Fast Escaping Set of Transcendental Semigroup

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Universal Journal of Applied Mathematics 7(1): 1-7, 2019
DOI: 10.13189/ujam.2019.070101
http://www.hrpub.org
Fast Escaping Set of Transcendental Semigroup
Bishnu Hari Subedi∗,Ajaya Singh
Central Department of Mathematics, Tribhuvan University, Kirtipur, Kathmandu, Nepal
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2019 by authors, all rights reserved. Authors agree that this article remains permanently
open access under the terms of the Creative Commons Attribution License 4.0 International License
Abstract In this paper, we study fast escaping sets of
transcendental semigroups. In particular, we discuss some
fundamental structure and properties of fast escaping sets.
We also show how far the classical dynamical theory of fast
escaping sets of transcendental entire functions applies to
general settings of transcendental semigroups and what new
phenomena can occur.
Keywords Tanscendental semigroup, escaping set, fast
escaping set, L-th levels of fast escaping set.
AMS Subject Classification (2010): 37F10, 30D05
1 Introduction
The principal aim of this paper is to study fast escaping sets
not of single transcendental entire maps of complex plane C,
but of semigroups generated by families of such maps under
composition. Let Fbe a collection of transcendental entire
maps on C. For any map f∈F, the complex plane Cis
naturally partitioned into two subsets: the set of normality and
its complement. The set of normality or Fatou set F(f)of fis
the largest open set on which the iterates fn=f◦f◦. . . ◦f
where n∈N, is a normal family. The complement of F(f)is
the Julia set J(f). A maximally connected subset of F(f)is a
Fatou component. The escaping set of any f∈Fis the set
I(f) = {z∈C:fn(z)→ ∞ as n→ ∞}
and any z∈I(S)is called escaping point. For any transcen-
dental entire function f, the I(f)was first studied by A. Ere-
menko [4]. He showed that I(f)6=∅,J(f) = ∂I (f),
I(f)∩J(f)6=∅, and I(f)has no bounded component. In
view of this last statement, he posed a conjecture:
Conjecture 1.1. Every component of I(f)is unboun-ded.
This conjecture is considered an important open problem of
transcendental dynamics, and nowadays it is famous as Ere-
menko’s conjecture. This conjecture 1.1 has been proved for
the fast escaping set A(f), which consists of points that tend
to infinity as fast as possible under iteration. This set is a sub-
set of I(S), and it was introduced first time by Bergweiler and
Hinkkanen [3] and defined in the following form by Rippon
and Stallard [15].
Definition 1.1. For a transcendental entire function f, the fast
escaping set is defined by
A(f) = {z∈C:∃L∈Nsuch that
|fn+L(z)| ≥ Mn(R)for n∈N}
where M(r) = max|z|=r|f(z)|, r > 0and Mn(r)denotes nth
iteration of M(r)with respect to r.R > 0can be taken to be
any number such that M(r)> r for r≥R.
Recall that
C(f) = {z∈C:f0(z)=0}
(where 0represents derivative of fwith respect to z) is the set
of critical points and
CV (f) = {w∈C:w=f(z)such that f0(z)=0}
is the set of critical values of f. 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 the set of asymptotic
values of fand
SV (f) = (C V (f)∪AV (f))
is the set of singular values of f. If SV (f)is finite, then fis
said to be of finite type. If S V (f)is bounded, 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 known as Speiser class and Eremenko- Lyu-
bich class. The main concern of transcendental iteration the-
ory is to describe the nature of the components of Fatou sets
and the structure and properties of the Julia sets, escaping sets,

2 Fast Escaping Set of Transcendental Semigroup
and fast escaping sets. We refer to the monograph: Dynamics
of Transcendental Entire Functions [7] and to the book: Ho-
lomorphic Dynamics [11] for basic facts concerning the Fatou
set, Julia set and escaping set of a transcendental entire functi-
ons. We refer to [14, 15, 16] for facts and results concerning
the fast escaping set of a transcendental entire functions.
Our particular interest is to study the dynamics of semi-
groups generated by a class of transcendental entire maps. For
a collection F={fα}α∈∆of such maps, let S=hfαibe a
transcendental semigroup generated by them. The index set ∆
is allowed to be infinite in general. Each f∈Sis a transcen-
dental entire function, and Sis closed under functional compo-
sition. 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. A semigroup generated by fini-
tely many transcendental entire functions fi,(i= 1,2, . . . , n)
is called finitely generated transcendental semigroup and we
write S=hf1, f2, . . . , fni. If Sis generated by only one
transcendental entire function f, then Sis a cyclic transcen-
dental semigroup and we write S=hfi. In this case, each
g∈Scan be written as g=fn. Note that in our study of
semigroup dynamics, we say S=hfiis a trivial semigroup.
The transcendental semigroup Sis abelian if fα◦fβ=fβ◦fα
for all generators fα,fβof S. The transcendental semigroup
Sis bounded type (finite type) if each of its generators fαis
bounded type (finite type).
The family Fof analytic maps forms a normal family in a
domain Dif every sequence of the member of Fcontains a
subsequence which converges or diverges locally uniformally
in D. If there is a neighborhood Uof the point z∈Csuch that
Fis normal family in U, then we say Fis normal at z.
Let fbe a transcendental entire map. We say that fis ite-
ratively divergent at z∈Cif fn(z)→ ∞ as n→ ∞. Semi-
group Sis iteratively divergent at zif fn(z)→ ∞ as n→ ∞
for all f∈S. A semigroup Sis said to be iteratively boun-
ded at zif there is f∈Swhich is not iteratively divergent of
z. Note that in a semigroup S=hfαiif f∈S, then for all
m∈N, f m∈S. So, fm=fα1◦fα2◦fα3◦ · · ·◦fαpfor some
p∈N. In this sense, any f∈Sis iteratively divergent at z
always means that there is a sequence (gn)n∈Nin Srepresen-
ting g1=f, g2=f2, . . . , gn=fn, . . . such that gn(z)→ ∞
as n→ ∞. In this sense, semigroup Sis iteratively divergent
at any z∈Cif every sequence in Shas a subsequence which
diverges to infinity at z.
Based on the Fatou-Julia-Eremenko theory of a transceden-
tal entire function, the Fatou set, Julia set and escaping set of
transcendental semigroup are defined as follows.
Definition 1.2 (Fatou set, Julia set and escaping set).Fatou
set of the transcendental semigroup Sis defined by
F(S) = {z∈C:Sis normal at z}
and the Julia set J(S)of Sis the compliment of F(S). 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.
It is obvious that F(S)is a largest open subset of Csuch
that semigroup Sis normal. Hence its compliment J(S)is a
closed set. Whereas the escaping set I(S)is neither an open
nor a closed set (if it is non-empty). Any maximally connected
subset Uof F(S)is called a Fatou component. 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).
In this paper, classical transcendental dynamics refers to the
iteration theory of single transcendental map and transcenden-
tal semigroup dynamics refers to the dynamical theory gene-
rated by a set of transcendental entire maps. In transcendental
semigroup dynamics, algebraic structure of semigroup natu-
rally attached to the dynamics and hence the situation is com-
plicated. The fundamental contrast between classical transcen-
dental dynamics and semigroup dynamics appears by different
algebraic structure of corresponding semigroups. In fact, non-
trivial semigroup may not be abelian in general, however trivial
semigroup is cyclic and therefore, abelian. As we discussed be-
fore, classical complex dynamics is a dynamical study of trivial
(cyclic) semigroup whereas semigroup dynamics is a dynami-
cal study of non-trivial semigroup.
2 Motivation and Statement of the Pro-
blem
The main motivation of this paper comes from seminal work
of Hinkkanen and Martin [6] on the dynamics of rational se-
migroups and the extension study of K. K. Poon [12] and
Zhigang Haung [25] to the dynamics of transcendental semi-
groups. Both of them naturally generalized some results of
classical complex dynamics to the semigroup dynamics as well
as they also investigated some new phenomena in semigroup
dynamical system. Another motivation of studying escaping
sets and some extra properties and structure of Fatou sets and
Julia sets of transcendental semigroups comes from the work
of Dinesh Kumar and Sanjay Kumar [8, 9, 10] where they de-
fined escaping set and discussed how far escaping set of clas-
sical transcendental dynamics can be generalized to semigroup
dynamics. In parallel, we also studied certain structure and
propperties of Fatou sets, Julia sets and escaping sets under se-
migroup dynamics in [17, 18, 19, 20, 21, 22, 23, 24]. From
these attempts, we again more motivated to the study of fast
escaping sets of transcendental semigroups.
In this paper, we introduce fast escaping set in transcenden-
tal semigroup dynamics. The principal aim of this paper is
to show how far classical transcendental dynamics applies to
transcendental semigroup dynamics and what new phenomena
appear in transcendental semigroup settings. Note that the fast
escaping set A(f)now plays a key role in classical transcen-
dental dynamics which was introduced first time by Bergweiler
and Hinkkanen [3] and studied in more depth by Rippon and
Stallard [15]. The set A(f)has some properties exactly similar
to those of I(f). For example A(f)6=∅, in fact, it is infinite
set ([3, Lemma 2]), ∂A(f) = J(f)([3, Lemma 3]) and ([15,

Universal Journal of Applied Mathematics 7(1): 1-7, 2019 3
Theorem 5.1 (b)]), A(f)∩J(f)6=∅([3, Lemma 3]) and ([15,
Theorem 5.1(a)]), The set A(f)is completely invariant under
f([15, Theorem 2.2 (a)]) and the set A(f)does not depend on
the choice of R > 0([15, Theorem 2.2 (b)]). However, in [14,
Theorem 1] and [15, Theorem 1.1], it is shown that all com-
ponents of A(f)are unbounded and since A(f)⊂I(f)so it
provides a partial answer to Eremenko’s conjecture 1.1. If U
is a Fatou component of fin A(f), then its boundary is also
in A(f), that is, U⊂A(f)([14, Theorem 2]) and ([15, Theo-
rem 1.2]). These are much stronger properties of A(f)than the
set I(f). In next sections 3, 4 and 5, we define and formulate
fast escaping set in transcendental semigroup and we also exa-
mine how far these stated properties of classical transcendental
dynamics can be generalized to semigroup dynamics.
3 Fast Escaping Set of Transcendental
Semigroup
There is no formulation of fast escaping sets in transcen-
dental semigroup dynamics. We have started to define fast es-
caping set and try to formulate some other related terms and
results.
Let Sbe a transcendental semigroup. Let us define a set
AR(S) = {z∈C:|fn(z)| ≥ Mn(R)
for all f∈Sand n∈N}(3.1)
where M(r) = max|z|=r|f(z)|and Mn(r)denotes the nth
iterates of M(r).R > 0can be taken any value such that
M(r)> r for r≥R. If ris sufficiently large then Mn(r)→
∞as n→ ∞. The set AR(S)is well defined in semigroup
Sbecause for any f∈S,fn∈Sfor all n∈N. From the
condition |fn(z)| ≥ Mn(R)for all f∈Sand n∈Nof
the set AR(S), we can also say that a point z∈Cis in AR(S)
if every sequence (gn)n∈Nin Shas a subsequence (gnk)nk∈N
which increases without bound at least as fast as the n-iterates
of the maximum modulus of each gnk.
Definition 3.1 (Fast escaping set).The fast escaping set A(S)
of a transcendental semigroup Sconsists the set AR(S)and all
its pre-images. In other words, fast escaping set is a set of the
form
A(S) = [
n≥0
f−n(AR(S)) (3.2)
where f−n(AR(S)) = {z∈C:fn(z)∈AR(S)}for all
f∈Sand n∈N.
We can do certain stratification of fast escaping set which
helps to make it more visible and provides a significant new
understanding of the structure and properties of this set. We
can write fast escaping set as a countable union of all its labels
as we define below.
Definition 3.2 (L th label of fast escaping set).Let A(S)be
a fast escaping set of transcendental semigroup S. For L∈Z,
then the set of the form
AL
R(S) = {z∈C:|fn(z)| ≥ Mn+L(R)
for all f∈S, n ∈Nand n+L≥0}(3.3)
is called Lth level of fast escaping set A(S).
Note that the set AR(S)defined above in 3.1 is the 0th level
of A(S). Since Mn+1 (R)> Mn(R)for all n≥0, so from
3.3, we get following chain of relation
. . . ⊂AL
R(S)⊂AL−1
R(S)⊂. . . ⊂A1
R(S)⊂AR(S)⊂
A−1
R(S)⊂A−2
R(S)⊂. . . ⊂A−(L−1)
R(S)⊂A−L
R(S)⊂. . .
(3.4)
From 3.2 and 3.4, the fast escaping set can also be written as
an expanding union of its labels.
A(S) = [
L∈N
A−L
R(S)(3.5)
Again from the definition 3.1, that is, from 3.2, if any z0∈
A(S), then z0∈f−n(AR(S)) for some n≥0. It gives
fn(z0)∈AR(S)for all f∈S. From 3.1, there is L∈N
such that |fL(fn(z0))|=|fn+L(z0)| ≥ Mn(R). With this
clause, the fast escaping set of transcendental semigroup Scan
now be written as
A(S) = {z∈C:there exists L∈Nsuch that
|fn+L(z)| ≥ Mn(R)for all f∈S, and n∈N}.(3.6)
We can use any one of the form 3.2 or 3.5 or 3.6 as a definition
of fast escaping set. Note that by the definition (3.2 or 3.5 or
3.6), fast escaping set A(S)of any transcendental semigroup S
is a subset of escaping set I(S). Since from [17, Theorem 1.2
(3)], we can say that I(S)may be empty. For any transcenden-
tal semigroup S, if I(S) = ∅, then we must have A(S) = ∅.
It is not known whether there is a transcendental semigroup
Ssuch that I(S)6=∅but A(S) = ∅. Note that in classical
transcendental dynamics, both of these sets are non-empty.
Example 3.1. Let Sbe a transcendental semigroup generated
by the functions f(z) = ezand g(z) = e−z. Since h=g◦
fn∈Sis iteratively bounded at any z∈C. So, I(S) = ∅and
A(S) = ∅.
Like escaping set I(S), fast escaping set A(S)is also neither
an open nor a closed set if it is non-empty. Similar to the result
[17, Theorem 1.2 (3)] of escaping set, the following result is
also clear from the definition of fast escaping set.
Theorem 3.1. Let Sbe a transcendental semigroup. Then
A(S)⊂A(f)for all f∈Sand hence A(S)⊂Tf∈SA(f).
We have mentioned several results and examples of
transcendental semigroups in [17, 18, 19, 21, 22, 23] where
escaping set is non-empty. One of the latest particular result in

4 Fast Escaping Set of Transcendental Semigroup
the field of transcendental semigroup dynamics is a condition
for which escaping set of the semigroup is same as the escaping
set of its each function. In such case, the fact would be obvious
from classical transcendental dynamics that the fast escaping
set is also non-empty.
4 Elementary Properties of Fast Esca-
ping Set
In this section, we check how far basic properties of fast
escaping set of classical transcendental dynamics can be ge-
neralized to fast escaping set of transcendental semigroup dy-
namics. In [18], we examined the contrast between classical
complex dynamics and semigroup dynamics in the invariant
features of Fatou sets, Julia sets and escaping sets. In this pa-
per, we see the same type of contrast in fast escaping set. Note
that in classical transcendental dynamics, the fast escaping set
is completely invariant.
Definition 4.1 (Forward, backward and completely invari-
ant 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.
We prove the following elementary results that are important
regarding the structure of fast escaping set A(S). These results
may also have more chances of leading further results concer-
ning the properties and structure of A(S). Indeed, it shows
certain connection and contrast between classical transcenden-
tal dynamics and transcendental semigroup dynamics and it is
also a nice generalization of classical transcendental dynamics
to semigroup dynamics.
Theorem 4.1. Let Sis a transcendental semigroup such that
A(S)6=∅. Then the following are hold.
1. A(S)is S-forward invariant.
2. A(S)is independent of R.
3. J(S) = ∂A(S).
4. J(S)⊂A(S).
5. A(S)∩J(S)6=∅.
Proof. (1). From the definition 3.2 (that is, from equation 3.3),
we can write
AL
R(S)⊂ {z∈C:|z| ≥ ML(R), L ≥0}.
So for any z0∈AL
R(S),
f(z0)∈ {z∈C:|f(z)| ≥ ML+1(R), L ≥0}=AL+1
R(S)
for all f∈Sand n∈N. This shows that
f(AL
R(S)) ⊂AL+1
R(S)
for all f∈S. However from relation 3.4,
AL+1
R(S)⊂AL
R(S).
Hence, we have
f(AL
R(S)) ⊂AL
R(S).
This fact together with equation 3.5, we can say that A(S)is
S-forward invariant.
(2) Choose R0> R, then from 3.4, we have
AL
R0(S)⊂AL
R(S)
for all L∈Zand so
[
L∈N
A−L
R0(S)⊂[
L∈N
A−L
R(S)
Since there is m∈Nsuch that Mm(R)> R0and so
[
L∈N
A−L
R(S)⊂[
L∈N
Am−L
R(S)
=[
L∈N
A−L
Mm(R)(S)⊂[
L∈N
A−L
R0(S)(4.1)
From above two inequality, we have
[
L∈N
A−L
R0(S) = [
L∈N
A−L
R(S) = A(S)
This proves A(S)is independent of R.
(3) We prove this statement by showing (A(S))0⊂F(S)
and (A(S))e⊂F(S)where (A(S))0and (A(S))eare re-
spectively interior and exterior of A(S). Since A(S)is S-
forward invariant, so fn(A(S)) ⊂A(S)for all f∈Sand
n∈N. Suppose z∈(A(S))0, then there is a neighborhood V
of zsuch that z∈V⊂A(S). Since A(S)contains no periodic
points, so |fn+L(z)| ≥ Mn(R)for all f∈S, and n∈N
and hence (fn)n∈Nis normal on Vby Montel’s theorem. Thus
z∈F(S). This proves (A(S))0⊂F(S).
By the theorem 3.2.3 of [1], the closure and complement of
A(S)are also forward invariant. So from fn(A(S))
⊂A(S), we can write
fn(C−A(S)) ⊂C−A(S).
for all n∈N. Since C−A(S) = (A(S))e. By the assumption
of non-empty A(S),A(S)is also a non-empty closed set. By
the definition, F(S)is a largest open set on which Sis normal
family, so we must
C−A(S) = (A(S))e⊂F(S)
.
(4) The proof follows from (3).
(5) By the theorem 3.1, A(S)⊂A(f)for all f∈S. A
Fatou component U⊂F(S)is also a component of F(f)for
each f∈S.
Case (i): If Uis multiply connected component of F(S), then

Universal Journal of Applied Mathematics 7(1): 1-7, 2019 5
by [14, Theorem 2 (a)] U⊂A(f)for all f∈S. Again by the
above same theorem 3.1, U⊂A(S). This shows that ∂U ⊂
A(S). Since ∂ U ⊂J(f)for all f∈S. By [12, Theorem 4.2],
we write ∂U ⊂J(S). This proves A(S)∩J(S)6=∅.
case (ii): If Uis simply connected component of F(S)that
meets A(S), then by [15, Theorem 1.2 (b)] U⊂A(f)for all
f∈S. So, as in case (i), U⊂A(S). By [15, Corollary 4.6],
if F(S)has only simply connected components, then ∂AL
R⊂
J(S)where ∂AL
R(S)is L-th label of F(S). From the equation
3.5, we conclude that A(S)∩J(S)6=∅.
There are many classes of functions from which we get
I(f)⊂J(f)and for such functions, we must have A(f)⊂
J(f). Dinesh Kumar and Sanjay Kumar [8, Theorem 4.5]
prove that I(S)⊂J(S)if transcendental semigroup Sis of
finite or bounded type. We prove the following similar result.
Theorem 4.2. Let Sbe a bounded or finite type transcendental
semigroup. Then A(S)⊂J(S)and J(S) = A(S).
Proof. For each f∈S, Eremenko and Lyubich [5] proved
that I(f)⊂J(f). K. K. Poon [12, Theorem 4.2] proved that
J(S) = Sf∈SJ(f). So for any f∈S, we have A(S)⊂
A(f)⊂J(f)⊂J(S). The second part follows from A(S)⊂
J(S)together with above theorem 4.1 (4).
There are many functions in the class B, the escaping set
I(f)consists of uncountable family of curves tending to infi-
nity. For example, function λsin z+γwith λ, γ ∈Cbelongs
to the class S⊂Band its escaping set is an uncountable
union of curves tending to infinity, the so-called Cantor bou-
quet. For the function f(z) = λez,0<λ<1/e, the Fatou set
is completely invariant attracting basin and Julia set is a Cantor
bouquet consisting of uncountably many disjoint simple cur-
ves, each of which has finite end point and other endpoint is
∞. The escaping set of such a function consists of open cur-
ves (without endpoints) together with some of their end points.
Note that for such a function, each point in the escaping set
can be connected to ∞by a curve in the escaping set. For such
functions, every point in such a curve belongs to fast escaping
set except possibly a finite endpoint. More generally, let fbe
a finite composition of functions of finite order in the class B
and let z0∈I(f). Then z0can be connected to ∞by a simple
curve Γ⊂I(f)such that Γ\ {z0} ⊂ A(f)(see for instance
[13, Theorem 1.2]).
There is a chance of similar result in semigroup dynamics
if semigroup Sis generated by the transcendental functions of
finite order in the class B. If so, then every f∈Sis a finite
composition of the functions of finite order in the class Band
for each of such function f,A(f)consists of curves Γ\ {z0}
with exception of some of the end points. Since A(S)⊂A(f)
for each f∈S, then A(S)may consist of curves Γ\ {z0}with
exception of some of the end points.
5 On the L-th Labels of A(S)
In this section, we concentrate on L-th label AL
R(S)of fast
escaping set A(S). Since fast escaping set can be written as
expanding union of L-th labels, so we hope that structure and
properties of each L-th label may determine structure and pro-
perties of fast escaping set. Again, we will also see contrast be-
tween fast escaping set and its label if there are. The following
result is a contrast. That is, analogous to classical transcenden-
tal dynamics [15], unlike the set A(S), each of its label is a
closed set.
Theorem 5.1. Let L∈Z, then for a transcendental semigroup
Ssuch that A(S)6=∅. Then the set AL
R(S)is closed and
unbounded for each L∈Zif it is non-empty.
Proof. From the definition 3.2, we can write
AL
R(S)⊂AL
R(f)
for all f∈S. This implies that
AL
R(S)⊂\
f∈S
AL
R(f)
Since for each L∈Z,AL
R(f)is a closed and unbounded set
and also by [15, Theorem 1.1] each component of AL
R(f)is
closed and unbounded for all f∈S. So, Tf∈SAL
R(f)is also
a closed and unbounded set and each of its component is clo-
sed and unbounded as well. Since AL
R(S)is a component of
Tf∈SAL
R(f), so it must be closed and unbounded.
On the light of this theorem 5.1 and equation 3.5, we have
obtained a new structure of fast escaping set A(S), a countable
union of closed and unbounded sets AL
R(S). This result also
provides a solution of Eremenko’s conjecture 1.1 in transcen-
dental semigroup dynamics. This generalizes the result of clas-
sical transcendental dynamics to transcendental semigroup dy-
namics.
Labels of fast escaping set A(S)can be used to show if Uis a
Fatou component in A(S), then boundary of Uis also in A(S).
There are variety of results on simply connected and multiply
connected Fatou components. Each of the Fatou component of
transcendental semigroup is either a stable (periodic) or unsta-
ble (wandering (non- periodic)) domain as defined below.
Definition 5.1 (Stabilizer, wandering component and stable
domains).For a transcendental semigroup S, let Ube a com-
ponent of the Fatou set F(S)and Ufbe a component of Fatou
set containing f(U)for some f∈S. The set of the form
SU={f∈S:Uf=U}
is called stabilizer of Uon S. If SUis non-empty, we say that
a component Usatisfying Uf=Uis called stable basin for
S. The component Uof F(S)is called wandering if the set
{Uf:f∈S}contains infinitely many elements. That is, Uis
a wandering domain if there is sequence of elements {fi}of S
such that Ufi6=Ufjfor i6=j. Furthermore, the component U
of F(S)is called strictly wandering if Uf=Ugimplies f=g.
A stable basin Uof a transcendental semigroup Sis
1. attracting if it is a subdomain of attracting basin of each
f∈SU

6 Fast Escaping Set of Transcendental Semigroup
2. supper attracting if it is a subdomain of supper attracting
basin of each f∈SU
3. parabolic if it is a subdomain of parabolic basin of each
f∈SU
4. Siegel if it is a subdomain of Siegel disk of each f∈SU
5. Baker if it is a subdomain of Baker domain of each f∈
SU
Note that stabilizer SUis a a subsemigroup of S([22,
Lemma 2.2]). Also, in classical case, a stable basin is one of
above type. For any Fatou component U, we prove the follo-
wing result which is analogous to [15, Theorem 1.2] of classi-
cal transcendental dynamics.
Theorem 5.2. Let Ube a Fatou component of transcendental
semigroup Sthat meets AL
R(S), where R > 0be such that
M(r, f)> r for r≥Rfor all f∈Sand L∈N. Then
1. U⊂AL−1
R(S);
2. if Uis simply connected, then U⊂AL
R(S)
Proof. Since U∩AL
R(S)6=∅. The fact AL
R(S)⊂AL
R(f)
for all f∈Simplies that U∩AL
R(f)6=∅for all f∈S.
So, from the theorem [15, Theorem 1.2 (a)], we always have
U⊂AL−1
R(f)for all f∈S. So U⊂AL−1
R(S). The second
part also follows similarly by using [15, Theorem 1.2 (b)].
By part (2) of above theorem 3.1, we can conclude that
U⊂AL
R(S)for all simply connected component Uof F(S).
So, if all components of F(S)are simply connected, then we
must ∂AL
R(S)⊂J(S)and hence interior of AL
R(S)is con-
tained in F(S). This theorem also generalizes the result of
classical transcendental dynamics to transcendental semigroup
dynamics. That is, whatever Fatou component (simply or mul-
tiply connected) Uof F(S)intersecting A(S), there is always
U⊂A(S). Again, another question may raise. Such a Fatou
component Uis periodic or wandering? Note that in classical
transcendental dynamics, such a Fatou component is always
wandering ([15, Corollary 4.2]). For transcendental semigroup
dynamics, such a Fatou component is again wandering domain.
For, if U∩A(S)6=∅, then U∩A(f)6=∅for all f∈S. In this
case Uis wandering domain of each f∈S, so it is wandering
domain of Sas well.
In above paragraph, we discussed about a Fatou component
intersecting the fast escaping set A(S). Are there any Fatou
components that lie in A(S)? In classical transcendental dyna-
mics, its answer is yes (see for instance [15, Theorem 4.4] and
[14, Theorem 2]). Indeed, in such case, the Fatou component
that lie in A(S)is a (closure of) multiply connected wandering
domain. Bergweiler constructed an example of transcendental
entire function ffor which A(f)contains simply connected
wandering domain ([2, Theorem 2]). This wandering domain
is simply connected bounded one which lie in between multi-
ply connected wandering domains and this one is only known
example of non-multiply connected Fatou component that lie
in A(f). The generalization of above discussion to semigroup
dynamics is also possible. For example, if Uis a multiply
connected wandering domain of F(S), then it is also multi-
ply connected wandering domains of every f∈S. In this
case, U⊂A(f)for all f∈S([15, Theorem 4.4]). Hence
U⊂A(S).
6 Conclusion and Final Remarks
We defined fast escaping set in transcendental semigroup dy-
namics and we described some elementary properties. We sho-
wed that most of the properties of fast escaping set of classical
transcendental dynamics are generalized to semigroup dyna-
mics and few new phenomena were also occurred. In addition,
there are some more things of classical transcendental dyna-
mics regarding the structure (such as spider’s web and Cantor
bouquet) of fast escaping set (see for instance in [14, 15, 16])
which are not still formulated in semigroup dynamics. We will
continue our study in this direction in coming days. Also, we
will investigate some concrete examples of transcendental se-
migroups that have non empty fast escaping sets with additio-
nal visible structure and properties.
Acknowledgements
We are very grateful to experts, reviewers and referees for
their appropriate and constructive suggestions to improve this
paper.
We are very grateful to Prof. G. B. Thapa, Tribhuvan Uni-
versity, Kathmandu, Nepal for his thorough reading over the
manuscript with fruitful discussion as well as valuable sugges-
tions and comments.
We are also thankful to University Grants Commission,
Nepal for PhD faculty fellowship and Research Support
Programs of the HERP (Programm No-9, Research Article
Publication) (2014-2020).
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