Transcendental Semigroup that has Simply Connected Fatou Components

Abstract

We prove that there exists a non-trivial transcendental semigroup S such that the wandering (or pre-periodic or periodic) components of Fatou set F(S) has at least a simply connected domain D.

Full text

arXiv:1804.01213v1 [math.DS] 4 Apr 2018
Transcendental Semigroup that has Simply
Connected Fatou Components
Bishnu Hari Subedi and Ajaya Singh
Abstract. We prove that there exists a non-trivial transcendental semigroup S
such that the wandering (or pre-periodic or periodic) components of Fatou set
F(S) has at least a simply connected domain D.
1. Introduction
We denote the complex plane by C,extended complex plane by C∞and set
of integers greater than zero by N. We assume the function f:C→Cis tran-
scendental entire function (TEF) unless otherwise stated. For any n∈N, f n
always denotes the nth iterates of f. If fn(z) = zfor some smallest n∈N, then
we say that zis periodic point of period n. In particular, if f(z) = z, then z
is a fixed point of f. If |(fn(z))′|<1, where ′represents complex differentia-
tion of fnwith respect to z, then zis called attracting periodic point. A family
F={f:fis meromorphic on some domain X of C∞}forms normal family if ev-
ery sequence (f)i∈Nof functions contains a subsequence which converges uniformly
to a finite limit or converges to ∞on every compact subset Dof X. The Fatou
set of fdenoted by F(f) is the set of points z∈Csuch that sequence (fn)n∈N
forms a normal family in some neighborhood of z. That is, z∈F(f) if zhas a
neighborhood Uon which the family Fis normal. By definition, Fatou set is open
and may or may not be empty. Fatou set is non-empty for every entire function
with attracting periodic points. If U⊂F(f) (Fatou component), then f(U) lies in
some component Vof F(f) and V−f(U) is a set which contains at most one point
(see for instance [3]). Let U⊂F(f) (a Fatou component) such that fn(U) for
some n∈N, is contained in some component of F(f), which is usually denoted by
Un. A Fatou component Uis called pre-periodic if there exist integers n > m >0
such that Un=Um. In particular, if Un=U0=U( that is, fn(U)⊂U) for
some smallest positive integer n>1, then Uis called periodic Fatou component
of period n and {U0, U1...,Un−1}is called the periodic cycle of U. In the case, if
2010 Mathematics Subject Classification. 37F10, 30D05.
Key words and phrases. Fatou set, pre-periodic component, periodic component, wandering
component, transcendental semigroup, Carleman set.
This research work of first author is supported by PhD faculty fellowship from University
Grants Commission, Nepal.
1

2 B. H. SUBEDI AND A. SINGH
U1=f(U)⊂U, then Uis called invariant domain. A component of Fatou set
F(f) which is not pre-periodic is called wandering domain.
For the complex plane C, let us denote the set of all holomorphic functions of C
by Hol(C). If f∈Hol(C), then fis a polynomial or transcendental entire function.
The set Hol(C) forms a semigroup with semigroup operation being the functional
composition.
Definition 1.1 (Transcendental semigroup).Let A={fi:i= 1,2,...} ⊂
Hol(C)be a set of transcendental entire functions fi:C→C. A transcendental
semigroup Sis a semigroup generated by the set Awith semigroup operation being
the functional composition. We denote this semigroup by S=hAi=hfi:i=
1,2,3,...ior simply by S=hf1, f2, f3,··· , fn,· · · i.
Here, each f∈Sis the transcendental entire function and Sis closed under
functional composition. Thus f∈Sis constructed through the composition of finite
number of functions fik,(k= 1,2,3,...,m). That is, f=fi1◦fi2◦fi3◦ · · · ◦ fim.
A semigroup generated by finitely many functions fi,(i= 1,2,3,...,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 or trivial
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=hfia trivial semigroup.
Based on the Fatou-Julia theory of a complex analytic function, The Fatou set
and Julia set in the settings of semigroup are defined as follows.
Definition 1.2 (Fatou set, Julia set).Fatou set of the transcendental semi-
group Sis defined by
F(S) = {z∈C:Sis normal in a neighborhood of z}
The connected component of Fatou set F(S)is called Fatou component. The Julia
set of Sis defined by J(S) = C−F(S).
From the definition 1.2, it is clear that F(S) is the open set and therefore, it
complement J(S) is closed set. Indeed, these definitions generalize the definitions
of Julia set and Fatou set of the iteration of single holomorphic map. If S=hfi,
then F(S) and J(S) are respectively the Fatou set and Julia set in classical iteration
theory of complex dynamics. In this situation we simply write: F(f) and J(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.
Note that for any semigroup S, we have
(1) F(S)⊂F(h) all h∈Sand hence F(S)⊂ ∩h∈SF(h).

TRANSCENDENTAL SEMIGROUP THAT HAS SIMPLY CONNECTED FATOU COMPONENTS3
(2) J(h)⊂J(S).
Since, in classical complex dynamics, Fatou set F(f) may be empty. So from the
above first relation, we can say that Fatou set F(S) of semigroup Smay also be
empty. In this paper we are interested to find a non-trivial semigroup Sthat
has non-empty Fatou set F(S). Basically, we prove that there is a non-trivial
transcendental semigroup that has simply connected Fatou component.
Proposition 1.1.There is a non trivial transcendental semigroup Ssuch that
the Fatou set F(S)has at least a simply connected component.
Note that if the semigroup Sis trivial, that is, semigroup S=hfigenerated by
a single transcendental entire function f, then Bergweiler[1] proved that the Fatou
set F(S) has both a simply and a multiply connected wandering domains. However,
in the case of non-trivial transcendental semigroup, the proof is not so easy. The
reason behind is that the dynamics of individual transcendental entire functions
differ largely from the dynamics of their composites.
2. Some Essential Lemmas
To workout a proof of the proposition 1.1, first of all we need a notion of approxi-
mation theory of entire functions. In our case, we can use the notion of Carleman set
from which we obtain approximation of any holomorphic map by entire functions.
Definition 2.1 (Carleman Set).Let Fbe a closed subset of Cand C(F) =
{f:F→C:fis continuous on Sand analytic in the interior of F◦of F}. Then
Fis called a Carleman set (for C) if for any g∈C(F)and any positive continuous
function ǫon F, there exists entire function hsuch that |g(z)−h(z)|< ǫ for all
z∈F.
The following important characterization of Carleman set has been proved by
A. Nersesjan in 1971 but we have been taken this result from [2].
Theorem 2.1 ([2, Theorem 4, page 157]).Let Fbe proper subset of C. Then
Fis a Carleman set for Cif and only if Fsatisfies:
(1) C∞−Fis connected;
(2) C∞−Fis locally connected at ∞;
(3) for every compact subset Kof C, there is a neighborhood Vof ∞in C∞
such that no component of F◦intersects both Kand V.
It is well known in classical complex analysis that the space C∞−Fis connected
if and only if each component Zof open set C−Fis unbounded. This fact together
with above theorem 2.1 will be a nice tool whether a set is a Carleman set for C.
The sets given in the following examples are Carlemen sets for C.
Example 2.1 ([2, Example-page 133]).The set E={z∈C:|z|= 1,Rez >
0} ∪ {z=x:x > 1} ∪ S∞
n=3{z=reiθ :r > 1, θ =π/n}is a Carleman set for C.

4 B. H. SUBEDI AND A. SINGH
Example 2.2 ([5, Set S, page-131]).The set E=G0∪S∞
k=1(Gk∪BK∪Lk∪
Mk), where G0={z∈C:|z−2|61};
Gk={z∈C:|z−(4k+ 2)|61} ∪ {z∈C:Rez= 4k+ 2,Imz>1}
∪{z∈C:Rez= 4k+ 2,Imz6−1},(k= 1,2,3,...);
Bk={z∈C:|z+ (4k+ 2)|61} ∪ {z∈C:Rez=−(4k+ 2),Imz>1} ∪
{z∈C:Rez=−(4k+ 2),Imz6−1},(k= 1,2,3,...);
Lk={z∈C:Rez= 4k},(k= 1,2,3,...);
and
Mk={z∈C:Rez=−4k},(k= 1,2,3,...)
is a Carleman set for Cby the theorem 2.1 .
From the help of the Carleman set of the example 2.2, A.P. Singh [5, Theorem
2] proved the following result.
Lemma 2.1.There are transcendental entire functions fand gsuch that there
exists a domain which lies in the wandering component of the F(f), F (g), F (f◦g)
and F(g◦f).
In fact, A. P. Singh [5] also had proved other results regarding the dynamics
of two individual functions and their composites (see for instance [5, Theorem 1,
Theorem 3 and Theorem 4]) which are also stricly based on the Carleman set of
example 2.2. Dinesh Kumar, Gopal Datt and Sanjay Kumar extended these result
of A.P. Singh in [4, Theorem 2.1 to Theorem 2.15]. For our purpose, we cite the
following two results from [4].
Lemma 2.2 (Theorem 2.2).There are TEFs fand gsuch that there exists
infinite number of domains which lie in the wandering component of the F(f), F (g),
F(f◦g)and F(g◦f).
Lemma 2.3 (Theorem 2.13).There are TEFs fand gsuch that there exist in-
finite number of domains which lie in the pre-periodic component of the F(f), F (g),
F(f◦g)and F(g◦f).
We extended above lemmas 2.1. 2.3 and 2.2 in [6, Theorem 1.1][7, Theorem
1.1] and [8, Theorem 1.1] to the following results:
Lemma 2.4.There are transcendental entire functions f,gand hsuch that
there exist infinite number of domains which lie in the wandering component of the
F(f), F (g), F (h), F (f◦g), F (g◦f), F (f◦h), F (g◦h), F (h◦f), F (h◦g), F (f◦g◦
h), F (f◦h◦g), F (g◦f◦h), F (g◦h◦f), F (h◦f◦g)and F(h◦g◦f).
Lemma 2.5.There are transcendental entire functions f,gand hsuch that
there exist infinite number of domains which lie in the pre-periodic component of
the F(f), F (g), F (h), F (f◦g), F (g◦f), F (f◦h), F (g◦h), F (h◦f), F (h◦g), F (f◦
g◦h), F (f◦h◦g), F (g◦f◦h), F (g◦h◦f), F (h◦f◦g)and F(h◦g◦f).

TRANSCENDENTAL SEMIGROUP THAT HAS SIMPLY CONNECTED FATOU COMPONENTS5
Lemma 2.6.There are transcendental entire functions f,gand hsuch that
there exist infinite number of domains which lie in the periodic component of the
F(f), F (g), F (h), F (f◦g), F (g◦f), F (f◦h), F (g◦h), F (h◦f), F (h◦g), F (f◦
g◦h), F (f◦h◦g), F (g◦f◦h), F (g◦h◦f), F (h◦f◦g)and F(h◦g◦f).
3. Proof of the Proposition 1.1
From all of above lemmas (Lemmas 2.1, 2.3, 2.2, 2.4, 2.5 and 2.6), we can say
that whatever domains that lie in the wandering, pre-periodic or periodic compo-
nents of F(f), F (g), F(h), F (f◦g), F (g◦f), F (f◦h), F (g◦h), F (h◦f), F (h◦
g), F (f◦g◦h), F (f◦h◦g), F (g◦f◦h), F (g◦h◦f), F (h◦f◦g) and F(h◦g◦f),
they also lie respectively in the wandering, pre-periodic or periodic components of
their successive composites. In this context, we can also prove the following two
results:
Lemma 3.1.If Dis a set which lies in the wandering (or pre-periodic or pe-
riodic) component of F(f), F (g), F (f◦g)and F(g◦f), then it also lies in wan-
dering (or pre-periodic or periodic) component of F(fnk◦gnk−1◦... ◦gn1)and
F(gnk◦fnk−1◦. . . ◦fn1), where nk,...n1∈N.
Proof. By the lemmas 2.1, 2.2 and 2.3, such a set Dexists. Since F(f) =
F(fn) and F(g) = F(gn) for any n∈N. So Dlies in the wandering (or pre-
periodic or periodic) component of F(fn) and F(gn) for all n∈N. As Dlies in the
wandering (or pre-periodic or periodic) component of F(f◦g), it also lies in the
wandering (or pre-periodic or periodic) component of F(fn◦gn) for all n∈N. By
the same argument we are using here, Dalso lies in the wandering (or pre-periodic
or periodic) component of F(f◦g)nfor all n∈N. Since F(f◦g)n=F(f◦g◦...◦f◦g)
(n -times f◦g), Dlies in the wandering (or pre-periodic or periodic) component of
F(fn◦gn◦...◦fn◦gn)(n-times fn◦gn) for all n∈N. Since n∈Nis arbitrary, so
we conclude that Dlies in the wandering (or pre-periodic or periodic) component
of F(fnk◦gnk−1◦...◦gn1) for all nk,...n1∈N. Similarly, we can show that Dlies
in the wandering (or pre-periodic or periodic) component of F(gnk◦fnk−1◦...◦fn1)
for all nk,...n1∈N.
Lemma 3.2.If Dis a set which lies in the wandering (or pre-periodic or peri-
odic) component of F(f), F (g), F (h),F(f◦g),F(g◦f), F (f◦h), F (g◦h), F (h◦
f), F (h◦g), F (f◦g◦h), F (f◦h◦g), F (g◦f◦h), F (g◦h◦f), F (h◦f◦g)
and F(h◦g◦f), then it also lies in the wandering (pre-periodic or periodic) com-
ponent of F(fnk◦gnk−1◦hnk−2... ◦fn1),F(gnk◦fnk−1◦hnk−2◦... ◦gn1)and
F(hnk◦fnk−1◦gnk−2◦...◦hn1)etc.
Proof. By lemmas 2.4, 2.5, 2.6, such a set Dexists. By the similar argument
of above lemma 3.1, the proof of this lemma follows. 
We prove the proposition 1.1 for a semigroup generated by two or three tran-
scendental entire functions as defined in above lemmas 2.1, 2.2, 2.3, 2.4, 2.5 and
2.6.

6 B. H. SUBEDI AND A. SINGH
Proof of the Proposition 1.1. Let Sbe a transcendental semigroup gen-
erated by two or three transcendental entire functions. If Sis generated by two tran-
scendental entire functions fand gas defined in the lemmas 2.1, 2.2 and 2.3, then by
the lemma 3.1, there is at least a domain which lies in the wandering (or pre-periodic
or periodic) component of the F(fnk◦gnk−1◦...◦gn1) and F(gnk◦fnk−1◦...◦fn1) for
all nk,...n1∈N. By the definition of transcendental semigroup, any h∈S=hf, gi
can be written in either of the form h=fnk◦gnk−1◦...◦gn1or h=gnk◦fnk−1◦...◦fn1
for all nk,...n1∈N. Therefore, there is a domain Dwhich lies in the wandering
(or pre-periodic or periodic) component of the Fatou set F(h) for every function h
of transcendental semigroup S. This shows that this domain lies in the wandering
(or pre-periodic or periodic) component of the Fatou set F(S). Since for transcen-
dental entire function, pre-peridic (or periodic) domains are simply connected and
so a domain within simply connected domains is also simply connected. In the
construction of functions in the lemmas 2.1, 2.2 and 2.3, the domain which lies
in the wandering domains is simply connected. If Sis generated by three tran-
scendental entire functions f,gand has defined in the lemmas 2.4, 2.5 and 2.6,
then by lemma 3.2 and similar argument as above, Fatou set F(S) contains simply
connected domain. 
We restricted our proof of the proposition 1.1 to the transcendental semigroup
generated by two or three transcendental entire functions . Rigorously, it is not
known that the essence of this proposition holds if a semigroup is generated by
more than three transcendental entire functions. We can only say intuitively, the
essence of this proposition may hold if semigroup Sis generated by n- transcendental
entire functions. There is another strong aspect of this proposition which is- if a
transcendental semigroup Sgenerated by such type of two or three transcendental
entire functions, then the Fatou set F(S) is non-empty.
References
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[3] Herring, M. E.: Mapping properties of Fatou components, Ann. Acad. Sci. Fenn. Math. 23
(1998), 263-274.
[4] Kumar, D. and Kumar, S.: Dynamics of composite entire functions, arXiv:
1207.5930v5[math.DS], 7 October, 2015.
[5] Singh, A. P.: On the dynamics of composite entire functions, Math. Proc. Camb. Phil. Soc.
134, (2003), 129-138.
[6] Subedi, B.H. and Singh, A.: Dynamics on the wandering components of the Fatou set of
three transcendental entire functions and their composites, arXiv: 1803.09259v1 [math. DS],
25 March, 2018.
[7] Subedi, B.H. and Singh, A.: Dynamics on the pre-periodic components of the Fatou set of
three transcendental entire functions and their composites, Preprint.
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TRANSCENDENTAL SEMIGROUP THAT HAS SIMPLY CONNECTED FATOU COMPONENTS7
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