Hinkkanen and Martin [4, Example1] provided
the following example that show that Fatou set F(S) need not be backward invariant
and Julia set J(S) need not be forward invariant.
Example 2.1.For a rational semigroup S=hz2, z2/ai, where a∈C,|a|>1,
the Fatou set F(S) = {z:|z|<1or |z|>|a|} is not S-backward invariant and
Julia set J(S) = {z: 1 6|z|6|a|} is not S-forward invariant.
Fatou [2] and Julia [5] independently proved that classical Fatou set and Julia
set of rational function are completely invariant. Note that all three sets F(f), J (f)
and I(f) of transcendental entire function are also completely invariant. This is also
fundamental contrast between classical complex dynamics and semigroup dynamics.
We prove under certain conditions, Fatou set is S-backward invariant and Julia
set is S-forward invariant of a rational semigroup S. First we define the notion of
abelian holomorphic semigroup.
Definition 2.3.Let S=hfαibe a holomorphic semigroup. We say the Sis
abelian if fα◦fβ=fβ◦fαfor all generators fαand fβof S.
Theorem 2.4.The Fatou set F(S)is S-backward invariant and Julia set J(S)
is S-forward invariant if Sis an abelian rational semigroup.
Proof. We prove that if g(z)∈F(S), then z∈F(S) for all g∈S. This
follows that g−1(F(S)) ⊂F(S) for all g∈S. Suppose, g(z)∈F(S). Let Ube a
neighborhood of g(z) such that U⊂F(S). Then there is a subsequence (fnj) such
that fnj(g(z)) →f(g(z)) uniformly on U, where fis rational or constant ∞. Since
Sis abelian, so we have g(fnj(z)) →g(f(z)) uniformly on Uas well. This shows
that g◦fnj→g◦funiformly on U. This proves z∈F(S) for all g∈S.
From the theorems 2.1 and 2.4, we can say that Fatou set F(S) and Julia set
J(S) are S-completely invariant if Sis an abelian rational semigroup. The following
example of Hinkkanen and Martin [4, Example 2] is best for the above theorem 2.4.
Example 2.2.The semigroup S=hTn(z) : n= 0,1,2,...igenerated by
Tchebyshev polynomials Tn(z)defined by T0(z) = 1, T1(z) = zand Tn+1(z) =
2zTn(z)−Tn−1(z)is abelian. Therefore, by above theorem 2.4, Fatou set F(S)
is S-forward invariant and Julia set J(S)is S-backward invariant.

6 B. H. SUBEDI AND A. SINGH
Under the same condition of rational semigroup, we prove that Fatou set is
S-backward invariant and Julia set is S-forward invariant if Sis a transcendental
semigroup.
Theorem 2.5.The Fatou set F(S)is S-backward invariant and Julia set J(S)
is S-forward invariant if Sis an abelian transcendental semigroup.
Proof. The proof is similar to theorem 2.4. 
In [8, Theorem 2.1], Dinesh Kumar and Sanjay Kumar provided the following
condition for backward invariance of I(S). Here, we give another proof based on
our definition 1.1 of escaping set.
Theorem 2.6.The escaping set I(S)of transcendental semigroup Sis S-
backward invariant if Sis an abelian transcendental semigroup.
Proof. We prove that if g(z)∈I(S), then z∈I(S) for all g∈S. This follows
that g−1(I(S)) ⊂I(S) for all g∈S. This will be proved if we are able to prove its
contrapositive statement: if z /∈I(S), then g(z)/∈I(S) for all g∈S. Let z /∈I(S)
there is some f∈Swhich is iteratively bounded at z. That is, fn(z)9∞as
n→ ∞. In this case, there exists a sequence (fk)k∈Nin Scontaining fwhich is
iteratively bounded at zand all subsequences of this sequence containing fare also
iteratively bounded at z. Now, for any g∈S, (fk◦g)k∈Nis a sequence in S. Since
Sis abelian and gis a transcendental entire function, so by the continuity of gat
z∈C, we can write that (fk◦g)(z) = (g◦fk)(z) for all k∈N. From which it follows
that the sequence (fk◦g)k∈Nis iteratively bounded at z. So, all subsequence of this
sequences are iteratively bounded at z. From the fact (fk◦g)(z) = (g◦fk)(z) for all
k∈N, we can say that all subsequences of the sequence (fk◦g)k∈Nare iteratively
bounded at g(z). That is, g(z)/∈I(S) for all g∈S. Therefore, g−1(I(S)) ⊂I(S)
for all g∈S. This proves that I(S) is backward invariant. 
From the result of above theorems 2.5 and 2.6, we can conclude that Fatou set
F(S), Julia set J(S) and escaping set I(S) are S-completely invariant if Sis an
abelian transcendental semigroup. For example, the following semigroups
(1) hz+γsin z, z +γsin z+ 2kπi,
(2) hz+γsin z, −z−γsin z+ 2kπi,
(3) hz+γez, z +γez+ 2kπii,
(4) hz−sin z, z −sin z+ 2πi,
(5) heγz , eγ z+2π i
γi, where 0 < γ < e−1,
are abelian transcendental semigroups, so their Fatou set, Julia set and escaping
sets are S-completely invariant.
The theorems 2.4, 2.5 and 2.6 give a kind of connection between classical com-
plex dynamics and semigroup dynamics. We got completely invariant structure
of Fatou set, Julia set and escaping set in both classical complex dynamics and
semigroup dynamics because of their associated abelian semigroups. This was our
expectation that abelian semigroup (rational and transcendental) must exhibit same
dynamical features as in classical complex dynamics. Note that it will not better to

INVARIANT PROPERTIES OF FATOU SET, JULIA SET AND ESCAPING SET ... 7
conclude that Fatou set, Julia set and escaping set can not be completely invariant
unless the semigroup is abelian.
If Sis a finitely generated rational semigroup, then Hiroki Sumi [14, Lemma
1.1.4 (2)] proved the following results.
Theorem 2.7.If S=hf1, f2,...,fniis a finitely generated rational semigroup,
then F(S) = Tn
i=1 f−1
i(F(S)) and J(S) = Sn
i=1 f−1
i(J(S))
In the case of finitely generated transcendental semigroup, we prove following
result which is analogous to above theorem 2.7.
Theorem 2.8.If S=hf1, f2,...,fniis a finitely generated transcendental semi-
group, then F(S) = Tn
i=1 f−1
i(F(S)) and J(S) = Sn
i=1 f−1
i(J(S))
Proof. The Fatou set F(S) is S-forward invariant in general (theorem 2.2).
So, fi(F(S)) ⊂F(S) and it follows F(S)⊂Tn
i=1 f−1
i(F(S)) for all i.
Next, let z0∈Tn
i=1 f−1
i(F(S)). Then (say) wi=fi(z0)∈F(S) for all i. The
semigroup Sis normal at wifor all i. In other words, every g∈Sis equicontinuous
at wifor all i. So, for any ǫ > 0, there is δ > 0 such that
d(g(w), g(wi)) < ǫ, whenever d(w, wi)< δ
for all w∈F(S) and i= 1,2,3,...,n. Where drepresents Euclidean metric on C.
For such δ, there is η > 0 such that
d(fi(z), fi(z0)) < δ, whenever d(z, z0)< η
for all z∈Tn
i=1 f−1
i(F(S)) and i= 1,2,3,...,n. Thus, ultimately, we conclude
that
d(g(fi(z)), g(fi(z0))) < ǫ whenever d(z, z0)< η
for all z∈Tn
i=1 f−1
i(F(S)) and i= 1,2,3,...,n. Here S=Sn
i=1(S◦fi). So Sis
equicontinuous at z0. That is, z0∈F(S). Hence, F(S) = Tn
i=1 f−1
i(F(S)). Second
part of the theorem easily follows as
J(S) = C−F(S) = C−
n
\
i=1
f−1
i(F(S)) =
n
[
i=1
(C−f−1
i(F(S)) =
n
[
i=1
f−1
i(J(S)).

The relation J(S) = Sn
i=1 f−1
i(J(S)) for Julia set is called backward self sim-
ilarity. With this property, dynamics of semigroup can be regarded as backward
iterated function systems.
The theorems 2.1 and 2.6 can be used directly to express the escaping set I(S) as
a finite intersection of all pre-images of itself under the generators of the semigroup
Sif it is finitely generated. The following result is due to Dinesh Kumar and Sanjay
Kumar [8, Theorem 2.6]. Here, we give an alternative proof.
Theorem 2.9.If S=hf1, f2,...,fniis a finitely generated transcendental semi-
group, then I(S) = Tn
i=1 f−1
i(I(S)) if Sis abelian semigroup.

8 B. H. SUBEDI AND A. SINGH
Proof. I(S) is forward invariant in general and under the assumption given in
the statement, it is also backward invariant (see for instance theorem 2.6). So, we
have fi(I(S)) ⊂I(S) and f−1
i(I(S)) ⊂I(S) for all 1 6i6n. From which we get
respectively I(S)⊂Tn
i=1 f−1
i(I(S)) and I(S)⊃Tn
i=1 f−1
i(I(S)) for all 1 6i6n.
Thus, we get I(S) = Tn
i=1 f−1
i(I(S)). 
Later, we found that the condition of abelian in semigroup Sis not required to
hold essence of above theorem 2.9. For I(S)⊂Tn
i=1 f−1
i(I(S)) as above. Next, let
z0∈Tn
i=1 f−1
i(I(S)). Then (say) wi=fi(z0)∈I(S) for all i. By the definition
1.1, semigroup Sis iteratively divergent at wifor all i. By the theorem 1.1, every
sequence (fj)j∈Nin Shas a subsequence (fjk)k∈Nwhich diverges to ∞at wi=fi(z0).
This proves that the sequence (fjk◦fi)k,i∈Ndiverges to ∞at z0. Hence the result.
3. Further results in abelian semigroup dynamics
We have some further expectations that semigroup dynamics behave just like
classical complex dynamics if corresponding semigroup is abelian. We see more
specific results in abelian semigroup dynamics. This also exhibits a nice connection
between classical complex dynamics and semigroup dynamics.
Theorem 3.1.Let Sbe an abelian rational semigroup. Then J(S) = J(f)for
all f∈Sof degree at least two.
We recall the following result of Fatou [3] and Julia [6] concerning commuting
rational functions.
Lemma 3.1.Let fand gbe two rational functions of degree at least two such
that f◦g=g◦f. Then J(f) = J(g).
Proof of the theorem 3.1. Since semigroup Sis abelian, so we have fi◦
fj=fj◦fifor all generators fiand fjwith i6=j. Then by above lemma 3.1,
J(fi) = J(fj) for all iand jwith i6=j. Also, every f∈Spermutes with each
generator fifor all i, so again by the same lemma 3.1, J(f) = J(fi) for all i. This
fact together with the fact of theorem 1.3, we can conclude that J(S) = J(f) for
all f∈S.
The analogous result in transcendental semigroup may not hold in general be-
cause of the essence of above lemma 3.1 is still unanswered for permutable transcen-
dental entire functions. Julia sets for two permutable entire functions were studied
in [12, 13, 15] where we found certain conditions from which we can get the essence
of above lemma 3.1. If we expose extra conditions in the statement, then result
analogous to the theorem 3.1 holds in the case of transcendental semigroup. One
of the analogous result of the theorem 3.1 was proved by K.K. Poon [11, Theorem
5.1]. Here, we only give sketch of the proof similar to above proof of the theorem
3.1.
Theorem 3.2.Let Sbe an abelian transcendental semigroup in which each
generator is of finite type. Then J(S) = J(f)for all f∈S.

INVARIANT PROPERTIES OF FATOU SET, JULIA SET AND ESCAPING SET ... 9
Before proving this theorem 3.2, We recall the notion of finite type, bounded
type functions and other related objects. Recall that the set CV (f) = {w∈C:
w=f(z) such that f′(z) = 0}is called the set of critical values. The set AV (f)
consisting of all w∈Csuch that there exists a curve Γ : [0,∞)→Cso that
Γ(t)→ ∞ and f(Γ(t)) →was t→ ∞ is called the set of asymptotic values of f
and the set SV (f) = (CV (f)∪AV (f)) is called the set of singular values of f. If
SV (f) has only finitely many elements, then fis said to be of finite type. If SV (f)
is a bounded set, then fis said to be of bounded type. The sets
S={f:fis of finite type}and B={f:fis of bounded type}
are respectively called Speiser class and Eremenko-Lyubich class. Again, we recall
the following result of K. K. Poon [11, Lemmas 5.1 and 5.2] concerning commuting
transcendental maps.
Lemma 3.2.Let fand gbe two transcendental entire functions of finite type.
Then f◦gis of finite type. Moreover, if fand gare permutable, then J(f) = J(g).
Proof of the theorem 3.2. Since semigroup Sis abelian, then by above
lemma 3.2, J(fi) = J(fj) for all generators fiand fjwith i6=j. Each f=
fi1◦fi2◦...fimis of bounded type by the same lemma 3.2 and J(f) = J(fi) for all
i. This fact together with the fact of theorem 1.4, we can conclude that J(S) = J(f)
for all f∈S.. 
We expect that the condition mentioned in the theorem 3.2 will also be enough
to hold I(S) = I(f) for all f∈S.
Theorem 3.3.Let Sis an abelian transcendental semigroup in which each
generator is of finite type (or bounded type). Then I(S) = I(f)for all f∈S.
Lemma 3.3.If fand gare permutable transcendental entire functions of finite
type (or bounded type), then I(f) = I(g).
Proof. As given in the statement of this lemma, Poon [11, Lemma 5.2] showed
that F(f) = F(g)(Theorem 3.2). Eremenko and Lyubich [1] proved that if tran-
scendental function f∈B, then I(f)⊂J(f), and J(f) = I(f). For any function
of finite type (or bounded type), we must have I(f) = I(g). This lemma will be
proved if we show J(f)−I(f) = J(g)−I(g). Let z∈J(f)−I(f). Then zis a
non-escaping point of J(f) and so the sequence (fn) has a bounded subsequence at
z.J(f) = J(g) implies that the sequence (gn) has also a bounded subsequence at
z. So z∈J(g)−I(g). Therefore, J(f)−I(f)⊂J(g)−I(g). By similar fashion,
we can show that J(g)−I(g)⊂J(f)−I(g). Hence, we got our claim.

Proof of the Theorem 3.3. Since semigroup Sis abelian, so we have fi◦
fj=fj◦fifor all generator fiand fjwith i6=j. So by above lemma 3.3, we
have I(fi) = I(fj). Any f∈Scan be written as f=fi1◦fi2◦fi3◦ · · · ◦ fim. By
permutability of each fi, we can rearrange fijand ultimately represented by
f=ft1
1◦ft2
2◦...◦ftn
n

10 B. H. SUBEDI AND A. SINGH
where each tk>0 is an integer for k= 1,2,...,n. The lemms 3.2 can be applied
repeatably to show each of ft1
1, f t2
2,...,ftn
nis of finite (or bounded) type and so
f=ft1
1◦ft2
2◦. . . ◦ftn
nis itself finite (or bounded) type. Since each fipermutes
with fand hence again by above lemma 3.3, I(fi) = I(f) for all f∈S. Therefore,
I(S) = I(f) for each f∈S.
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Central Department of Mathematics, Institute of Science and Technology,
Tribhuvan University, Kirtipur, Kathmandu, Nepal
E-mail address:subedi.abs@gmail.com / subedi bh@cdmathtu.edu.np
Central Department of Mathematics, Institute of Science and Technology,
Tribhuvan University, Kirtipur, Kathmandu, Nepal
E-mail address:singh.ajaya1@gmail.com / singh a@cdmathtu.edu.np