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arXiv:1809.06743v1 [math.DS] 17 Sep 2018
MANUSCRIPT
Fast Escaping Set of Transcendental Semigroup
Bishnu Hari Subedi and Ajaya Singh
Abstract. In this paper, we study fast escaping set of transcendental semi-
group. We discuss the structure and properties of fast escaping set of transcen-
dental semigroup. We also see how far the classical theory of fast escaping set of
transcendental entire function applies to general settings of transcendental semi-
groups and what new phenomena can occur.
1. Introduction
The principal aim of this paper is to study fast escaping set not for iterates of
single transcendental entire maps of complex plane Cbut for the composite of the
family Fof such maps. Let Fbe a space of transcendental entire maps on C. For
any map f∈F,Cis naturally partitioned into two subsets: the set of normality
and its complement. The set of normality or Fatou set F(f) of f∈Fis the largest
open set on which the iterates fn=f◦f◦... ◦f(n-fold composition of fwith
itself, n∈N) is a normal family. The complement of Fatou set in Cis the Julia set
J(f). A maximally connected subset of the Fatou set 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 point z∈I(S) is called escaping point. For transcendental entire function
f, the escaping set I(f) was first studied by A. Eremenko [4]. He showed that
I(f)6=∅;J(f) = ∂I (f); I(f)∩J(f)6=∅; and I(f) has no bounded component.
By motivating from this last statement, he posed a conjecture:
Conjecture 1.1.Every component of I(f) unbounded.
This conjecture is considered as an important open problem of transcendental
dynamics and nowadays it is famous as Eremenko’s conjecture. The Eremenko’s
conjecture 1.1 in general case has been proved by using the fast escaping set A(f),
which consists of points whose iterates tends to infinity as fast as possible. This
set is a subset of escaping set and it was introduced first time by Bergweiler and
2010 Mathematics Subject Classification. 37F10, 30D05.
Key words and phrases. Escaping set, Fast escaping set, levels etc.
This research work of first author is supported by PhD faculty fellowship of University Grants
Commission, Nepal.
1
2 B. H. SUBEDI AND A. SINGH
Hinkkanen [3] and defined in the following form by Rippon and Stallard [14]. For
a transcendental entire function f, the fast escaping set is a set of the form:
A(f) = {z∈C:∃L∈Nsuch that |fn+L(z)|>Mn(R) for n∈N}
where M(r) = max|z|=r|f(z)|, r > 0 and Mn(r) denotes nth iteration of M(r) with
respect to r.R > 0 can be taken any value such that M(r)> r for r>R.
Recall that the set C(f) = {z∈C:f′(z) = 0}is the set of critical points
of the transcendental entire function fand 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 (asymptotic path) Γ : [0,∞)→
Cso that Γ(t)→ ∞ and f(Γ(t)) →was t→ ∞ is called the set of asymptotic
values of fand the set SV (f) = (CV (f)∪AV (f)) is called the 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.
The main concern of such a transcendental iteration theory is to describe the
nature of the components of Fatou set and the structure and properties of the
Julia set, escaping set and fast escaping set. We use monograph: dynamics of
transcendental entire functions [7] and book: holomorphic dynamics [10] for basic
facts concerning the Fatou set, Julia set and escaping set of a transcendental entire
function. We use [13, 14, 15] for facts and results concerning the fast escaping set
of a transcendental entire function.
Our particular interest is to study of the dynamics of the families that are
semigroups generated by the class Fof transcendental entire maps. For a collection
F={fα}α∈∆of such maps, let S=hfαibe a transcendental semigroup generated
by them. The index set ∆ to which αbelongs is allowed to be infinite in general
unless otherwise stated. Here, each f∈Sis a transcendental entire 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. A semigroup generated by finitely many transcendental entire
functions fi,(i= 1,2,...,n) is called finitely generated transcendental semigroup.
We write S=hf1, f2,...,fni. If Sis generated by only one transcendental entire
function f, then Sis cyclic transcendental 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 transcendental semigroup Sis abelian if fi◦fj=fj◦fifor all
generators fiand fjof S. The transcendental semigroup Sis bounded type (or
finite type) if each of its generators fiis bounded type (or finite type).
The family Fof complex analytic maps forms a normal family in a domain
Dif given any composition sequence (fα) generated by the member of F, there
is a subsequence (fαk) which is uniformly convergent or divergent on all compact
subsets of D. If there is a neighborhood Uof the point z∈Csuch that Fis normal
FAST ESCAPING SET OF TRANSCENDENTAL SEMIGROUP 3
family in U, then we say Fis normal at z. If Fis a family of members from the
semigroup S, then we simply say that Sis normal in the neighborhood of zor Sis
normal at z.
Let fbe a transcendental entire map. We say that fiteratively divergent at
z∈Cif fn(z)→ ∞ as n→ ∞. Semigroup Sis iteratively divergent at zif
fn(z)→ ∞ as n→ ∞ for all f∈S. Otherwise, a function fand semigroup Sare
said to be iteratively bounded at 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 any z∈Calways means that
there is a sequence (gn)n∈Nin Srepresenting g1=f, g2=f2,...,gn=fn,... such
that gn(z)→ ∞ as n→ ∞ or there a sequence in Swhich contains (gn)n∈Nas
a subsequence such that gn(z)→ ∞ as n→ ∞. More generally, semigroup Sis
iteratively divergent at any z∈Calways means that every f∈Sis iteratively
divergent at z. That is, every sequence in Shas a subsequence which diverges to
infinity at z.
Based the Fatou-Julia-Eremenko theory, the Fatou set, Julia set and escaping
set in the settings of transcendental semigroup are defined as follows.
Definition 1.1 (Fatou set, Julia set and escaping set).Fatou set of the
holomorphic 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.
It is obvious that F(S) is the largest open subset of Con which the family Fin
S(or semigroup Sitself) is normal. Hence its compliment J(S) is a smallest closed
set for any transcendental semigroup S. Whereas the escaping set I(S) is neither an
open nor a closed set (if it is non-empty) for any transcendental semigroup S. Any
maximally connected subset Uof the Fatou set 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 our study, classical transcendental dynamics refers the iteration theory of
single transcendental map and transcendental semigroup dynamics refers the dy-
namical theory generated by a set of transcendental entire maps. In transcendental
semigroup dynamics, algebraic structure of semigroup naturally attached to the
dynamics and hence the situation is largely complicated. The principal aim of this
paper is to see how far classical transcendental dynamics applies to transcendental
semigroup dynamics and what new phenomena appear in transcendental semigroup
settings.
4 B. H. SUBEDI AND A. SINGH
The fundamental contrast between classical transcendental dynamics and semi-
group dynamics appears by different algebraic structure of corresponding semi-
groups. In fact, non-trivial semigroup need not be, and most often will not be
abelian. However, trivial semigroup is cyclic and therefore abelian. As we dis-
cussed before, classical complex dynamics is a dynamical study of trivial (cyclic)
semigroup whereas semigroup dynamics is a dynamical study of non-trivial tran-
scendental semigroup.
The main motivation of this paper comes from seminal work of Hinkkanen and
Martin [6] on the dynamics of rational semigroup and the extension study of K. K.
Poon [11] to the dynamics of transcendental semigroup. Both of them naturally
generalized classical complex dynamics to the dynamics of the sequence of different
functions by means of composition. Another motivation of studying escaping set of
transcendental semigroup comes from the work of Dinesh Kumar and Sanjay Kumar
[8, 9] where they defined escaping set and discussed how far escaping set of classical
transcendental dynamics can be generalized to semigroup dynamics. In parallel, we
also studied structure and propperties of Fatou set, Julia set and escaping under
semigroup dynamics in [16, 17, 18, 19, 20, 21, 22, 23]. From these attempts, we
again more motivate to study fast escaping set of transcendental semigroup.
In this paper, we introduce fast escaping set in transcendental semigroup set-
tings which is a main concern of our study. Note that the fast escaping set A(f) in
classical transcendental dynamics introduced first time by Bergweiler and Hinkka-
nen [3] and studied in more depth by Rippon and Stallard [14]. Indeed, it is a
set consisting of points whose iterates tends to infinity as fast as possible and now
plays a key role in classical transcendental dynamics. The set A(f) has some prop-
erties 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( [14, Theorem 5.1 (b)]),
A(f)∩J(f)6=∅([3, Lemma 3]) and([14, Theorem 5.1(a)]), The set A(f) is com-
pletely invariant under f([14, Theorem 2.2 (a)]) and the set A(f) does not depend
on the choice of R > 0 ([14, Theorem 2.2 (b)]). However, in [13, Theorem 1] and
[14, Theorem 1.1], it is shown that all components of A(f) are unbounded and since
A(f)⊂I(f) provides a partial answer to Eremenko’s conjecture in normal form.
If Uis a Fatou component of fin A(f), then its boundary is also in A(f), that
is, U⊂A(f) ([13, Theorem 2]) and ([14, Theorem 1.2]). These are much stronger
properties of A(f) than those of escaping set I(f).
2. Fast Escaping Set of Transcendental Semigroup
There is no equivalent formulation of fast escaping set in semigroup settings.
We have started to define fast escaping set and try to formulate some other related
terms and results. Note that it is our seminal work on the study of fast escaping
set in transcendental semigroup settings.
Let Sbe a transcendental semigroup. Let us define a set of the form
(2.1) AR(S) = {z∈C:|fn(z)|>Mn(R) for all f∈Sand n∈N}
FAST ESCAPING SET OF TRANSCENDENTAL SEMIGROUP 5
where M(r) = max|z|=r|f(z)|, with r>R > 0 and Mn(r) denotes the nth iterates
of M(r) with itself. R > 0 can 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∈Nwhich increases without bound at least as fast as the n-iterates of the
maximum modulus of each gnk:M(r) = max|z|=r|gnk(z)|. Where gnk=fnsuch
that |fn(z)|>Mn(R).
Definition 2.1 (Fast escaping set).The fast escaping set A(S)of a transcen-
dental semigroup Sconsists the set AR(S)and all its pre-images. In other words,
fast escaping set is the set of the form
(2.2) A(S) = [
n>0
f−n(AR(S))
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 2.2 (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
(2.3) AL
R(S) = {z∈C:|fn(z)|>Mn+L(R)for all f∈S,
n∈Nand n+L>0}
is called Lth level of fast escaping set A(S).
Note that the set AR(S) defined above in 2.1 is the 0th level of fast escaping set
A(S). Since Mn+1(R)> M n(R) for all n>0, so from 2.3, we get following chain
of relation
(2.4) ...⊂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)⊂...
From 2.2 and 2.4, the fast escaping set can also be written as an expanding union
of its labels.
(2.5) A(S) = [
L∈N
A−L
R(S)
Again from the definition 2.1, that is, from 2.2, 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 2.1, there
is L∈Nsuch that |fL(fn(z0))|=|fn+L(z0)|>Mn(R). With this clause, the fast
escaping set of transcendental semigroup Scan now be written as
6 B. H. SUBEDI AND A. SINGH
(2.6) A(S) = {z∈C: there exists L∈Nsuch that
|fn+L(z)|>Mn(R) for all f∈S, and n∈N}.
We can use any one of the form 2.2 or 2.5 or 2.6 as a definition of fast escaping
set in our forth coming study. Note that by the definition (2.2 or 2.5 or 2.6), fast
escaping set A(S) of any transcendental semigroup Sis a subset of escaping set
I(S). Since from [16, Theorem 1.2 (3)], we can say that I(S) may be empty. For
any transcendental 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 2.1.Let Sbe a transcendental semigroup generated by 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 open nor closed set
if it is non-empty. Similar to the result [16, Theorem 1.2 (3)] of escaping set, the
following result is also clear from the definition of fast escaping set.
Theorem 2.1.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 [16, 17, 18, 20, 21, 22] where escaping set is non-empty. One of important
particular result in this regards is a condition for which escaping set of a transcen-
dental semigroup is same as escaping set of its each element. In such case, the
fact would be obvious from classical transcendental dynamics that the fast escap-
ing set is also non-empty. In our fourth coming study, we always talk with such a
semigroup whose fast escaping set is non-empty.
3. Elementary Properties of Fast Escaping Set
In this section, we check how far basic properties of fast escaping set of classical
transcendental dynamics can be generalize to fast escaping set of transcendental
semigroup dynamics. In [17], we examined the contrast between classical complex
dynamics and semigroup dynamics in the invariant features of Fatou set, Julia set
and escaping set. In this paper, we see the same type contrast in fast escaping set.
Note that in classical transcendental dynamics, the fast escaping set is completely
invariant.
Definition 3.1 (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.
FAST ESCAPING SET OF TRANSCENDENTAL SEMIGROUP 7
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 concerning the properties and structure of A(S). Indeed,
it shows certain connection and contrast between classical transcendental dynam-
ics and transcendental semigroup dynamics and it is also a nice generalization of
classical transcendental dynamics to semigroup dynamics.
Theorem 3.1.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 2.2 (that is, from equation 2.3), 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 2.4, AL+1
R(S)⊂AL
R(S).Hence, we have f(AL
R(S)) ⊂AL
R(S).
This fact together with equation 2.5, we can say that A(S) is S-forward invariant.
(2) Choose R0> R, then from 2.4, 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)
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 respectively 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 Vof 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).
8 B. H. SUBEDI AND A. SINGH
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 2.1, 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 by [13, Theorem 2
(a)] U⊂A(f) for all f∈S. Again by the above same theorem 2.1, U⊂A(S).
This shows that ∂U ⊂A(S). Since ∂U ⊂J(f) for all f∈S. By [11, 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 [14,
Theorem 1.2 (b)] U⊂A(f) for all f∈S. So, as in case (i), U⊂A(S). By [14,
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 2.5, 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 3.2.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 [11, 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 3.1 (4).
There are many functions in the class B, the escaping set I(f) consists of
uncountable family of curves tending to infinity. For example, function λsin z+γ
with λ, γ ∈Cbelongs to the class S⊂Band its escaping set is an uncountable
union of curves tending to infinity, the so-called Cantor bouquet. 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
curves, each of which has finite end point and other endpoint is ∞. The escaping
set of such a function consists of open curves (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
FAST ESCAPING SET OF TRANSCENDENTAL SEMIGROUP 9
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
Band let z0∈I(f). Then z0can be connected to ∞by a simple curve Γ ⊂I(f)
such that Γ r{z0} ⊂ A(f) (see for instance [12, Theorem 1.2]).
There may 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 B
and for each of such function f,A(f) consists of curves Γ r{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 r{z0}with exception of some of the end points.
4. On the L-th Labels of A(S)
In this section, we more 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 properties of fast escaping set. Again, we will also see contrast between fast
escaping set and and its label if there are. The following result is a contrast. That
is, analogous to classical transcendental dynamics [14], unlike the set A(S), each of
its label is a closed set.
Theorem 4.1.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 2.2, we can write AL
R(S)⊂AL
R(f) for all f∈S.
This implies that AL
R(S)⊂Tf∈SAL
R(f). Since for each L∈Z,AL
R(f) is a closed and
unbounded set and also by [14, 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 closed 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 4.1 and equation 2.5, 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 tran-
scendental semigroup settings. This generalizes the result of classical transcendental
dynamics to transcendental semigroup dynamics.
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 unstable
(wandering (non- periodic)) domain as defined below.
Definition 4.1 (Stabilizer, wandering component and stable domains).
For a transcendental semigroup S, let Ube a component of the Fatou set F(S)and
Ufbe a component of Fatou set containing f(U)for some f∈S. The set of the
10 B. H. SUBEDI AND A. SINGH
form
SU={f∈S:Uf=U}
is called stabilizer of Uon S. If SUis non-empty, we say that a component U
satisfying 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 Ssuch that Ufi6=Ufj
for i6=j. Furthermore, the component Uof 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
(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([21, Lemma 2.2]). Also, in
classical case, a stable basin is one of above type. For any Fatou component U,
we prove the following result which is analogous to [14, Theorem 1.2] of classical
transcendental dynamics.
Theorem 4.2.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 [14, 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 [14, Theorem 1.2 (b)].
By part (2) of above theorem 2.1, 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 con-
nected, then we must ∂AL
R(S)⊂J(S) and hence interior of AL
R(S) is contained in
F(S). This theorem also generalizes the result of classical transcendental dynamics
to transcendental semigroup dynamics. That is, whatever Fatou component (sim-
ply or multiply 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 wan-
dering? Note that in classical transcendental dynamics, such a Fatou component
is always wandering ([14, 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.
Whatever discussion we have done above was about a Fatou component inter-
secting the fast escaping set A(S) of a transcendental semigroup S. Are there any
FAST ESCAPING SET OF TRANSCENDENTAL SEMIGROUP 11
Fatou components that are obviously known to lie in A(S)? In classical transcen-
dental dynamics, its answer is yes (see for instance [14, Theorem 4.4] and [13,
Theorem 2]). Indeed, in such case, the Fatou component that obviously 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 multiply 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 of
course possible. For example, if Uis a multiply connected wandering domain of
F(S), then it also multiply connected wandering domains of every f∈S. In this
case, U⊂A(f) for all f∈S([14, Theorem 4.4]). Hence U⊂A(S).
References
[1] Beardon, A. F.: Iteration of rational functions, Complex analytic dynamical systems, Spinger-
Verlag, New York, Inc, 1991.
[2] Bergweiler, W.: An entire functions with simply and multiply connected wandering domains,
Pure Appl. Math. Quarterly, 7 (2011), 107-120.
[3] Bergweiler, W. and Hinkkanen, A.: On semi-conjugations of entire functions, Math. Proc.
Camb. Phil. Soc. 126 (1999), no. 3, 565- 574.
[4] Eremenko, A.: On the iterations of entire functions, Dynamical System and Ergodic Theory,
Banach Center Publication Volume 23, Warsaw, Poland, (1989).
[5] Eremenko, A., and Lyubich, M.Y.: Dynamical Properties of Some Classes of Entire Func-
tions,Ann. Inst. Fourier, Grenoble, 42 (1992), 989-1020.
[6] Hinkkanen, A. and Martin, G.J.: The dynamics of semigroups of rational functions- I, Proc.
London Math. Soc. (3) 73, 358-384, (1996).
[7] Hua, X.H. and Yang, C.C.: Dynamic of transcendental functions, Gordon and Breach Science
Publication, (1998).
[8] Kumar, D. and Kumar, S.: The dynamics of semigroups of transcendental entire functions-II,
arXiv: 1401.0425 v3 (math.DS), May 22, 2014.
[9] Kumar, D. and Kumar, S.: Escaping set and Julia set of transcendental semigroups,
arXiv:141.2747 v3 (math. DS) October 10, 2014.
[10] Morosawa, S., Nishimur, Y., Taniguchi, M. and Ueda, T.: Holomorphic Dynamics, Cambridge
University Press,Cambridge, UK, 2000.
[11] Poon, K.K.: Fatou-Julia theory on transcendental semigroups, Bull. Austral. Math. Soc. Vol-
58(1998) PP 403-410.
[12] Rempe, L., Rippon, P.J., and Stallard, G.M.:Are Devaney hairs fast escaping?, Diff. eq. and
Appl. , 16 (2010), 739-762.
[13] Rippon, P.J. and Stallard, G. M.: On the questions of Fatou and Eremenko,Proc.Amer. Math.
Soc. 133, (2005), 1119-1126.
[14] Rippon, P.J., and Stallard, G.M.: Fast escaping points of entire functions, Proc. London
Math. Soc. (3) 105 (2012), no 4, 787-720.
[15] Sixsmith, D.J.:Topics in transcendental dynamics, PhD Thesis, The Open University, UK,
(2013).
[16] Subedi, B. H. and Singh, A.: A short comparison of classical complex dynamics and holo-
morphic semigroup dynamics, arXiv: 1803.08368v1[math. DS] 22 March 2018.
[17] Subedi, B. H. and Singh, A.: Invariant properties of Fatou set, Julia set and escaping set of
holomorphic semigroup, arXiv: 1803.09662v1[math. DS] 23 March 2018.
12 B. H. SUBEDI AND A. SINGH
[18] Subedi, B. H. and Singh, A.: Escaping set of hyperbolic semigroup, arXiv:1803.10381v1 [math
DS], 28 March, 2018.
[19] Subedi, B. H. and Singh, A.: On completely invariant Julia set of transcendental semigroup,
arXiv: 1804. 03554v1 [math. DS], 10 April 2018.
[20] Subedi, B. H. and Singh, A.: Completely invariant escaping set of transcendental semigroup,
arXiv: 1804. 11252v1 [math.DS], 30 April 2018.
[21] Subedi, B. H. and Singh, A.: Fatou set, Julia set and escaping set in holomorphic subsemi-
group dynamics, arXiv:1807.04499v1 [math. DS], 12 July, 2018.
[22] Subedi, B. H. and Singh, A.: A study of Fatou Set, Julia set and escaping set in nearly abelian
transcendental semigroup, arXiv: 1808. 00857v1 [math. DS], 1 August 2018.
[23] Subedi, B. H. and Singh, A.: A study of Fatou Set, Julia set and escaping set in conjugate
transcendental semigroup, arXiv: 1808.03275v1 [math DS], 9 August 2018.
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
